UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
INSTITUTO DE CIENCIAS NUCLEARES
Numerical spacetimes with
fundamental field sources
T E S I S
QUE PARA OPTAR POR EL GRADO DE
DOCTOR EN CIENCIAS (FÍSICA)
PRESENTA
VÍCTOR MANUEL JARAMILLO PÉREZ
TUTOR PRINCIPAL
DR. DARÍO NÚÑEZ ZÚÑIGA
INSTITUTO DE CIENCIAS NUCLEARES
MIEMBROS DEL COMITÉ TUTOR
DR. JUAN CARLOS HIDALGO CUÉLLAR
INSTITUTO DE CIENCIAS FÍSICAS
DR. MIGUEL ALCUBIERRE MOYA
INSTITUTO DE CIENCIAS NUCLEARES
CIUDAD UNIVERSITARIA, CD. MX., JUNIO 2023
 
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A mis abuelos Estela y Salvador,
AGRADECIMIENTOS
A mi asesor Dr. Darío Núñez por el incalculable apoyo que he recibido, por las
ideas originales que comparte. Por crear un ambiente cordial en el que da gusto
estar.
Al Dr. Juan Carlos Degollado por su apoyo en la preparación de mi examen de
candidatura y en la discusión (y paciente corrección) de muchos de los resultados
contenidos aquí.
Al Dr. Olivier Sarbach, Dra. Argelia Bernal y Dr. Juan Barranco por su gran
apoyo. Agradezco las discusiones sobre física.
A los colaboradores y coautores de los trabajos que aquí se presentan. Gracias
en particular a Erik Jiménez, por su ayuda técnica con el uso del cluster y sus
consejos en temas numéricos.
Agradezco el tiempo de cómputo otorgado en el cluster Tochtli del Instituto de
Ciencias Nucleares de la UNAM.
Al Dr. Carlos Herdeiro, Dr. Eugen Radu, Etevaldo Costa, Jianzhi Yang, Sergio
Xavier y los demás por su hospitalidad en mi estancia en Portugal.
A mis excompañeros y profesores de la escuela de música y en particular a
Luis Cibrián y Joel Rodríguez Almaguer, sus enseñanzas atraviesan disciplinas
y las llevo todos los días a todas partes.
Al Consejo Nacional de Ciencia y Tecnología (Conacyt) por la beca otorgada
durante la realización de mi doctorado. De igual manera agradezco el apoyo
económico brindado por el proyecto No. 376127 “Sombras, lentes y ondas grav-
itatorias generadas por objetos compactos astrofísicos” para la presentación de
trabajos y asistencia en congresos.
i
ii
A la Coordinación General de Estudios de Posgrado por el apoyo recibido para
la estancia de investigación.
A Alexandra y José en Tonalco. Gracias por la gran hospitalidad y ayuda
en los primeros años del doctorado. A Alexandra G. por su paciencia con mis
experimentos pedagógicos que terminaron bien.
A mis ciberamig@s Johann, Abigail y Sofía.
A mis tíos Fabiola y Eric y mis primos por siempre invitarme. Sin esas pausas
para comer y jugar cartas me habría quedado sin energía muchas veces.
Por supuesto a toda mi familia y en particular a mis Padres y a mi hermano.
Esta tesis también está dedicada a ellos.
A mis abuelos. Mientras escribo los agradecimientos sólo puedo pensar en que
me gustaría estar en su comedor platicando, comiendo tortas de papa, arroz, una
salsa de morita y tomando tepache.
Resumen
Al día de hoy, las pruebas prácticas y observacionales indican que la Relatividad
General continúa siendo la mejor descripción para la interacción gravitatoria. La
extrapolación de este modelo físico-matemático a escalas cosmológicas forma una
gran parte de la base teórica de varios modelos cosmológicos, incluido el modelo
estándar actual. Dentro de dicho modelo la materia oscura y la energía oscura
son hipótesis complementarias, necesarias por consistencia y pueden entenderse
como predicciones de este paradigma, sin embargo, aún ésta descripción del uni-
verso presenta problemas a escalas cosmológicas pequeñas, abriendo la posibil-
idad para explorar formas hipotéticas diferentes de materia y energía oscuras.
En este sentido, los campos clásicos acoplados a la gravedad han resultado una
excelente alternativa para modelar una o las dos componentes oscuras.
Esperaríamos que en la evolución del universo ésta forma de sustancia se
aglomere y forme configuraciones autogravitantes. Por este motivo presenta in-
terés el estudio de soluciones a las ecuaciones de Einstein con campos escalares
como fuente del campo gravitatorio. Dicha tarea puede dividirse en dos partes:
La primera, que corresponde a la construcción de soluciones estacionarias, en
donde la materia se encuentre en equilibrio o bien en movimiento perpetuo reg-
ular. La segunda, constituida por los escenarios dinámicos como por ejemplo las
evoluciones cosmológicas, las perturbaciones a las soluciones estacionarias y los
choques de objetos compactos. Esta última con aplicaciones en la generación de
ondas gravitatorias y su posterior comparación con los datos obtenidos oportuna-
mente por los observatorios de ondas gravitatorias.
En esta tesis se utiliza la potencia de diferentes métodos numéricos para con-
struir soluciones relativistas estacionarias en una dimensión temporal y en una
o dos dimensiones espaciales, así como evoluciones temporales completamente
no lineales con tres dimensiones espaciales. En particular se presentan config-
uraciones de estrellas de bosones con forma de cáscara y masas arbitrariamente
grandes, soluciones formadas de múltiples campos, estrellas de bosones magnéti-
cas, agujeros de gusano esféricos pero formados por campos con dependencia an-
gular no trivial, soluciones formadas por un campo canónico y un campo fantasma
cuya compacidad excede el límite de Buchdahl, perturbaciones genéricas a las es-
trellas de bosones multicampos y colisiones frontales de las mismas. El desarrollo
de códigos propios, así como el manejo de códigos públicos, tales como Cactus y
Kadath, abre camino al desarrollo de trabajos futuros.
Palabras clave: Relatividad General, Relatividad Numérica, Teoría Clásica de
Campos, Campo Escalar, Objetos Compactos.
iii
Abstract
To date, practical and observational evidence indicates that General Relativity re-
mains the best description for the gravitational interaction. The extrapolation of
this mathematical model to cosmological scales forms a large part of the theoret-
ical basis for several cosmological models, including the current standard model.
In this model, the dark matter and dark energy are additional hypothesis neces-
sary for consistency and can also be understood as predictions of this paradigm,
however even this description of the universe presents problems at small cosmo-
logical scales, opening the possibility to explore different hypothetical forms of
dark matter and dark energy. In this sense, classical fields coupled to gravity
have proved to be an excellent alternative for modeling one or both of the dark
components.
We would expect that in the evolution of the universe this form of substance
will agglomerate and form self-gravitating configurations. For this reason it is
of interest to study solutions to the Einstein equations with scalar fields as the
source of the gravitational field. This task can be divided into two parts: The
first, which corresponds to the construction of stationary solutions, where matter
is either in equilibrium or in regular perpetual motion. The second, constituted by
dynamical scenarios such as cosmological evolutions, perturbations to stationary
solutions and collisions of compact objects. The latter with applications in the
generation of gravitational signals and their subsequent comparison with the data
obtained by gravitational wave observatories.
In this thesis the power of different numerical methods is used to construct
stationary relativistic solutions in one time dimension and in one or two spatial
dimensions as well as fully nonlinear time evolutions with three spatial dimen-
sions. In particular, we present configurations of shell-shaped boson stars with
arbitrarily large masses, solutions formed of multiple fields, magnetic boson stars,
spherical wormholes but formed by fields with nontrivial angular dependence, so-
lutions formed by a canonical field and a ghost field whose compactness exceeds
the Buchdahl limit, generic perturbations to multifield boson stars, and head-on
collisions of them. The development of our own codes as well as the usage of public
codes, such as Cactus and Kadath, paves the way for the development of future
work.
Keywords: General Relativity, Numerical Relativity, Classical Field Theory, Scalar
Fields, Compact Objects.
iv
Declaration
I hereby declare that this thesis was composed by myself and without use of oth-
ers than the indicated sources. I confirm that the work submitted is my own,
except where work which has formed part of jointly-authored publications has
been included. Where the work was done in collaboration with others, a signifi-
cant contribution was made by the author, which is explicitly indicated below in
the List of Publications. This work has not been submitted for any other degree
or professional qualification.
Víctor Manuel Jaramillo Pérez, May 2023
v
CONTENTS
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Resumen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction to this manuscript 3
1 Introduction 3
1.1 Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Cosmological motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Compact objects and fundamental fields . . . . . . . . . . . . . . . . . 10
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Spatially-homogeneous cosmological solutions 15
2.1 Dynamics of the homogeneous and isotropic Universe . . . . . . . . . 16
2.2 Scalar field dark matter with two components . . . . . . . . . . . . . . 18
2.3 Complex quintessence exact solution . . . . . . . . . . . . . . . . . . . 28
I Spherically-symmetric solutions 37
3 Extreme ℓ-boson stars 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 ℓ-boson stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Extreme ℓ-boson stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 Scaling properties for large ℓ . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Confined ghost 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Theoretical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
vi
vii
4.3 Equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 Traversable ℓ-wormholes 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Stationary Wormhole equations . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Numerical wormhole solutions . . . . . . . . . . . . . . . . . . . . . . . 107
5.5 Embedding diagrams and geodesic motion . . . . . . . . . . . . . . . . 112
5.6 Discussion and concluding remarks . . . . . . . . . . . . . . . . . . . . 116
II Axisymmetric solutions 119
6 Magnetostatic boson star 121
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.4 Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7 Two boson stars in equilibrium 141
7.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Some results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
III Numerical evolutions 149
8 Stability 151
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.4 Time evolution and numerical results . . . . . . . . . . . . . . . . . . . 158
8.5 Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9 Collisions 171
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.2 Models for ℓ-boson star binaries . . . . . . . . . . . . . . . . . . . . . . 174
9.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.4 Head-on Collision Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 180
viii
9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.6 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Conclusions 209
Appendices 213
A Black hole solutions with magnetic dipole moment 213
Bibliography 219
LIST OF PUBLICATIONS
This doctoral thesis is based on the following publications. After each reference I
indicate my particular contribution.
[A] Víctor Jaramillo, Erik Jiménez-Vásquez, Darío Núñez
Confinement of exotic matter. Static solutions
Physical Review D 107, 064061 (2023)
D.N. and me conceived the project and performed the analytical calculations, I wrote
the code. All three authors wrote the manuscript.
This work is presented in Chap. 4.
[B] Víctor Jaramillo and Darío Núñez
Magnetostatic boson stars
Physical Review D 106, 104023 (2022)
I conceived the project and numerically solved the system of PDEs. Together with
D.N. (advisor) performed the analytical calculations and wrote the manuscript.
This work is presented in Chap. 6.
[C] Víctor Jaramillo, Nicolas Sanchis-Gual, Juan Barranco, Argelia Bernal,
Juan Carlos Degollado, Carlos Herdeiro, Miguel Megevand, Darío Núñez
Head-on collisions of ℓ-boson stars
Physical Review D 105, 104057 (2022)
I adapted the Einstein Toolkit thorn to interpolate initial data of two multi-field bo-
son stars, evolved the cases presented in the manuscript. All authors analyzed the
data and wrote the manuscript.
This work is presented in Chap. 9.
ix
x
[D] Miguel Alcubierre, Juan Barranco, Argelia Bernal, Juan Carlos Degollado,
Alberto Diez-Tejedor, Víctor Jaramillo, Miguel Megevand, Darío Núñez, Olivier
Sarbach
Extreme ℓ-boson stars
Classical and Quantum Gravity, 39, 094001 (2022)
I wrote the spectral code and constructed all the 25< ℓ<∞ solutions. Together with
O.S. obtained the effective equations for high ℓ, solved the corresponding equations
and wrote section 4 of the manuscript.
This work is presented in Chap. 3.
[E] Eréndira Gutiérrez-Luna, Belen Carvente, Víctor Jaramillo, Juan Barranco,
Celia Escamilla-Rivera, Catalina Espinoza, Myriam Mondragón, Darío Núñez
Scalar field dark matter with two components: Combined approach from
particle physics and cosmology
Physical Review D 105, 083533 (2022)
This study was conceived by all the authors. I obtained the evolution equations,
wrote the code to solve them, obtained the values for the constraints, wrote section
3, a significant part of section 4 and collaborated in the drafting of the other sections
of the paper except for section 2.
This work is presented in Sec. 2.2 of Chap. 2.
[F] Belen Carvente, Víctor Jaramillo, Celia Escamilla-Rivera, Darío Núñez
Observational constrains on complex quintessence with attractive self-interaction
Monthly Notices of the Royal Astronomical Society 503, 4008-4015 (2021)
This study was conceived by all the authors. I performed the analytical calculations
of section 3 of the manuscript and collaborated in the drafting of the entire article,
except for section 4.
Part of this work is presented in Sec. 2.3 of Chap. 2.
[G] Víctor Jaramillo, Nicolas Sanchis-Gual, Juan Barranco, Argelia Bernal,
Juan Carlos Degollado, Carlos Herdeiro, Darío Núñez
Dynamical ℓ-boson stars: Generic stability and evidence for nonspherical
solutions
Physical Review D 101, 124020 (2020)
This study was conceived originally by N.S.G., J.C.D., D.N. (advisor) and myself.
Jointly with N.S. implemented the calculation of global quantities in the evolution
code. Together with J.C.D. chose the perturbations and evolved the cases presented
in the manuscript. All authors analyzed the data and wrote the manuscript.
This work is presented in Chap. 8.
xi
[H] Belen Carvente, Víctor Jaramillo, Juan Carlos Degollado, Darío Núñez,
Olivier Sarbach
Traversable ℓ-wormholes supported by ghost scalar fields
Classical and Quantum Gravity 36, 235005 (2019)
This study was conceived by all authors. I found the trick to construct spherically
symmetric solutions in this self-interacting model and wrote the code. Together with
B.C. constructed and analyzed the solutions presented in the manuscript. All au-
thors wrote the manuscript.
This work is presented in Chap. 5.
During my PhD I was also an author of the following works which are not
discussed in this thesis:
[I] Alejandro Aguilar-Nieto, Víctor Jaramillo, Juan Barranco, Argelia Bernal,
Juan Carlos Degollado, Darío Núñez
Self-interacting scalar field distributions around Schwarzschild black holes
Physical Review D 107, 044070 (2023)
[J] Víctor Jaramillo, Daniel Martínez-Carvajal, Juan Carlos Degollado, Darío
Núñez
Born-Infeld boson stars
arXiv:2303.13666
[K] Laura O. Villegas, Eduardo Ramirez-Codiz, Víctor Jaramillo, Juan Carlos
Degollado, Claudia Moreno, Darío Núñez, Fernando J. Romero-Cruz Deter-
mination of the angular momentum of the Kerr black hole from equatorial
geodesic motion
arXiv:2211.10464

Introduction to this
manuscript

CHAPTER 1
INTRODUCTION
Contents
1.1 Numerical Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Cosmological motivation . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Compact objects and fundamental fields . . . . . . . . . . . . . . 10
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
G
ravity is one of the four known to exist fundamental interactions of na-
ture. Different mathematical models have been proposed over the years
to describe this interaction and discover its essence. The first mathe-
matical description of gravity, proposed by Newton in 1687 [1], considered that
between different elements of nature acts a force of attraction whose magnitude
is proportional to the product of the mass of the objects and inversely proportional
to the square of the distance that separates them.
In the 17th century this theory was enough to describe the motion of terrestrial
and celestial bodies, however at the end of the 20th century some discrepancies
in observations were discovered with respect to the Newtonian model of gravity.
Together with the appearance of the theory of Special Relativity at that time,
which is incompatible with Newton’s theory, the urgency of constructing a new
theory of gravity arose. The theory of General Relativity proposed by Einstein in
1915 [2] succeeded, beautifully and in a single mathematical model, to reconcile
the compatibility of the gravitational interaction model with Special Relativity
and at the same time managed to accurately predict the advance of Mercury’s
perihelion.
In General Relativity, gravity is described by a rank 2 tensor field gµν which
contains the information of the metric properties of a geometric four-dimensional
entity, called spacetime. The equations of motion for gµν are the Einstein equa-
tions
Rµν−
1
2
R gµν+Λgµν = 8πGTµν, (1.1)
with Rµν the Ricci tensor, R the Ricci scalar, Λ the cosmological constant and Tµν
the energy momentum tensor. The Einstein equations (1.1) can be derived using
3
4 1.1 Numerical Relativity
a variational approach from the Einstein-Hilbert action
S =
1
16πG
∫
d4x
p
−g (R−2Λ)+Smatter, (1.2)
with Smatter the action for the matter fields and g the determinant of the ten-
sor field gµν. The energy momentum tensor appearing in Eqs. (1.1) given by the
variation of Smatter with respect to gµν,
Tµν =−2
1
p−g
δSmatter
δgµν
. (1.3)
To date there is no evidence against the predictions of Einsteinian gravitation.
On the contrary, in addition to the accurate description of the orbit of the planet
Mercury and the proof of light deflection in the event of an Eclipse [3], gravita-
tional waves, predicted by General Relativity, have recently been detected [4].
Other, no less impressive predictions of General Relativity are the existence of
black holes, recently confirmed by the first image of Sagittarius A∗, at the center of
our Galaxy [5] and the supermassive black hole in the center of the giant elliptical
galaxy M87 [6]. However, one must not lose sight of the fact that this theory,
however perfect it may be, is still a mathematical model of reality that does not
exhaust matter, infinite in extension and depth.
The electron is as inexhaustible as the atom,
nature is infinite, but it exists infinitely.
Lenin, Materialism and Empirio-Criticism
(1908)
The main objective of this thesis is the construction of solutions of compact ob-
jects within this theory of gravitation, when the matter component refers neither
to barotropic fluids nor to electromagnetism but to other (classical) fields moti-
vated by physical cosmology. Obtaining solutions in the context we are interested,
requires (as a rule) the use of numerical methods and only in a few exceptions
is it possible to obtain solutions in closed form. Following this line of thought,
the introduction consists of brief summary of the basics of Numerical Relativity
in Sec. 1.1, followed in Sec. 1.2, by the cosmological motivation of the classical
(mainly scalar) fields to be used in the thesis; then in Sec. 1.3 a brief summary on
the solutions of compact objects in different field theories and finally, in Sec. 1.4 a
summary of the thesis content by chapter.
1.1 Numerical Relativity
The spacetime of Special and General Relativity contains in dialectical unity time
and space. However, this unity is partially broken on purpose in order to have
Introduction 5
more control over the mathematical modeling and to put the differential equa-
tions contained in the tensor equation (1.1) in the form of an initial value problem
(Cauchy problem), suitable to describe how the gravity fields evolve from an initial
configuration toward the future.
The 3+1 formalism allows to formulate the problem of solving the Einstein
equations as an initial value problem. A great achievement from the mathemati-
cal point of view was the demonstration of well-posedness (existence and unique-
ness) in the Cauchy problem in General Relativity [7–9].
Considering a space-time (M, g), i.e. a Lorentzian manifold M endowed with
a metric g, it is said to be globally hyperbolic if there exists a space-like hyper-
surface Σ such that the union of the causal future and causal past of Σ coincides
with M. If the spacetime under consideration is globally hyperbolic, then it can be
shown that it is possible to foliate M into space-likeΣ surfaces that are themselves
manifolds endowed with a metric. Proper distances on this hyper-surface (with
coordinates xi) are measured precisely by this induced three-dimensional metric
γ: dl2 = γi j dxi dx j.
To each of these hyper-surfaces composing spacetime, we associate a constant
value to a certain scalar function t: Σt. We define (see e.g., [10]) Eulerian ob-
servers as those observers moving in the direction normal to Σt, and define the
lapse function, α, as the proper time span of an Eulerian observer between the
hyper-surface Σt and Σt+dt, i.e. dτ=α(t, xi)dt.
There is no reason to expect Eulerian observers to necessarily follow trajecto-
ries with constant xi (coordinate observers), so a third 3-tensor, in this case the
shift vector βi, is needed to specify the foliation of (M, g): xi
t+dt
= xi
t −βi(t, xi)dt.
In the adapted Cauchy framework [11], where we make a foliation-adapted choice
for the vector basis at each point of the manifold, the unit vector nmu normal to
the hyper-surfaces t = constant has components
nµ =
(
1/α,−βi/α
)
. (1.4)
And the metric tensor, symbolically reads
ds2 =
(
−α2 +βkβ
k
)
dt2 +2βi dtdxi +γi j dxi dx j. (1.5)
In addition to the curvature induced by g and the foliation in Σt, we can also
define the curvature, in this case called extrinsic, of the hyper-surfaces when im-
mersed in M,
Kµν =−Pα
µ∇αnν, (1.6)
where Pµν is the projection operator into the Σt hyper-surface, defined as Pλ
σ =
δλσ + nλnσ. It can be shown that its nontrivial components satisfy an evolution
equation for the extrinsic curvature:
∂tγi j =−2αK i j +D iβ j +D jβi (1.7)
6 1.1 Numerical Relativity
with D i, the covariant derivative compatible with γi j.
Now, one can define
E = nµnνTµν, (1.8)
Sα = −PαµnνTµν, (1.9)
Sαβ = P
µ
αPν
βTµν, (1.10)
interpreted as the energy density, momentum density and the stress tensor (re-
spectively) as measured by an Eulerian observer.
It can be shown now [11], that in this system of coordinates, using the above
definitions together with identities from differential geometry the Einstein equa-
tions (1.1) are split into two: a set of equations without time derivatives, called
constraints, and another set including time derivatives, called evolution equa-
tions. The Cauchy problem in General Relativity then reduces, in principle, to
providing an initial data solving the Hamiltonian and momentum constraints
(3)R+K2 −KµνKµν = 16πG E, (1.11)
Dµ(Kαµ−γαµK)= 8πG Sα, (1.12)
respectively and, evolving using (1.7) and the following evolution equation for the
extrinsic curvature,
∂tK i j =βk∂kK i j +Kki∂ jβ
k +Kk j∂iβ
k
−D iD jα+α
[
(3)Ri j +KK i j −2K ikKk
j
]
+4πGα[γi j(S−E)−2Si j].
(1.13)
The required equations for evolving the initial data as presented, constitutes a
system of equations that turns out to be not strongly hyperbolic, which in practice
results in unstable simulations. The reformulation of the above equations (called
ADM-York equations) is necessary. In this manuscript we will be interested in
the one (initially) made by Nakamura, Oohara and Kojima [12] and refined over
the years. An extensive discussion of the details of the 3+1 formalism and its
numerical implementation can be found in the books [10,13].
To date there are some numerical codes and computational infrastructures
created to solve the equations of numerical relativity. In the case where no con-
tinuous symmetries are assumed in the 3+1 formulation (i.e., solving in its full
3+1 dimensions) we find the code Einstein Toolkit [14,15] which in turn uses the
Cactus infrastructure and therefore is composed of several thorns that solve dif-
ferent aspects of the physical problem. For example, some thorns will be in charge
of implementing the material part of the Einstein equations and others will take
care of the evolution of the metric. For the case of interest in this work, for which
classical fields are included, the particular implementation will depend on the
Introduction 7
mathematical characteristics of the field and the chosen coupling with gravita-
tional and electromagnetic fields.
For the major part of this manuscript (Sec. I and Sec. II) we will be solving only
the constraint equations and for this purpose we will use our own spectral solver
for the spherically symmetric problems in section I, and the Kadath library [16,17]
for the axially symmetric problems. Finally, the evolutions presented at the final
part of this manuscript (Sec. III), will be performed in the Einstein Toolkit by
implementing different thorns in the chapters for the corresponding problem in
turn.
The particular choices of the type of matter that we will consider both to con-
struct global solutions/initial data and to evolve in this manuscript have a moti-
vation from the field of physical cosmology, which we will give in the next section.
1.2 Cosmological motivation
The aforementioned virtues of the modern (and yet classical) theory of gravita-
tion have motivated physicists over the last century to extrapolate this particular,
mesoscopic-astronomical laws of physics and apply them to cosmological scales
i.e., to systems of the size of the observable Universe. Today, thanks to the ap-
pearance on the scene of general relativity, quantum field theory and the new pos-
sibilities to obtain observations of cosmic processes with considerable precision,
cosmology has emerged from the realm of pseudoscience and has established itself
as one of the most active fields of theoretical and observational research.
The standard cosmological model, at the time of writing this thesis, describes
a spatially homogeneous and isotropic Universe at large scales, whose main ma-
terial components are photons, neutrinos, baryonic matter, cold dark matter and
a cosmological constant Λ (see Eq. (1.1)) also called dark energy. In this model,
called ΛCDM, the curvature of spacetime -not just space- and therefore its expan-
sion are linked to the presence of these types of matter precisely through Einstein
equations.
Returning to the discussion on the predictions of the physical theories, one
may ask whether the dark components of the Universe are a prediction of Gen-
eral of Relativity rather than simply hypotheses to be added by hand to the model
for consistency. At first, it might seem that the extrapolation of local laws to gi-
gantic scales may be a mistake, however the current cosmological paradigm is
unexpectedly consistent with itself, at least in most of the model’s content1. Of
course, there are alternative models of gravity that modify some or several prin-
1However, on the other hand, this model still has an important set of problems/challenges to be
solved. Such are the Hubble tension, the S8 tension, the lithium problem and others. There are also
the problems related to the results carried by the assumptions made for dark matter, which will be
discussed below.
8 1.2 Cosmological motivation
ciples at the basis of the theory and that interestingly do not require one or both
of the two dark components, but we will not discuss them in this manuscript.
Dark matter
Evidence of the existence of dark matter has accumulated during the last decades
by different means, but in all cases via gravitational effects, however there is no
information on what this component is in terms of its properties at fundamental
level and as a consequence there is a wide variety of models for the concrete type
of matter that constitutes cold dark matter. On the other hand some of these
models seem to fail in the small-scale description of the cosmos [18, 19], such is
the case of the ΛCDM model and other scenarios that do not modify the power
spectrum, either primordially or through direct modification of the properties of
dark matter [20]. The missing satellites problem [21], the core-cusp problem [22]
and the too-big-to-fail problem [23] consist of discrepancies between the results of
the ΛCDM model N-body simulations and observations of galaxies.
Although these problems may probably be due to an oversimplification of the
baryonic (i.e. non-dark) matter physics, it is unlikely that these multiple discrep-
ancies can be fixed simultaneously by considering more complex baryonic matter
models given the existence of galaxies with different compositions and particu-
larly when some of them contain mainly dark matter.
One might then raise the possibility that standard cold dark matter is irrecon-
cilable with these problems and therefore a different model for dark matter need
to be proposed. Indeed, there is a solution to these issues in the scalar field dark
matter model [24–26] also called fuzzy dark matter and ultralight dark matter in
other contexts. At the action level the scalar field dark matter model substitutes
the ΛCDM dark matter fluid with a scalar field in the Smatter part of Eq. (1.2),
Ssfdm =−
1
2
∫
d4x
p
−g
(
∇αφ∇αφ+V (φ)
)
, (1.14)
where φ ∈R and V (φ) is the scalar potential, which is usually that of a free scalar
field, V = µ2φ2. The parameter µ is related to the mass of the proposed dark
matter (ultralight) particles, and is considered to be of the order of 10−22 eV [27],
or in terms of the characteristic (Compton wave) length λC ∼ 1000 kpc, which is
about the scale at which ΛCDM starts to have trouble.
The equations of motion for the scalar field, the fluid components and for the
spacetime are obtained from the variation of the total action S and details (in the
homogeneous background case) will be given in Chap. 2.
Dark energy
In ΛCDM the density of dark energy, coming from Λ, constitutes about 70% of the
Universe content at present with a constant value over time: ρΛ = Λ
8πG
= ρcritΩΛ
Introduction 9
with ρcrit the critical density and ΩΛ ≈ 0.7 [28,29]. Dark energy is responsible for
the current accelerated expansion of the Universe, and it is only very recently2
that it has become the dominant component of the cosmos. Beyond speculation,
there is no information on what substance makes up dark energy. When treated
as a fluid with pressure p and a constant equation of state p = wE, then according
to the latest data of the cosmic microwave background obtained from the Planck
satellite [29] it is obtained w =−1.03±0.3.
Quintessence [30] on the other hand, is a dynamical model for dark energy
consisting of a scalar field coupled to gravity. The beautiful name comes from the
fifth element, after matter, dark matter, photons and neutrinos [31]. It is also
very interesting that such a form of matter could imply the existence of a fifth
fundamental interaction in nature [32].
In the simpler case, when the scalar field is real and the coupling to gravity
is minimal the Quintessence action is also of the form in Eq. (1.14) and the asso-
ciated pressures and densities3 of the scalar field are p := Sr
r = (dφ/dt)2 −V (φ)
and E = (dφ/dt)2 +V (φ) in a homogeneous background. Therefore, the equation
of state for the quintessence is
w =
(dφ/dt)2 −V (φ)
(dφ/dt)2 +V (φ)
, (1.15)
so that a slowly varying φ along V (slow-roll) lead to an accelerated expansion
of the Universe with w dynamical and close to −1. Different models with dif-
ferent motivations have been appearing in the literature, for example the impor-
tant tracker solutions are obtained with inverse power-law potentials of the form
V (φ) ∝ φ−q (q > 0). Exponential potentials V (φ) ∝ e−qφ, double exponential po-
tentials, to name a few, have also been considered.
Massive and self-interacting potential of the form V (φ)=µ2|φ|2+λ |φ|4, used
in many areas of physics, from Bose-Einstein condensates to the Standard Model
of particle physics, also lead to quintessence solutions and in particular when
φ ∈ C . In Chap. 2 we will present closed form solutions to the Friedmann equa-
tions and analyses to evaluate the viability of these models in the light of recent
cosmological data.
Several models proposed for describing dark energy, even the Λ cosmological
constant one, violate the strong energy condition. Some dark energy models other
than quintessence fail to satisfy the null energy condition which in turn implies
the violation of the weak energy condition [33], meaning that at least some time-
like observer measure negative energy densities. The phantom models [34], with
the simplest cases consisting of a scalar field with a negative kinetic term in the
2Cosmologically speaking, since in fact the epoch of matter-energy equality occurred at a redshift
of z = 0.3 i.e., billions of years ago
3As measured by a comoving observer with nµ∂µ = ∂t.
10 1.3 Compact objects and fundamental fields
Lagrangian, are an example of this. On the other hand, quintessence and scalar
field dark matter models described above, which encompass scalar fields with a
canonical kinetic term, always satisfy the null energy condition [35].
As can be obtained from Eq. (1.15), in quintessence, w stays above the cos-
mological constant value, w ≥ −1, while for the phantom field w ≤ −1. In de-
scribing the dark energy with a dynamical equation of state, cosmological data
analyses suggest that the boundary w =−1 is crossed in the evolution of the Uni-
verse [36,37]. Models with this kind of behaviour, require two scalar field (or fluid)
components [36,38] and are called quintom4 (see Ref. [39] for a review).
Self-gravitating scalar fields that violate the null energy condition, such as the
phantom field, are the constituent material from with hypothetical objects called
wormholes are constructed. In Chap. 5 we will discuss some of these non-trivial
topology solutions to the Einstein equations. On the other hand, to our knowledge,
compact objects made of quintom fields are less common in literature and we will
review two solutions with this type of matter in Chap. 4.
1.3 Compact objects and fundamental fields
Electrovacuum
Birkhoff theorem states that the more general spherically symmetric electrovac-
uum5 spacetime is static. This greatly restricts the kind of spherical solutions
to the Einstein-Maxwell equations, where the metric is given by the Reissner-
Nordström metric [40]
gµνdxµdxν = −N(r) dt2 +
1
N(r)
dr2 + r2
(
dθ2 +sin2θdϕ2
)
, (1.17)
Aµdxµ =
Q
4πǫ0r
dt , (1.18)
where we defined
N(r)= 1−
rs
r
+
r2
Q
r2
, rs = 2GM , r2
Q =
Q2G
4πǫ0
. (1.19)
with ǫ0 = 1/µ0 being the vacuum permittivity, the parameters M and Q are the
mass and charge of the solution and the coordinate r takes the values 0 ≤ r <∞.
Together with the spacetime metric solution gµν the Reisner-Nordström solution
4From quintessence + phantom.
5 Source-free Einstein-Maxwell equations, i.e., Eq. (1.1) and ∇νFµν = 0, also obtained from the
variation of the action
S =
∫
d4x
p
−g
(
R
16πG
−
1
4µ0
FabFab
)
. (1.16)
with Fab =∇a Ab −∇a Ab , Aa the electromagnetic four-potential and µ0 the vacuum permeability.
Introduction 11
gives an explicit form for the electromagnetic four-potential, as given in Eq. (1.18)
in Boyer-Lindquist coordinates. This restriction is somewhat expected as it is
similar to what happens in Newtonian gravitation and in the Coulomb solution,
however an outstanding result in General Relativity are the (somewhat converse
of Birkhoff theorem) black holes uniqueness theorems which have no analogy with
the classical physics areas mentioned but do have an analogy with thermodynam-
ics in the sense that a small set of parameters fully determine the properties of a
physical system.
Also let us remind that there is no analogy of Birkhoff theorem for rotating
axisymmetric spacetimes however, on the other hand, the uniqueness theorems
strongly restrict the possible solutions if the spacetime contains a black hole.
For electrovacuum (the system presented in footnote 5) the black hole unique-
ness theorem states that any, four-dimensional, asymptotically flat, stationary,
black hole spacetime6 with regular horizon has an exterior isometric to the Kerr-
Newman spacetime, which in Boyer-Lindquist coordinates reads [40]:
gµνdxµdxν = −
∆
Σ
(dt−asin2θdϕ)2 (1.20)
+
sin2θ
Σ
[
(r2 +a2)dϕ−adt
]2
+Σ
(
dr2
∆
+dθ2
)
,
Aµdxµ = −
rQ
4πǫ0Σ
(
dt−asin2θdϕ
)
, (1.21)
with
a =
J
M
, (1.22)
Σ= r2 +a2 cos2θ , (1.23)
∆= r2 − rsr+a2 + r2
Q , (1.24)
rs, rQ defined in Eq. (1.19) and J is the angular momentum of the black hole.
Further discussion of the Kerr-Newman spacetime can be found in Appendix A of
this manuscript.
The uniqueness theorem as stated above, do not assume axisymmetry or static-
ity, so it also asserts that (electrovacuum) “black holes have no hair”, meaning
that the complete mathematical solution is completely determined by a small set
of parameters, namely M, J and Q.
Beyond classical electrodynamics coupled to gravity, there are cases that present
interesting physical scenarios. In nonlinear electrodynamics for instance there is
an exact solution of a regular black hole [41].
6In the precise mathematical statement, the theorem also requires analyticity of the spacetime
(M, g), which is physically difficult to justify.
12 1.3 Compact objects and fundamental fields
Other self-gravitating matter fields
The electromagnetic field and the gravitational field are classical field prototypes
which successfully describe a wide variety of phenomena in nature. Other hy-
pothetical classical field are the scalar dark matter and quintessence, both pre-
sented in the previous section. However, it is also worthwhile to take into con-
sideration other kind of matter -say motivated by quantum field theory or some
particle model- and couple them with gravity. For example, the Yang-Mills theory
presents a historically important case in the development of the general non-hair
conjecture. The first particle-like everywhere regular solution to the Einstein-
Yang-Mills equations7 was obtained in 1988 and today is known as the Bartnik-
McKinnon solution [42]. One year later Mikhail S. Volkov and Dimitri V. Gal’tsov
in the Soviet Union found a generalization containing a horizon i.e., a non-Abelian
Einstein-Yang-Mills black hole [43].
Besides Einstein-Yang-Mills Bartnik-McKinnon spacetime, other particle-like
everywhere regular solutions, also called gravitational solitons (and this is how we
will refer to them from now on) have been found using different types of matter;
from the Oppenheimer-Volkoff fluid solutions to wormholes composed of exotic
fields. There is no Birkhoff theorem outside electrovacuum and many spherical
solutions have been found. A solution of central importance to the purposes of
this manuscript are boson stars [44, 45]: gravitational solitons made of complex
scalar field. These are static configurations with harmonic time dependence,
Φ(t,r)=φ(r)eiωt. (1.26)
Boson stars possess robust dynamical properties [46] and even a formation
mechanism from clouds of initially unbound states is known [47]. Furthermore,
if the substance that gives body to dark matter is an scalar field, as the fuzzy
dark matter model states, boson stars with their wide range of shapes, sizes and
masses could model galactic dark matter halos [48–51].
An important role of boson stars is that they are the starting point, in terms of
simplicity, to study solutions with other kinds of fields (vector, spinor, etc.) coupled
to Einstein’s theory of gravity. Recent results on global solutions and dynamical
analysis in numerical relativity show similarities between the scalar (spin 0) field
and other fields with higher spin. Different to what happens for the electrovacuum
case, an example of a vector field which manages to constitute stable regular self-
7 Described by Einstein-Hilbert action plus Smatter =
∫
d4x
p−gLYang−Mills, with
LYang−Mills =−
1
2
Tr F2; and F = dA+ A∧ A, (1.25)
A being the gauge potential.
Introduction 13
gravitating objects, is the Proca field, Smatter =
∫
d4x
p−gLProca with,
LProca =−
1
4
FαβF
αβ∗−
1
2
µ2
AαA
α∗
, (1.27)
µ corresponds to the mass parameter of the Proca field, F = dA and A being
the complex (Proca) four-potential, where, similarly to boson stars a harmonic de-
composition (Eq. (1.26)) is implemented in the fields A . The equations of motion
derived from the Proca action, imply that the vector field, when µ 6= 0, must sat-
isfy the (Lorentz) condition ∇αAα = 0, which breaks gauge invariance. From the
quantum field theory point of view, the Proca field is prone to certain problems,
among them, the non-renormalizability of the field, however at the classical level,
the Proca Lagrangian could be thought of as an effective Lagrangian. In fact,
(classical) gravitational solitons made of this type of fields also allows to build
a family of solutions called Proca stars, introduced in [52]. It is also possible to
form solitonic configurations with a Dirac field (spin 1/2) [53], however the quan-
tum fermionic nature of the Dirac field compels the particle number8 to be equal to
1, which in turn implies a discrete distribution of solutions of microscopic nature,
which are not useful if the idea is to form compact objects [54].
The stability of “mini” boson stars (scalar potential V (|Φ|)=µ2|Φ|2), has been
extensively studied from different points of view. At first from linear perturba-
tion analysis [55] and later in a complete way through nonlinear evolutions [56].
These studies showed that in the boson star family of solutions there is a stable
and an unstable set of solutions which have three final fates: black hole, scat-
tering, and migration to a stable solution, depending on the perturbations and
more concretely, on the total mass and binding energy of the configuration. But
in essence one has a set of solutions, called a stable branch, where equilibrium
configurations neither collapse nor scatter and whose final state is a configuration
very close to the unperturbed configuration.
In addition to the question of stability, a central question is that of the dynam-
ical conformation of these self-gravitating configurations. In order to consider
seriously the possible existence of such stars in nature, a mechanism for their for-
mation should exist; in the paper [47] they showed the existence of a mechanism
(in spherical symmetry) called gravitational cooling, which consists of a dissipa-
tion of the “excess” of scalar field and the subsequent collapse to a boson star.
Other studies carried out on the dynamics of boson stars are frontal collisions
[57] and orbital collisions (see e.g., [58] for boson stars with a potential somewhat
different from the free field case). Also, of relevance to this work is the article [59]
where the frontal and orbital collisions of dark stars9 is studied. Further details
8 Related to the value of the conserved charge associated with the U(1) symmetry of the field under
consideration
9 Which refer to a case where the interaction between the stars in the binary system, is purely
gravitational. This case will be explored in Chap. 9.
14 1.4 Thesis outline
on the stability and head-on collisions of boson stars will be given in Sec. III.
Similar results for Proca stars, which are also formed by bosonic but now vec-
tor fields, have been obtained recently. We make reference to [60–62] where non-
linear analysis of their stability and a study of their formation and frontal/orbital
collisions are performed. An interesting peculiarity of Proca stars is that their
rotating configurations, unlike the case of rotating Boson Stars, do not possess a
non-axisymmetric instability [63], indicating in fact that Proca-rotating stars are
“robust” configurations.
1.4 Thesis outline
The organization of this thesis is as follows. Chap. 2 concerning homogeneous
solutions to the Einstein equations with scalar fields is part of the introduction to
the manuscript, since the cosmological evolution of complex scalar fields is part
of the motivation of this work, this chapter includes results from publications
[E] and [F]. Part I concerns the solutions of compact objects in spherical symme-
try; In Chap. 3, extracted from publication [D], we construct solutions for ℓ-boson
star solutions with ℓ ≫ 1, Chap. 4, extracted from [A], discusses new E -boson
star solutions, characterized by an exotic matter core and very high compactness.
Chap. 5, based on publication [H], concerns numerical construction of multifield
wormhole solutions. Part II concerns the solutions of compact objects in axial
symmetry; Chap. 6, extracted from publication [B], deals with solutions with two
scalar fields coupled to the electromagnetic field and representing a magnetized
boson star, in Chap. 7 we discuss two-boson star solutions that we are currently
constructing in a system composed of a canonical complex scalar field and a real
field that gives mass to it. Finally, in Part III make numerical evolutions of ℓ-
boson stars; a perturbation analysis in Chap. 8, which is extracted from [G] and,
head-on collisions in Chap. 9, based on publication [C]. Unless otherwise stated,
we will continue to use units such that c = 1. Throughout the manuscript we will
remain using the metric signature (−,+,+,+).
CHAPTER 2
SPATIALLY-HOMOGENEOUS
COSMOLOGICAL SOLUTIONS
Contents
2.1 Dynamics of the homogeneous and isotropic Universe . . . . . 16
2.2 Scalar field dark matter with two components . . . . . . . . . . 18
2.3 Complex quintessence exact solution . . . . . . . . . . . . . . . . 28
C
osmological scenarios such as the ones presented in Chap. 1 motivate
the incorporation of scalar fields as the dark matter and dark energy
components. Although the real scalar fields are most commonly preferred
in quintessence and ultralight dark matter models, complex fields in certain cases
considerably simplify obtaining spatially homogeneous and isotropic solutions to
the Einstein equations [64,65].
In this chapter we present two of such solutions. The first one corresponding
to a global two scalar field model describing dark matter and the second case is
a quintessence-like (spintessence) model, which has an exact solution in a region
of the parameters of the solutions.
This chapter is organized as follows. In Sec. 2.1 the mathematical framework
in which homogeneous and isotropic cosmologies are constructed. Also, we obtain
general equations that will apply to the following two scalar fields examples. In
Sec. 2.2 we construct solutions of a two scalar-field dark matter model and present
a simple constraint in its parameters. In Sec. 2.3 we present solutions using an-
other scalar field cosmological model, this time corresponding to a dark energy.
Regarding the constraints in both models, in the dark matter case the free
parameters of the model are constrained by comparison with the observed abun-
dance of light elements from big bang nucleosynthesis, while in the dark energy
case, the model is simplified by using an approximation and which lead to an
equation of state with two free parameters which are then constrained by statis-
tical methods. The motivations for the scalar potential V (|Φ|) in both model comes
from different areas of physics. We will review briefly the context in which such
scalar field description arise.
15
16 2.1 Dynamics of the homogeneous and isotropic Universe
2.1 Dynamics of the homogeneous and isotropic Universe
The spacetime (M, g) is said to be spatially1 homogeneous if there exists a par-
ticular foliation where the hypersurfaces Σt of constant t such that for any points
p, q ∈ Σt, there exist an isometry of g which takes p into q. Isotropy means that
it should be impossible to find any preferred vectors on Σt. Let uµ be the tangent
vector of the timelike curves followed by the particular observers for which the
Universe looks isotropic. In coordinates where Σt is labeled by the proper time
t measured by the isotropic coordinates and assigning fixed spatial coordinates
(ψ,θ,ϕ) to each of these observers it can be shown (see e.g. [66], etc.) that the
metric reduce to one of the three possibilities
g
3−sphere
µν dxµdxν = −dt2 +a2(t)
[
dψ2 +sin2ψ
(
dθ2 +sin2θdϕ2
)
]
, (2.1)
g
spat.−flat
µν dxµdxν = −dt2 +a2(t)
[
dψ2 +ψ2
(
dθ2 +sin2θdϕ2
)
]
, (2.2)
g
hyperboloid
µν dxµdxν = −dt2 +a2(t)
[
dψ2 +sinh2ψ
(
dθ2 +sin2θdϕ2
)
]
, (2.3)
for an arbitrary function a(t).
The Friedmann metric (2.1) can be expressed using the “areal-radius” coordi-
nate r:
gµνdxµdxν =−dt2 +a2(t)
[
dr2
1−kr2
+ r2
(
dθ2 +sin2θdϕ2
)
]
(2.4)
where k, which has units of (length)−2, is positive, zero or negative in the 3-sphere,
spat.-flat and hyperboloid cases of Eq. (2.1), respectively. In this convention a(t) is
unitless. Also, if t0 is the age of the Universe2, we choose the convention in which
a(t0)= 1.
We now make use of the Einstein equations to determine the unique metric
coefficient a(t). If the matter and energy are modeled by a barotropic fluid3 at rest
in the comoving r,θ,ϕ coordinates with Tµ
ν = diag(ρ, p, p, p).
Combination of the µ
ν =t
t and r
r components of the Einstein equations (1.1)
lead to the Friedmann equations [67]
(
ȧ
a
)2
=
8πG
3
ρ−
k
a2
+
Λ
3
(2.5)
and
ä
a
=−
4πG
3
(ρ+3p)+
Λ
3
, (2.6)
1 There also exist spacetimes which are homogeneous throughout space and time (see e.g. Chap.
12 in Ref. [40]). They are also called maximally symmetric Universes in some other references.
213.8×109 years.
3 The resulting Friedmann equations will be the same if the energy-momentum tensor is that of a
scalar field which in an isotropic background is also of the form Tµ
ν = diag(ρ, p, p, p).
Spatially-homogeneous cosmological solutions 17
where ˙= d/dt and ¨= d2/dt2.
These equations where first derived by the Soviet physicist and mathemati-
cian Alexander Alexandrovich Friedmann in 1922, even before the discovery of
Hubble’s law (1929).
Knowledge of ρ as a function of a and substitution in Eq. (2.5) would be enough
to finish solving Einstein equations. To this end we obtain the zero component of
the energy-momentum tensor conservation equation, from which we obtain ρ̇+
3ȧ(ρ+ p)/a. Now, if we assume a constant in time equation of state ρ = wp, which
obviously will not work for the case of a general scalar field but will work for
barotropic fluids relevant in cosmology, we obtain ln ρ̇+3(1+w) ln ȧ = 0 which can
be integrated to obtain
ρ∝ a−3(1+w) (w constant). (2.7)
Now we define some of the cosmological parameters which will be useful be-
low: First, the Hubble parameter H := ȧ/a, which characterize the rate of expan-
sion of the Universe. ρΛ = Λ
8πG
being the (constant) density associated with dark
energy, as already defined in Chap.1. The critical density is ρcrit := 3H2/(8πG).
And finally the density parameter of the substance x is defined as Ωx := ρ/ρcrit,
x = r,m,Λ being radiation, total matter (baryionic + dark) and the cosmologi-
cal constant, respectively. With these definitions in hand and the solutions at
Eq. (2.7) the Friedmann equation (2.5) can be written as follows:
H2
H2
0
=
Ωr
a4
+
Ωm
a3
+ΩΛ (ΛCDM model). (2.8)
In the last equation we have not included the curvature parameter. According
to the latest results [29], the curvature parameter k is consistent with the spa-
tially flat case and will be considered as such for the remainder of this chapter,
however the question of the size and finiteness-infiniteness of the Universe is a
fascinating topic worth addressing (for a philosophical discussion from dialectical
materialism see e.g. [68,69]),
k = 0 . (2.9)
Let us now turn to the scalar field dark matter/energy models. For the case of
a complex scalar field minimally coupled to gravity, with (strictly real) mass term
µ and quartic (positive or negative) self-interaction λ, the total action reads
S =
∫
d4x
p
−g
(
1
16πG
R−
1
2
∇µ
Φ
∗∇µΦ−
1
2
µ2|Φ|2 −
λ
4
|Φ|4
)
+Sother−matter, (2.10)
where the other–matter term includes barotropic fluid contributions. Then the
scalar field energy momentum tensor T
(Φ)
µν can be computed from Eq. (1.3), ob-
taining:
T(Φ)µ
ν =gµη∂(ηΦ
∗∂ν)Φ−
δ
µ
ν
2
(
gαβ∂αΦ
∗∂βΦ+µ|Φ|2 +
λ
2
|Φ|4
)
, (2.11)
18 2.2 Scalar field dark matter with two components
for the mixed components.
Using the background metric (2.4), there are only two independent contribu-
tion of T
(Φ)
µν ; these are the energy density and the “pressure” terms,
ρΦ =
1
2
|∂tΦ|2 +
1
2
µ2|Φ|2 +
1
4
λ|Φ|4, (2.12)
pΦ =
1
2
|∂tΦ|2 −
1
2
µ2|Φ|2 −
1
4
λ|Φ|4, (2.13)
which enter the Friedman in a fluid type fashion:
H2 =
8πG
3
[ρother−matter +ρΦ], (2.14)
On the other hand, the conservation equation ∇µT
(Φ)
µν = 0 leads4 to the equa-
tion of motion for the field Φ
Φ−µ2
Φ−λ|Φ|2Φ= 0, (2.15)
also known as the Klein-Gordon equation.  :=∇µ∇µ is known as the d’Alembert
operator. In particular, applying the Friedmann ansatz for gµν and the defini-
tions for ρΦ and pΦ, Eq.(2.15) implies that ∂tρΦ+3H(ρΦ+ pΦ)= 0, which is iden-
tical to the conservation equation for a cosmological barotropic fluid, however, it
is important to remember that this parallelism occurs only for the homogeneous
case; in scalar field case we should solve only for the function Φ with the Klein-
Gordon equation, without assuming fixed a priori values for the equation of state
wΦ = pΦ/ρΦ.
2.2 Scalar field dark matter with two components5
Bohua Li et al. analyzed in Ref. [64] the hypothesis of dark matter comprised by
an ultralight complex and self-interacting scalar field. They found by analyzing
only the background equations, without the need to make the step to perturbation
theory, some constrains that the complex field ultralight particle should satisfy
even at this level in order to be consistent the big bang nucleosynthesis and with
the expected equation of state of dark matter at the present time. Interestingly,
they found that the free parameters of the model are very well constrained with
this method. Specifically, the terms they obtained that if the scalar field mass
4 Equivalently, can be obtained from the variation of S (Eq. (2.10)) with respect to Φ
∗.
5 This section is partially extracted with minor revisions from Ref. [70], which was written in col-
laboration with E. Gutiérrez-Luna, B. Carvente, J. Barranco, C. Escamilla-Rivera, C. Espinoza, M.
Mondragón and D. Núñez. © American Physical Society. Reproduced with permission. All rights
reserved.
Spatially-homogeneous cosmological solutions 19
parameter and the self-interaction satisfy6
µ~& 5×10−21 eV; (2.16)
8×10−4 eV−4 .
λ
(µ~)4
. 10−2 eV−4, (2.17)
then the complex scalar field dark matter will have an equation of state sufficiently
close to the dust-type material value |w| < 0.001 at the epoch of matter-radiation
equality and an effective number of neutrino species within the inferred range
from observed primordial light element abundances.
Now, the study made by Li et al. was performed without specifying or restrict-
ing to a particular elementary particle model extension. Also, there is no good
reason to expect that dark matter should be composed by a single type of matter;
ultimately its contribution to the energy density of the Universe (today) is several
times greater than that of known ordinary matter. With these two ideas in mind,
in [70] we addressed a situation with two scalar fields, one motivated by particle
physics and the other (classical) motivated by Li et al. study [64] of the possibility
of increasing the number of relativistic degrees of freedom by the correct amount
indicated by nucleosynthesis.
At the time of writing this manuscript, there has been no success in experi-
ments to detect any of the most popularly proposed dark matter particles, this has
prompted the proposal of other solutions to this problem. An example of this are
the scalar field models in which we are interested. The two most popular scalar
dark matter candidates are the axion and Higgs-like particles. In section II of
Ref. [70] we have reviewed some essential properties of the axion/axion-like par-
ticles and the Higgs-like particle consisting of an inert scalar SU(2) doublet. For
this manuscript we are interested in the final result of the discussion presented
there, which indicates that in both extensions of the standard model of particle
physics, there are ways in which at a classical level both the axion Φa, which at
the quantum level is a real field, and the Higgs-like particle Φh, which at the fun-
damental level is a bicomponent field, can be described by complex scalar fields.
It turns out that before coupling the fundamental fields with gravity, a classical
limit must be performed in order to be able to introduce the semi-classical energy-
momentum tensor in the Einstein equations (see section II in [70]), which looks
as follows for the scalar potentials
Vh = m2
h(Φ
†
h
Φh)+
λh
2
(Φ
†
h
Φh)2 −→ Vh(|Φh|)= m2
h|Φh|2 +
λh
2
|Φh|4 ;
Va = m2
a f 2
a

1−cos
(
Φa
fa
)

 −→ Va(|Φa|)= m2
a|Φa|2 −
m2
a
12 f 2
a
|Φa|4 ,
6 An update of the obtained constraints was made in Ref. [71]
20 2.2 Scalar field dark matter with two components
where fa is the scale of spontaneous symmetry breaking of the axion field which is
related to the (negative) self interaction by means of the relationλa/2= m2
a/(12 f 2
a ).
In the equations above and in the remainder of this section we will use units in
which we also assume ~= 1. In the two-scalar field models we also considered a
classical scalar field with potential Vc(|Φc|) = m2
c|Φc|2 + λc
2
|Φc|4. Before moving
on to the composite systems, let us note that there are already certain a priori
ranges or orders of magnitude of the single field models to be considered. Table
2.1 shows the range of these parameters as well as some representative cases that
will be useful below.
Table 2.1: Single scalar field models. Three single scalar field models with their
free parameters and the validity intervals of m and λ parameters. The repre-
sentative cases are the specific values of the parameters explored in the figures
within this section. Adapted from [70]
SINGLE MODEL Free m λ Representative
param. cases (m, λ)
Axion (Φa) fa 5.7
(
106 GeV
fa
)
eV −m2
a/(6 f 2
a ) (6×10−13 eV,−5×10−82)
Higgs (Φh) mh, λh ∼ 100GeV (−4π,4π) (100GeV,1)
Classical (Φc) mc, λc . 1eV > 0 (3×10−21 eV,4×10−86)
All that remains is to solve the Friedmann (2.14) and Klein-Gordon (2.15)
equations for a couple of self-interacting scalar fields. To integrate the equations
we follow the procedure presented by Li et al. in [64]. To this end we will only
consider cases where the two complex scalar fields behave like dust at the present
time, which will be the case if the complex phases ω of both fields satisfy
ω
H
≫ 1, (2.18)
known as the fast oscillation regime for scalar fields. We also ask the interaction
between the two fields to be negligible, then, the scalar field dark matter equations
presented in Sec. 2.1 are trivially modified. For example, the energy momentum
tensor in Eq. (2.11) will be the sum of two (functionally) equal parts and the same
happens for the density and pressure terms, while on the other hand we will have
two Klein-Gordon equations (2.15), one for Φ1 and one for Φ2.
We introduce the variables A1 = ρ1 − p1, A2 = ρ2 − p2 and B1 = m2
1
∂t|Φ1|2,
B2 = m2
2
∂t|Φ2|2. Then, the Friedmann equation (2.14) become
ȧ = aH0
√
Ωr
a4
+
Ωb
a3
+ΩΛ+
ρ1
ρcrit
+
ρ2
ρcrit
, (2.19)
Spatially-homogeneous cosmological solutions 21
and, the Klein-Gordon equation (2.15) for the field Φ1 splits up into
dρ1
da
= −3
2ρ1 − A1
a
, (2.20)
dA1
da
= ±
B1
ȧ
√
1+
2λ1
m4
1
A1, (2.21)
dB1
da
= −3
B1
a
+2m2
1
1
ȧ



2(ρ1 − A1)−
m4
1
2λ1


√
1+
2λ1
m4
1
A1 ∓1


2



. (2.22)
where if λ1 > 0 the upper signs are used and, if λ1 < 0 both signs are possible.
Similar equations hold for the Φ2 field. The density parameter for dark matter,
ΩDM fixes the total energy density ρ1+ρ2 at the present, a = 1, therefore we write
ρ1(a = 1)= η ΩDMρcrit, (2.23)
ρ2(a = 1)= (1−η) ΩDMρcrit, (2.24)
with 0≤ η≤ 1 representing the fraction of Φ1 with respect to Φ2 as of today.
In order to obtain evolutions in terms of the scale factor, it is only necessary
to solve Eqs. (2.20–2.22). To this end we will solve backwards in time the differ-
ential equations subject to the conditions (2.23). This might seem as a straight-
forward task, however, since the complex scalar field must be oscillating fast at
a = 1, Eq. (2.18), solving the system numerically becomes intractable. To solve
this problem, the system of differential equations is solved separately and ap-
proximately in the interval in which Eq. (2.18) is valid. As shown in [64, 65], the
system (2.20–2.22) reduce to the two equations for the field Φ1,
p1 =
m4
1
9λ1

1∓
√
1+
3λ1
m4
1
ρ1


2
, (2.25)
dρ1
da
=−3
ρ1 + p1
a
, (2.26)
and similarly for Φ2, whenever ω≫ H. The same previous rule for signs applies
here, however, from now on we will choose a sign, because as showed in [65] only
the upper sign give rise to dark matter behavior (the lower signs will be addressed
in the next section).
To evaluate when the fast oscillation regime stops to be valid for, say, Φ1 we
use the expression [64]
ω1 :=
d
dt
arg(Φ1)= m1
√
√
√
√
1
3
−
2
3
√
1−
3λ1
m1
1
ρ1, (2.27)
and make the switch to the full system (2.20–2.22) when ω1/H is sufficiently big.
At this point, a = ae, we must provide a non-trivial matching value for B1, which
22 2.2 Scalar field dark matter with two components
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
Ω
i
Radiation
Baryonic Matter
Λ
Classical (fiducial)
Higgs, Axion, CDM
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
w
(a
)
Classical (fiducial)
Higgs, CDM
Axion - Planck scale, fa = 1019GeV
Axion - GUT scale, fa = 1016GeV
Figure 2.1: Single scalar field representative cases of Table 2.1. Left panel: All
solid lines correspond to the classical positive self-interaction fiducial cosmol-
ogy [64]. The dashed lines are the reference CDM Universe, which happens to
coincide in this plot to the Higgs and axion (GUT, Planck) cases. Right panel:
Equations of state. While the positive self-interaction classical field undergoes
three phases, the negative self-interaction case undergoes two and the Higgs field
remains indistinguishable from standard cold dark matter. Adapted from [70].
can be shown to take the value
B1(ae)=−H(ae)
ρ1(ae)+ p1(ae)
√
1−2
λ1
m4
1
(ρ1(ae)− p1(ae))





2+
1
√
1−3
λ1
m4
1
ρ1(ae)





. (2.28)
We are now ready to solve the Einstein-Klein-Gordon system by means of nu-
merical integration. To this end we will use a simple implementation of the fourth
order Runge-Kutta method. Before proceeding to use this solution and constrain
we present typical solutions for the single and double scalar field models. In
Fig. 2.1 we show the equation of state for both, the representative single mod-
els, shown in the last column of Table 2.1 and the energy fraction parameters
Ωx = ρx/ρcrit for those cosmologies.
In this figure we note in particular that the axion has essentially two stages,
a stiff-matter era, with w = 1 and then a matterlike dust period w = 0, with this
transition occurring later in cosmological time as the scale of symmetry breaking
fa increases. On the other hand, the classical field has three characteristic stages:
consecutively w = 1, then w = 1/3 (radiation) and finally w = 0, matter-like. Fi-
nally, the Higgs-like field have constant w = 0 behavior for the considered range
of the scale factor.
There are three different types of scalar fields for the two-scalar field model,
hence we have taken the three possible combinations and display them in Ta-
ble 2.2. The analysis of the double model III is an ongoing project and it was not
Spatially-homogeneous cosmological solutions 23
Table 2.2: Double scalar field models: Three possible two-scalar field models with
the corresponding combinations at the description. The η constraint is referred
to the minimum fraction of the energy density of the lightest field at the present
(a = 1) with respect to the total dark matter density. The viability of the models
is reported in the last column.
DOUBLE MODEL Description η constraint Viability
I Classical + Higgs & 0.423 X
II Axion + Higgs × ×
III Classical + Axion - X
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
Ω
i
Φc +Φh
Φc
Φh
Radiation
Baryonic Matter
Λ
(a) Two scalar field model I. For η= 0.25
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
Ω
i
Φc +Φh
Φc
Φh
Radiation
Baryonic Matter
Λ
(b) Two scalar field model I. For η= 0.75
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
Ω
i
Φa +Φh
Φa
Φh
Radiation
Baryonic Matter
Λ
(c) Two scalar field model II. For η= 0.25
10−14 10−12 10−10 10−8 10−6 10−4 10−2 100
a
0.0
0.2
0.4
0.6
0.8
1.0
Ω
i
Φa +Φh
Φa
Φh
Radiation
Baryonic Matter
Λ
(d) Two scalar field model II. For η= 0.75
Figure 2.2: Evolution of the density parameters. All solid lines correspond to
the scalar field dark matter model with two components and the dashed lines
represent the rest of the density contributions. Top panel: Two scalar field model
I (Classical + Higgs). Bottom panel: Two scalar field model II (axion+Higgs).
Adapted from [70].
24 2.2 Scalar field dark matter with two components
presented either in Ref. [70] or in this manuscript. However, we have used the
double models I, II in Fig. 2.2 to present the fraction of energy of these scenarios.
In the top panels we plot the classical + Higgs case, with the same values for the
mass and self-interaction as the ones presented in the single model Table 2.1. In
the bottom panels of Fig. 2.2 we show the axion + Higgs case, again with the pa-
rameters given by the Table 2.1. Two different values for the fraction η of field 1
with respect to the total dark matter have been selected for each case.
We note that no fully relativistic solutions for a complex axion in a homoge-
neous background have been reported in the literature before, however cases com-
posed of two (real) dark matter scalar fields have already been reported in other
works (see e.g. [72–74]) and new papers continue to appear. We now proceed to
explain the last two columns of Table II which are the central result reported in
this section and mostly based in Sec. V B in [70].
Constraints from Neff and zeq
Among the parameters that determine the production of light elements at
BBN we have the expansion rate H. This is a period where every component
other than radiation is subdominant, therefore the presence of extra relativistic
degrees of freedom, beyond the Standard Model implies a modification to H with
respect to its ΛCDM profile. This can be quantified inside the effective number of
neutrino species Neff as a contribution to the ΛCDM value N0
eff
through a param-
eter known as number of equivalent neutrinos ∆Nν, although its source does not
necessarily come from a neutrino. It is defined by
∆Nν =
ρξ
ρν
, (2.29)
where ρν is the energy density of the standard model neutrino (per neutrino
specie) and ρξ is the energy density of the additional relativistic fields in consider-
ation, this contribution could correspond to the positive self-interaction (classical)
scalar field or to the negative self-interaction axion field for those cases when the
energy contribution is important in order to modify H, that is, when they behave
as radiation and/or stiff matter during BBN. With the previous definition, the
total radiation energy density divided by the photon energy density, ργ, is
ρr
ργ
= 1+
ρν
ργ
(
3+∆Nν
)
. (2.30)
If it is assumed that neutrinos are completely decoupled from the electromag-
netic plasma at the electron-positron annihilation, then the temperature of the
photons increases with respect to that of the neutrinos by (Tν/Tγ)3 = 4/11. Now,
the density ratio ρν/ργ = 7/8(Tν/Tγ)4, implies that
ρr
ργ
= 1+
7
8
(
4
11
)4/3
Neff, (2.31)
Spatially-homogeneous cosmological solutions 25
with
Neff = N0
eff
(
1+
∆Nν
3
)
; N0
eff = 3


11
4
(
Tν
Tγ
)3


4/3
. (2.32)
Where in this case N0
eff
= 3. However, if it is not assumed that the neutrinos
are completely decoupled when the electron-positron pairs annihilate, then N0
eff
=
3.046 [75].
The total Neff enters through H to the equations that determine the primordial
light element abundances (solved by BBN codes) and if for example the lepton
asymmetry is neglected, then the BBN primordial abundances can be confronted
with astronomical observations of the abundances of (mainly) deuterium D [76]
and the isotope 4He [77]. These constraints on the observed elements can be
traduced in constraints over Neff as well as Ωb [78,79].
In 2015 reference [79], obtained Neff = 3.56±0.23 or
∆Nν = 0.5±0.23. (2.33)
This value certainly excludes the possibility of a new neutrino as well as the
standard N0
eff
case. Nevertheless the parameters of a complex scalar field with
positive λ can be constrained to be consistent with this measurement as showed
by Li et al. [64,71] if a time dependent ∆Nν(a) is assumed rather than a relatively
late time fixed value. The constraint (2.33) is applied through BBN, between the
neutron to proton freeze-out and the first nuclei production, at an/p and anuc re-
spectively.
In our numerical analysis, if Φ1 is not subdominant at BBN, then the con-
straint (2.33) is implemented with the formula,
Neff =
N0
eff
2



1+
Ω1
Ωr
+
√
√
√
√
(
1+
Ω1
Ωr
)2
+
Ω1
Ωr
32
7
(
11
4
)4/3 1
N0
eff



, (2.34)
which is a result of inserting ργ from (2.31) into (2.32) along with the definition
(2.29) and ρν =Ωr −Ωγ, notice that in this expression, Ωr contains the γ and ν
contributions only, which are evolved separately from Φ in the code.
Additional to the BBN constraints discussed so far, there is a need to make
the relativistic and stiff matter scalar field solutions reach a matterlike behavior
in w at the latest in the matter-radiation equality zeq ≈ 3365. This condition is
imposed in the code by setting w(zeq)< 0.001.
The results of this BBN+zeq analysis for the single λ1 > 0 scalar field, was
reported first by Li et al. in [64] and later an update was made within their
work [71]. We recover their result in Eqs. (2.16) and (2.17).
If we repeat this analysis now including the single scalar axion case, we should
be able to obtain a constraint on the single parameter fa particularly for the cases
26 2.2 Scalar field dark matter with two components
with big values of this parameter, which as shown in the previous section, are the
models that affect expansion the most. It should be mentioned that the general
λ< 0 case cannot be solved in all the cases, particularly in those where the slow
oscillation regime appears closer to a = 1 and the square root arguments in (2.21)
and (2.22) become negative at certain point ai which corresponds to a place where
the scalar field “turns on” [65]. Luckily, numerical experimentation on solutions
for the axion field (where the mass and self-interaction have a specific dependence
on fa) shows that this is never the case and therefore, no discontinuities in the
Einstein equations appear.
However, the situation occurs when the stiff matter stage of the axion affects
Neff(a) very drastically, and not in the “stepped" way in which it happens for the
λ> 0 case. If the limits are kept to 1σ in equation (2.33) then there is no value of
fa for which Neff(a) is kept inside these limits, not even at 2σ. It happens that if
the Neff(a) enters into the limits (2.33) in an/p at the beginning of nucleosynthesis,
then it no longer enters at the end of it, at anuc, and vice versa.
Therefore, the single axion model is discarded in relation to this cosmological
constraint.
It is possible to repeat this analysis for the two scalar field cases. We are
interested in exploring Model I and Model II (Table 2.2). Both of them include
the Higgs-like field, which as has been said is similar to CDM fluid regardless of
the specific values that mh and λh assume. Therefore, in Model I we have a three
parameter model and in Model II we have just two parameters.
• Model I. We fix the value of η, (i.e. the fraction of the energy density of Φc at
a = 1 with respect to total dark matter density, (2.23)), and explore the exis-
tence of possible values of m1 and λ1 consistent with the 1σ BBN+zeq anal-
ysis. The case η= 1 coincides with the single case constraints in (2.16,2.17).
If we begin to decrease the value of η, the range of the parameters consis-
tent with the constraint also decrease in size, as shown in Figure 2.3, until
a critical value is reached, after which no value is allowed. This constraint
on η, gives
η& 0.423 . (2.35)
That is, an upper bound of ∼ 58% for the Higgs (or w = 0 fluid) component
can be considered in order to be consistent with these constraints. In the
critical case, where η takes values near 0.423, we have that the (m,λ/m4)
parameter space narrows to the values m & 2× 10−21eV and λ/m4 ∼ 3×
10−2eV−1.
• Model II. In this simpler case, a joint analysis over η and fa can be made.
We find that no 1− η ratio of the Higgs field is capable of smoothing the
Neff(a) evolution dictated by the axion, during BBN. And since the Higgs
Spatially-homogeneous cosmological solutions 27
10−14 10−12 10−10 10−8 10−6 10−4 10−2
λ/m4 [eV−4]
10−22
10−21
10−20
10−19
10−18
m
[e
V
]
anuc
an/p
A
llo
w
ed
R
eg
io
n
η = 1
(a)
10−14 10−12 10−10 10−8 10−6 10−4 10−2
λ/m4 [eV−4]
10−22
10−21
10−20
10−19
10−18
m
[e
V
]
anuc
an/p
A
llo
w
ed
R
eg
io
n
η = 0.75
(b)
10−14 10−12 10−10 10−8 10−6 10−4 10−2
λ/m4 [eV−4]
10−22
10−21
10−20
10−19
10−18
m
[e
V
]
anuc
an/p
η = 0.5
(c)
10−14 10−12 10−10 10−8 10−6 10−4 10−2
λ/m4 [eV−4]
10−22
10−21
10−20
10−19
10−18
m
[e
V
]
anuc
an/p
η = 0.25
(d)
Figure 2.3: Constraints from zeq and Neff within 1σ for the two scalar field Model
I. η is the fraction of the classical field with respect to the total dark matter com-
ponents and its values are (a) η = 1, (b) η = 0.75, (c) η = 0.5, (d) η = 0.25. The
crosshatched region that appears on the right side of all figures, represents the
values of the scalar field parameters not allowed by the zeq constraint. The green
and yellow bands are the allowed regions from the Neff constraint, (2.33), at an/p
and anuc respectively. The red band is the region of the parameter space that
is consistent with both the zeq and Neff, throughout BBN, constraints. Adapted
from [70].
28 2.3 Complex quintessence exact solution
field has a contribution to Neff of 0 with respect to N0
eff
, this two fields case
(like the single axion case), is discarded in the sense that there is no set of
parameters such that (2.33) is satisfied. Relaxing the constraint to 2σ in
(2.33) no allowed values are found either.
• Model III. The classical (λ> 0) + axion case has 4 relevant free parameters
and a higher complexity. It is found that for all axions with fa < 1019GeV, for
which the equation of state is 0 before the start of BBN, as showed in Fig. 2.1,
the axion scalar field behaves effectively as a w = 0 fluid for the purposes of
this restriction, and therefore the same restriction as for case I would apply
here. A complete analysis of this case will be reported elsewhere.
2.3 Complex quintessence exact solution7
Complex scalar fields should be considered since such fields (unlike the real case)
have been invoked in many different sectors of particle physics (as discussed in
previous section) and interestingly in the scene of ultra cold gases [81]; they can
be used to construct static distributions as boson stars, and also configurations
surrounding a black hole, the so–called wigs [82]. Furthermore, a real quantized
scalar field yields the same field equations as those obtained by using a classi-
cal complex scalar field [83]. These reasons motivated us to consider a dark en-
ergy model described by a massive quintessence–complex scalar field with attrac-
tive self interaction. Such field was formerly studied in [65] and, in the present
work, we revisited the idea focusing in the so-called peculiar branch solution of
the Einstein-Klein-Gordon equations in order to obtain parameter restrictions of
the potential consistent with the current precision observations. Although we
are aware of the latest results regarding the possible dynamical behavior of the
EoS [37] and the impossibility for a single canonical field to evolve crossing over
w =−1 because of the no-go theorem [39], it is interesting to explore in detail the
properties of the previously mentioned branch and in computing best fit values of
their parameters, in order to have a quantitative description of the model and a
clearer picture of what the model needs in order to be consistent with such a dy-
namical behavior of the dark energy. In this section we will use units that recover
the speed of light c.
We use the evolution described as a starting point, and introduce a complex
scalar field in order to model dark energy. Our proposal is based in the fact that
the scalar potential V (
∣
∣Φ
∣
∣
2
), has a quartic-form with a negative scattering length
7 This section is partially extracted with minor revisions from Ref. [80], which was written in collab-
oration with B. Carvente, C. Escamilla-Rivera and D. Núñez. © Oxford University Press. Reproduced
with permission. All rights reserved.
Spatially-homogeneous cosmological solutions 29
as
V (
∣
∣Φ
∣
∣
2
)=
m2c2
2~2
∣
∣Φ
∣
∣
2 −
2πAsm
~2
∣
∣Φ
∣
∣
4
, (2.36)
where m is the complex scalar field mass, As the absolute value of the scatter-
ing length and ~ the reduced Planck constant. This scalar potential describes,
for instance, a relativistic Bose-Einstein condensate at zero temperature with at-
tractive self-interaction, and it is also similar to the Higgs potential of particle
physics but with an overall opposite sign.
The evolution of this complex scalar field in the cosmological scenario de-
scribed above is given by the Klein-Gordon equation
1
c2
d2
Φ
dt2
+
3H
c2
dΦ
dt
+2
dV
d
∣
∣Φ
∣
∣
2
Φ= 0, (2.37)
from where we can express the complex scalar field as
Φ= |Φ|eiθ. (2.38)
Solutions to the Einstein-Klein-Gordon equations would require in total six pa-
rameters related to initial conditions for the real and imaginary parts of Φ and
their first time derivative together with the scalar field values for m and As. A
further simplification can be made within this model when, consistently with dark
energy-like behavior, we assume that the field is oscillating rapidly, this leads to
a three-parameter model.
In our proposal, we are going to follow the procedure given in [65], of which
we summarize some key points. Using (2.38) in (2.37), the Klein-Gordon equation
can be divided into a real and an imaginary part, from which the second leads to
the equation:
Q =−
1
~c2
a3
∣
∣Φ
∣
∣
2 dθ
dt
, (2.39)
where Q a is constant8 and a the scale factor.
From the real part, and using the conserved charge Q explicitly in this equa-
tion, we obtain
1
c2


d2
∣
∣Φ
∣
∣
dt2
−
Q2
~
2c4
a6
∣
∣Φ
∣
∣
3

+
3H
c2
d
∣
∣Φ
∣
∣
dt
+2
dV
d
∣
∣Φ
∣
∣
2
∣
∣Φ
∣
∣= 0. (2.40)
The term containing Q2 is usually related to a centrifugal force when making
the analogy of this equation with that of fictitious particle with radial coordinate
∣
∣Φ
∣
∣, hence the name spintessence for that model [65,84].
8After integration, the imaginary part of the Klein-Gordon equation leads to a conserved
quantity, which corresponds to the conserved charge of a complex scalar field, given by Q =
1
c2~
∫
dx3p−g Im(Φ∂tΦ
∗).
30 2.3 Complex quintessence exact solution
In the real case, with a quartic potential analogous to (2.36), we have θ = 0,
therefore the conserved quantity Q in (2.39) is equal to zero, implying among
other things, that the solution must have a rapidly oscillating behavior with an
equation of state also oscillating around w = 0 [24] and the solutions to the equa-
tion of motion must be obtained by numerical integration in an appropriate set of
variables. The quartic potential is not the only possibility, for instance taking a
massless scalar field (µ= 0, λ= 0) the equation of state stays trivially at the value
w = 1. Other (real) scalar fields, describing quintessence potentials, as the ones
listed in the introduction, may have dynamical EoS some of which also oscillate
in time. In this work we take the opposite approach, namely Q ≫ 0, leading to an
exact solution of the problem which is useful in the implementation of tests for the
model with cosmological analyzes. To compute the energy density and pressure
of the complex scalar field, we consider the following expressions:
ǫ=
1
2c2
∣
∣
∣
∣
dΦ
dt
∣
∣
∣
∣
2
+V (
∣
∣Φ
∣
∣
2
), (2.41)
P =
1
2c2
∣
∣
∣
∣
dΦ
dt
∣
∣
∣
∣
2
−V (
∣
∣Φ
∣
∣
2
). (2.42)
Notice how we can connect these equations to the ones presented in Sec.2.1 where
the quantity ǫ will replace the ΛCDM quantity ρcritΩΛ in the Fridman equation.
From the equations (2.37), (2.41) and (2.42) we can obtain a useful equation for
the energy density that resembles the continuity equation for a barotropic fluid
dǫ
da
+
3
a
(ǫ+P)= 0. (2.43)
With these equations, now we are ready to study particular solutions of the
Einstein-Klein-Gordon system evolving with a complex scalar field mimicking the
dark energy component. As mentioned, this particular model in the fast oscilla-
tion regime and its homogeneous solution have already been presented previously
by [65], and we extend the study in order to obtain analytical expressions for most
of the quantities of the solution, including w(z).
Dark Energy Solution in the fast oscillation regime
In [65] was found that in the fast-oscillation regime, i.e., when the oscillation fre-
quency of the scalar field is much larger than the value of the Hubble function, the
solution of the Einstein-Klein-Gordon equations for the case of a complex scalar
field with an attractive self interaction potential (2.36) has two different solutions.
One solution (called normal branch) resembles to a dark matter scalar field, while
the other solution (called peculiar branch) corresponds to a quintessence model.
This solution only exists in the fast oscillation regime, in which the scalar field
suddenly emerges and behaves as dark energy at late times.
Spatially-homogeneous cosmological solutions 31
Following the same logic, in this paper we propose a deduction of an exact so-
lution for the equation of state of the quintessence field. Once with this equation,
we explore their possible constraints by using current observational data.
Peculiar branch solution in the fast oscillation approximation
To establish the fast oscillation regime mentioned above, we consider the following
condition which needs to be satisfied during the evolution of the scalar field
ω=
dθ
dt
≫ H. (2.44)
In addition to the latter condition, we will impose that the magnitude of the scalar
field change slowly on time respect to the angular frequency of oscillation ω as:
1
∣
∣Φ
∣
∣
d
∣
∣Φ
∣
∣
dt
≪ω. (2.45)
Conditions (2.44)-(2.45) set the so-called fast oscillation regime of the Klein-
Gordon equation (2.37). Following this prescription, (2.40) can be reduce to
ω2 = 2c2 dV
d
∣
∣Φ
∣
∣
2
. (2.46)
This allows us to write the fast oscillation condition in terms of the charge Q
defined in (2.39), which becomes
Q2
~
2c4
a6
∣
∣Φ
∣
∣
4
= 2c2 dV
d
∣
∣Φ
∣
∣
2
. (2.47)
Using the expression for the scalar field potential (2.36), we can approximate
(2.41) using the condition (2.46) as
ǫ =
1
2c2


(
d
∣
∣Φ
∣
∣
dt
)2
+ω2
∣
∣Φ
∣
∣
2

+
m2c2
2~2
∣
∣Φ
∣
∣
2 −
2πAsm
~2
∣
∣Φ
∣
∣
4
≈
m2c2
~2
∣
∣Φ
∣
∣
2 −
6πAsm
~2
∣
∣Φ
∣
∣
4
, (2.48)
By a similar approach, the scalar pressure from (2.43) can take the approximate
form
P ≈−
2πAsm
~2
∣
∣Φ
∣
∣
4
. (2.49)
Solving (2.48) for
∣
∣Φ
∣
∣
2, we obtain two possible branches that correspond to so-
lutions of the Einstein-Klein-Gordon system in the fast oscillation approximation
∣
∣Φ
∣
∣
2 =
c2m
12πAs



1±
√
1−
24πAs~
2
m3c4
ǫ



. (2.50)
32 2.3 Complex quintessence exact solution
Notice that this is a different result in comparison to the repulsive self-interaction
case [64], where there is an unique branch in the solution since only the (+) sign of
the square root is possible. Furthermore, in [65] was shown that for the attractive
self-interaction case (2.50) and when we take the negative sign, the scalar field
undergoes a matter-like phase (and even an inflation epoch). While for the positive
branch, the solution behaves as dark energy. From this point forward we will take
the positive sign, to focus on that particular branch.
Therefore, by using (2.50) in (2.49) we obtain
P(ǫ)=−
m3c4
72πAs~
2



1+
√
1−
24πAs~
2
m3c4
ǫ



2
. (2.51)
Physical solutions of this latter equation correspond to those values of ǫ smaller
than a certain ǫi:
ǫi =
m3c4
24πAs~
2
. (2.52)
From the two latter expressions, notice that P(ǫi) = − m3 c4
72πAs~
2 , implies that
wi = P(ǫi)
ǫi
=−1/3.
The scale factor for which the energy density takes the value ǫi can be calcu-
lated by inserting the value of
∣
∣Φ
∣
∣
2 evaluated in ǫi, and taking the result on the
fast oscillation condition (2.47):
ai =
3
√
12
p
3πAs~
2|Q|
m2c2
. (2.53)
For convenience, we re-define a dimensionless quantity in terms of the differential
equation for the energy density as
ǭ=
ǫ
ǫi
, (2.54)
therefore (2.43) can be written as
dǭ
da
=−
3
a
[
ǭ−
1
3
(
1+
p
1− ǭ
)2
]
. (2.55)
Evaluating in ǫ = ǫi, we can see that dǭ/da, takes a negative value of − 2
ai
,
therefore for a < ai the solution is not valid. The value ai indicates the scale factor
at the time when the scalar field turns on. Furthermore, at a →∞, ǫ approaches
to a constant value.
Now, taking the fast oscillation equation (2.47) and inserting
∣
∣Φ
∣
∣
2 from (2.50)
we obtain
(
ai
a
)6
= 3
(
1+
p
1− ǭ
)2
−2
(
1+
p
1− ǭ
)3
. (2.56)
Spatially-homogeneous cosmological solutions 33
In order to find the asymptotic value of ǫ, when a →∞, we should consider the
fast oscillation equation (2.47), which for potential (2.36) takes the form
Q~c2
a3
=
p
2c
∣
∣Φ
∣
∣
2
√
m2c2
2~2
−
4πAsm
~2
∣
∣Φ
∣
∣
2
. (2.57)
Since ǫ decreases with a, then
∣
∣Φ
∣
∣
2 increases as a → ∞ as we can notice from
(2.50), therefore the term inside the square root in (2.57) should vanish as a →∞,
leading to an asymptotic value of
∣
∣ΦΛ
∣
∣
2 =
mc2
8πAs
. (2.58)
Using (2.48) and (2.51) we can obtain
ǫΛ =
m3c4
32πAs~
2
=
3
4
ǫi, (2.59)
P(ǫΛ)=−ǫΛ. (2.60)
Notice how in the limit a →∞, the scalar field has an EoS with a value wΛ =−1.
Therefore, the EoS interpolates between the values−1/3 and−1. This is a result of
both the rapidly oscillating behavior of the field and the chosen peculiar branch,
although not a general property of a homogeneous complex cosmological scalar
field nor a direct consequence of having a non-zero conserved quantity Q. This
result is very different from the one that would have been obtained for the other
branch of the solution or even for the real case. In those cases we would not have
a scalar field solution with w < 0 that turns on at a certain scale factor ai and not
before.
Exact solution for the dark energy term
To obtain an expression for ǫ in terms of the scale factor, we have to solve the
equation (2.56). This can be obtained making the change of variable
ζ=
p
1− ǭ+
1
2
. (2.61)
The latter leads to an expression in terms of a cubic equation
ζ3 −
3
4
ζ+
1
2


a6
i
a6
−
1
2

= 0, (2.62)
34 2.3 Complex quintessence exact solution
which has three real solutions. However, it must satisfy the conditions ζ(ai) = 1
2
and ζ(a →∞)= 1. The only solution that satisfy these conditions is
ζ(a)= cos



1
3
arccos

1−2
a6
i
a6





, (2.63)
in terms of this function ζ(a), the energy density and the EoS parameter are given
by the following expressions
ǫ(a)=
[
1−
(
ζ(a)−
1
2
)2
]
ǫi, w(a)=−
(
ζ(a)+ 1
2
)2
3−3
(
ζ(a)− 1
2
)2
. (2.64)
This is the so-called Complex Scalar Field Dark Energy (CSFDE) model. These
solutions should be considered only in certain region ai < a < ae of the evolution
of the Universe, the upper limit ae is defined as the scale factor when the fast
oscillation regime ceases to be valid, which we will calculate below. This is evident
from (2.47), since ω get suppressed by the term a6, while
∣
∣Φ
∣
∣ goes to a constant
value. From now on, a will only be referred to this range. First, we must make
sure that the solution at ai satisfy the fast oscillation approximation described in
the latter section.
Under these ideas, the fast oscillation condition ω≫ H is given by
Q2
~
2c4
a6
∣
∣Φ
∣
∣
4
≫
8πG
3c2
(ρm +ǫ), (2.65)
where ρm = Ωm/ρcrit. By performing the substitution of
∣
∣Φ
∣
∣
2 using (2.50), re-
writing it in terms of ǭ and, finally, taking ǫ≫ ρm, we can obtain
(
ai
a
)2
≫
mG
3c2 As
ǭ(1+
p
1− ǭ)2. (2.66)
This condition will be satisfied initially if
3c2 As
mG
≫ 1. (2.67)
In order to compute the value ae > ai, where the solution is no longer valid, we
will consider the end of the fast oscillation regime when ω = NH (with N = 200
analogous to [64]). If ae ≫ 1 and also ae ≫ ai, in order to be able to make the
approximations ǫ ≫ ρm and ai/ae ≪ 1 in (2.66), then the end value of the scale
factor will be
ae ≈
6
√
768
N2 π
2 A3
s~
4Q2
Gm5c2
. (2.68)
Spatially-homogeneous cosmological solutions 35
Figure 2.4: Evolution of the w Eq. (2.64) as a function of the scale factor, a, in the
quintessence model. Adapter from [80].
In Fig. 2.4 we show an example for the evolution of the equation of state
parameter w between the values for the scale factor ai and ae, determined by
specific values of m, As and Q. In this example we take9 ai to be the value
amin = 0.1< 1/(1+ zmax) where zmax = 2.26 corresponds to the maximum redshift
used in the multiple data sets within the analysis described in the next section. In
this way, we ensure that the scalar field is present throughout the a range of the
analysis. We have restricted this example to the case where ae = 1, thus ensuring
that the limit of rapid oscillations and therefore the cosmological constant type
behaviour continues to be valid today.
9This particular choice of ai and ae in our example reduces the dimension of the free parameter
space from 3 to 2, thus we can put Q and As in terms of m:
Q =
4a9
minmc4
27
p
3πNG~4
, As =
9N2G~
2m
16a6
minc2
. (2.69)

Part I
Spherically-symmetric
solutions

CHAPTER 3
EXTREME ℓ-BOSON STARS1
Contents
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 ℓ-boson stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Extreme ℓ-boson stars . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.3 Geodesic motion . . . . . . . . . . . . . . . . . . . . . . . 52
3.4 Scaling properties for large ℓ . . . . . . . . . . . . . . . . . . . . 54
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A
new class of complex scalar field objects, which generalize the well known
boson stars, was recently found as solutions to the Einstein-Klein-Gordon
system. The generalization consists in incorporating some of the effects
of angular momentum, while still maintaining the spacetime’s spherical symme-
try. These new solutions depend on an (integer) angular parameter ℓ, and hence
were named ℓ-boson stars. Like the standard ℓ = 0 boson stars these configu-
rations admit a stable branch in the solution space; however, contrary to them
they have a morphology that presents a shell-like structure with a “hole” in the
internal region. In this chapter we perform a thorough exploration of the param-
eter space, concentrating particularly on the extreme cases with large values of
ℓ. We show that the shells grow in size with the angular parameter, doing so
linearly for large values, with the size growing faster than the thickness. Their
mass also increases with ℓ, but in such a way that their compactness, while also
growing monotonically, converges to a finite value corresponding to about one half
of the Buchdahl limit for stable configurations. Furthermore, we show that (the
1This chapter is extracted with minor revisions from Ref. [85], which was written in collaboration
with M. Alcubierre, J. Barranco, A. Bernal, J. C. Degollado, A. Diez-Tejedor, M. Megevand, D. Núñez
and O. Sarbach. © IOP Publishing. Reproduced with permission. All rights reserved
39
40 3.1 Introduction
pressure components of the) ℓ-boson stars can be highly anisotropic, with the ra-
dial pressure diminishing relative to the tangential pressure for large ℓ, reducing
asymptotically to zero, and with the maximum density also approaching zero. We
show that these properties can be understood by analyzing the asymptotic limit
ℓ→∞ of the field equations and their solutions. We also analyze the existence
and characteristics of both timelike and null circular orbits, especially for very
compact solutions.
3.1 Introduction
The possibility that dark matter can be described by a scalar field has recently
found an increasing interest, either through the study of models with a particle
physics motivation [86], or through the description of lighter fields with the poten-
tial to alleviate some possible tensions in the standard cosmological scenario on
small scales [26, 87–91]. Gravitationally bound bosonic structures appearing as
the consequence of these fields may be relevant in astrophysics, as they could de-
velop dark matter halos and/or very compact objects, depending on the particular
choice of the parameters of the model. In the high compactness regime, bosonic
structures can approach the Buchdahl limit [92] and form objects similar in size
and mass to neutron stars or even black holes. Like other compact objects [93],
boson stars may form bound binary systems emitting gravitational waves of dis-
tinctive features. The dynamics of these systems has been studied for instance
in references [58, 59, 94] (see also [95] where the waveforms calculated from the
head-on collision between two Proca stars is confronted with gravitational wave
observations). On the other hand, in the low compactness regime, gravitation-
ally bound structures can be used to describe dark matter halos, although some
controversies have arisen regarding the non-compatibility on the required val-
ues of the field mass when combining different data sets. For example, the char-
acteristic masses needed to describe the internal kinematics of the Milky Way
dwarf spheroidal satellites are in tension when faced with cosmology [96], and
even at local scales the mass density profiles of dwarf spheroidal and ultra-faint
dwarf galaxies suggest different values for the field mass. Furthermore, for larger
galaxies the dark matter halos could be even more cuspy than the standard cold
dark matter Navarro-Frenk-White profiles [97]. These problems emerge when
fitting the observations to the dark matter halo profile predicted by a standard
boson star, in some cases enlarged with an external Navarro-Frenk-White pro-
file as suggested by numerical cosmological simulations [98]. However, in recent
years, it has been argued that more general stable, self-gravitating scalar field
objects could exist in nature, and this may affect the previous conclusions.
An interesting example of such configurations are the ℓ-boson stars we have
Extreme ℓ-boson stars 41
presented in previous work [99]. Based on similar ideas used previously in the
context of gravitational collapse [100], ℓ-boson stars incorporate some effects of
the angular momentum into the scalar fields while maintaining the spherical
symmetry of the spacetime, which results in a relatively simple model for their de-
scription. In the particular case where ℓ= 0 the standard boson stars by Kaup [44]
and Ruffini and Bonazzola [45] are recovered. However, in general, in addition to
the parameters that characterize the standard ℓ = 0 solutions, there is an “an-
gular momentum number” ℓ that provides a model with a richer structure that
could potentially be relevant for the description of dark matter halos and com-
pact objects. In particular, and as we further explore in this chapter, boson stars
with ℓ> 0 can be more compact than standard ones. It turns out that the maxi-
mum mass of these objects increases greatly with ℓ, giving masses that are orders
of magnitude larger than for the ℓ= 0 case. Even if these configurations are also
larger in size than the standard ones, the growth in mass is faster than the growth
in size in such a way that the compactness increases.
The stability of ℓ-boson stars under spherical perturbations has first been
studied in [101] by performing numerical evolutions of the Einstein-Klein-Gordon
equations in spherical symmetry, and later also in [102] based on a more formal
study of the linearized system. These analyses have revealed that ℓ-boson stars
show stability characteristics that are qualitatively similar to those of the ℓ = 0
case, where for each value of ℓ there exist a stable and an unstable branch with
the transition point given by the solution of maximum total mass. For other stud-
ies addressing the stability of ℓ-boson stars which are based on full nonlinear
numerical evolutions without symmetries see [103,104] (see also [105] for a study
of the Newtonian regime in axial symmetry). In particular, in [104] it was shown
that ℓ-boson stars assume a privileged role among other stationary solutions of
the multi-field, multi-frequency scalar field scenario as far as their stability is
concerned.
In the present work we perform an exhaustive exploration of the ℓ-boson stars’
parameter space, focusing in particular on solutions with very large values of ℓ,
including the ℓ→∞ limit. Our analysis covers the stars’ morphology, anisotropy
and compactness, the characteristics of the circular orbits (including the null ones,
also known as light rings), as well as the scaling properties of the fields and rel-
evant physical quantities with respect to ℓ. We start in section 3.2 with a brief
review of ℓ-boson stars, presenting the main equations and properties, includ-
ing the definitions of density, pressure, anisotropy and compactness, and present
the equations for geodesic motion, particularly those describing circular causal
geodesics. Next, in section 3.3, we present our solutions, analyzing in each case
the role played by the angular momentum parameter ℓ on various of their proper-
ties, and paying particular attention to the large ℓ regime. We accomplish this by
42 3.2 ℓ-boson stars
numerically obtaining and analyzing hundreds of solutions. The observed scal-
ing properties of the fields for large ℓ motivate the in-depth study of section 3.4,
where we obtain effective equations which describe the asymptotic behavior of the
fields in the limit ℓ→∞. Conclusions and an overview of our results are given in
section 3.5. Technical details and tables summarizing our notation and numerical
data are included in appendix 3.6.
Throughout this work we use the signature convention (−,+,+,+) for the space-
time metric and Planck units such that G = c = ~= 1. We present our results in a
form that is independent of the scalar field mass µ. The rescaling rules in µ are
summarized in table 3.2 of appendix 3.6.
3.2 ℓ-boson stars
In this section we summarize the relevant equations that describe ℓ-boson stars,
as well as some of their most significant properties. Additional information can
be found in our previous works [99, 101, 102]. ℓ-Boson stars are self-gravitating
objects that consist of an odd number N = 2ℓ+ 1 of complex scalar fields Φℓm,
m =−ℓ, . . . ,ℓ of equal mass µ and the same radial profile. The dynamics of these
fields is described by the following Lagrangian
L =
R
16π
−
1
2
ℓ
∑
m=−ℓ
(
∇µΦℓm∇µ
Φ
∗
ℓm +µ2|Φℓm|2
)
, (3.1)
where R is the Ricci scalar and the scalar fields have the form:
Φℓm(t, r,ϑ,ϕ)= eiωtψℓ(r)Y ℓm(ϑ,ϕ), (3.2)
with ω a real frequency and ψℓ a real-valued radial function which is indepen-
dent of m. As usual, Y ℓm denote the standard spherical harmonics with angular
momentum numbers ℓ and m. By applying the addition theorem for spherical
harmonics one can see (as shown in [99]) that in the absence of self-interactions,
the total stress energy-momentum tensor
Tµν =
1
2
ℓ
∑
m=−ℓ
[
∇µΦ
∗
ℓm∇νΦℓm +∇µΦℓm∇νΦ
∗
ℓm − gµν
(
∇αΦ
∗
ℓm∇α
Φℓm +µ2
Φ
∗
ℓmΦℓm
)
]
(3.3)
is spherically symmetric, even if ℓ > 0 (N > 1) and the individual fields have an-
gular momentum.
The spacetime metric is parameterized according to
ds2 =−α2(r)dt2 +γ2(r)dr2 + r2dΩ2, γ2(r) :=
1
1− 2M(r)
r
, (3.4)
where α and M denote the lapse and the Misner-Sharp mass functions, respec-
tively, r is the areal radius and dΩ2 is the standard metric on the unit two-sphere.
Extreme ℓ-boson stars 43
The field equations are obtained from the Einstein-Klein-Gordon system and take
the form [99]:
M′ =
κℓr2
2


ψ′2
ℓ
γ2
+
(
µ2 +
ω2
α2
+
ℓ(ℓ+1)
r2
)
ψ2
ℓ

= 4πr2ρ, (3.5a)
(αγ)′
αγ3
= κℓr


ψ′2
ℓ
γ2
+
ω2
α2
ψ2
ℓ

= 4πr(ρ+ pr), (3.5b)
1
r2αγ
(
r2α
γ
ψ′
ℓ
)′
=
(
µ2 −
ω2
α2
+
ℓ(ℓ+1)
r2
)
ψℓ, (3.5c)
whith κℓ := 2ℓ+1, and where we have introduced the energy density, radial pres-
sure and tangential pressure defined as:
ρ :=−T t
t =
κℓ
8π


ψ′
ℓ
2
γ2
+
ω2
α2
ψ2
ℓ+
(
µ2 +
ℓ(ℓ+1)
r2
)
ψ2
ℓ

 , (3.6a)
pr := Tr
r =
κℓ
8π


ψ′
ℓ
2
γ2
+
ω2
α2
ψ2
ℓ−
(
µ2 +
ℓ(ℓ+1)
r2
)
ψ2
ℓ

 , (3.6b)
pT := Tθ
θ = Tϕ
ϕ =
κℓ
8π

−
ψ′
ℓ
2
γ2
+
ω2
α2
ψ2
ℓ−µ2ψ2
ℓ

 . (3.6c)
We denote by MT the total mass of the object, given by the limit r → ∞ of the
function M(r) = 4π
∫r
0 r̃2ρ(r̃) dr̃. In the case of our numerical solutions, we ap-
proximate MT by evaluating M(r) a the outer boundary of the numerical domain
(after ensuring that the mass variation is negligible near that boundary).
Each ℓ-boson star solution is uniquely determined by a given set of the param-
eters ℓ, µ, u0, and a discrete set of values ω, with u0 given by ψℓ/rℓ evaluated at
r = 02. Given ℓ and µ, u0 is a free parameter (which reduces to the central scalar
field amplitude ψc =ψ(r = 0) for ℓ= 0), and the ω’s are the frequency eigenvalues
obtained by demanding that the field vanishes at infinity and that the solution re-
mains regular at r = 0. In this work we only consider the ground state for which
ψℓ has no nodes in the open interval r ∈ (0,∞), hence fixing ω for each ℓ and u0.
Finally, solutions with different µ are related to each other by a simple rescaling
(see table 3.2 in appendix 3.6). Consequently, for each ℓ it is sufficient to study a
one-parameter family of solutions, usually parameterized by u0 or (equivalently)
by α0 :=α(r = 0).
Since boson stars do not have a well-defined boundary, one usually describes
their size by the R99 radius, defined as the (areal) radius of the sphere containing
2Note that ψℓ = Arℓ+O (rℓ+2) with constant A, such that ψℓ/rℓ is regular at r = 0. In practice u0
is evaluated either by taking the limit r → 0 or by directly evaluating u0 = A.
44 3.2 ℓ-boson stars
99% of the total mass MT . In addition, we use two different measures for the
star’s compactness:
C99 :=
MT
R99
, (3.7a)
and
Cm :=max
r>0
{
M(r)
r
}
=:
Mm
Rm
, (3.7b)
where we also defined Rm as the point r of maximum M(r)/r, and Mm as M(r =
Rm). To help better understand the meaning of these definitions we highlight
their differences in the top panel of Fig. 3.1, where some density profiles are
shown, together with vertical lines indicating the radii R99 and Rm for each star.
As one can appreciate from this figure, some solutions (see for instance the
purple and green lines) can be interpreted as having two parts: A very compact
“core”, located mostly to the left of r/Rm = 1, plus a less dense “halo” to the right
of that point. Note that the halo is much wider than the central region (a fact
that might be unnoticed at a first glance since the horizontal axis is in logarith-
mic scale). We clearly see that the definition C99 is a proper indicator of the whole
object’s compactness, while the definition Cm is more representative of the cen-
tral region’s compactness. However, as we will see later, the sets of definitions
[
R99, MT ,C99
]
and
[
Rm, Mm,Cm
]
tend to coincide for larger ℓ’s.3 Although we
have found the core-and-halo structure only for configurations lying on the unsta-
ble branches, the mentioned differences between these two sets are seen for stable
as well as unstable solutions.
The stress tensor of a perfect fluid is isotropic, and pressure is the same in all
directions of a fluid star. Even if common for some materials, isotropy is not a nat-
ural consequence of the underlying spacetime symmetries, and there exist static
and spherical configurations that exhibit fractional anisotropy, defined as the rel-
ative difference between the radial and tangential components of the pressure:
f a =
pr − pT
pr
. (3.8)
From the right-hand sides of equations (3.6b) and (3.6c) one can see that ℓ-boson
stars (including the standard ℓ = 0 boson stars) are anisotropic. Furthermore,
one might suspect that solutions with higher anisotropy will exist for the cases
with non-vanishing angular momentum number, due to the presence of the cen-
trifugal term ℓ(ℓ+1)/r2 in pr. We will corroborate this assertion in the next sec-
tions. This is not just a curious fact, since configurations with larger fractional
anisotropy have been identified to be stable up to higher values of the central den-
sity [106], hence leading to more compact objects [107]. This enhancement in the
3See also Fig. 3.7 for noticeable differences between the two sets of definitions.
Extreme ℓ-boson stars 45
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1 1 10
←
unstable
stable →
C99 Cm
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4 SLR ULR
δ = 0
0
0.2
0.4
0.6
0.8
1
0.1 1 10
SCO UCO SCO
δ = 1
µL = 1
4π
r2
ρ
0.108 0.360
0.096 0.350
0.141 0.306
0.118 0.149
0.038 0.047
0.010 0.013
V
eff
/(
µ
L
)2
V
eff
/(
µ
L
)2
r/Rm
Figure 3.1: Top panel: For various configurations with ℓ= 1, we show the rescaled
density profile with respect to the rescaled radial coordinate r → r/Rm. This
rescaling provides an easier way to compare the different curves between them-
selves, while also allowing to easily locate Rm, given by r/Rm = 1 (black vertical
line), as well as R99 (vertical lines with the same color as the corresponding so-
lution). Middle panel: Effective potentials for circular null geodesics (δ = 0) for
the same solutions as in the top panel, displaying cases without light rings, with
a pair of (stable and unstable) light rings, and the transition solution for which
the (degenerate) light rings first appear. Bottom panel: Effective potentials for
circular timelike geodesics (δ = 1) with µL = 1 for the same solutions as in the
other panels. The local extrema correspond to stable (minimum) and unstable
(maximum) circular orbits.
46 3.2 ℓ-boson stars
allowed compactness within the stable branch is an interesting property that is
also satisfied for ℓ-boson stars, as we discuss later.
It will also be helpful to identify some general properties of the motion of test
particles propagating in the spacetime associated with the ℓ-boson stars, and in
particular to determine whether the solutions admit innermost stable circular
orbits (ISCOs) and/or light rings [93] and, if so, to find their location. Given the
spacetime symmetries we can obtain the geodesics with the help of conserved
quantities using the expression [66]
(
dr
dλ
)2
=
E2
α2γ2
−
1
γ2
(
δ+
L2
r2
)
, (3.9)
where E and L are constants of motion (associated with the particle’s energy and
total angular momentum), and δ = 0 for null geodesics, while δ = 1 for timelike
geodesics. It is convenient to introduce the effective potential
Veff(r) :=α2
(
δ+
L2
r2
)
, (3.10)
leading to an equation of motion that resembles a point particle moving in a
one-dimensional potential.4 Then, orbiting particles are restricted to the regions
where Veff(r) < E2. Circular orbits can be obtained when E2 equals a local ex-
tremum of Veff, and those orbits are stable (unstable) if said extremum is a mini-
mum (maximum).
In the null case, the condition for circular orbits is
α− rα′ = 0, (3.11)
where the sign of the second derivative of the lapse function evaluated at the
light ring radius determines the stability of the orbit: it is stable if α′′ is negative
and unstable otherwise [108]. In the timelike case the energy and total angular
momentum per unit rest mass of a particle in circular motion at radius r must
satisfy
E =
√
α3
α− rα′ , L =
√
r3α′
α− rα′ . (3.12)
These orbits are stable wherever L(r) grows with r, whereas they are unstable
otherwise [108]. For regular configurations light rings can appear only in pairs,
4Defining x :=
∫r
0 α(r)γ(r)dr we can rewrite equation (3.9) as
(
dx
dτ
)2
= E2 −Ueff(x), where Ueff(x) :=
Veff[r(x)]. Hence, in analogy with Classical Mechanics we can infer that the orbits are restricted to the
regions where Ueff ≤ E2, with the equality being satisfied at the turning points. Circular orbits are
obtained where E2 equals an extremum of Ueff, and their stability depends on whether the extremum
is a maximum or a minimum. Given that the transformation x = x(r) is monotonic, the same conditions
are satisfied for Veff(r).
Extreme ℓ-boson stars 47
one of them being stable and the other unstable. Note, however, that not all stars
admit light rings. On the other hand, there always exist stable circular orbits of
massive particles. In particular, the existence of stable orbits is guaranteed both
at large distances and close enough to the center. However, regions of instability
may exist too, being delimited by innermost stable circular orbits (ISCOs) and
outermost stable circular orbits (OSCOs). In a similar way, the light ring pairs
delimit a region where α− rα′ is negative and circular orbits are not allowed at
all.5 We will now give more explicit details about these assertions.
The central panel of Fig. 3.1 illustrates distinct cases regarding the existence
of light rings, as determined by equation (3.10), all with ℓ= 1: (i) Potentials with-
out local extrema (besides at r = 0). These solutions cannot have light rings.
(ii) Potentials with a local minimum at some r = rin and with a local maximum
at some other r = rout, such that rin < rout. These solutions have a pair of light
rings, a stable one at rin and an unstable one at rout. (iii) The transition case,
in which the potential have an inflection point, giving rise to degenerate light
ring solutions with rin = rout. Note that cases (ii) and (iii) only occur for unstable
spacetimes [101,102]. In a similar way, in the bottom panel of this figure we illus-
trate different cases regarding the existence of unstable circular orbits of massive
particles with µL = 1.
Finally, we give an expression for the test particle’s speed moving on a circular
orbit (more precisely, the magnitude of its three-velocity as measure by a static
observer located at the corresponding radius):
v(r) := r
dφ
dt
=
√
rα′(r)
α(r)
, (3.13)
which will be used in the next section to show some rotation curves.
3.3 Extreme ℓ-boson stars
In this section we present and analyze our results. For all integer ℓ from 0 to
15, and for ℓ = 20, 25, 50, 75, 100, 200, 400 and 1600, we constructed solutions,
tens of them in some cases, that correspond to different values of the central pa-
rameter u0. The parameters and main properties of some of the most relevant
solutions that we have obtained are displayed in table 3.3 of appendix 3.6, which
also includes a reference to the figures in which they are used. In addition, in
the next section we obtain general expressions that are applicable for the limiting
case in which ℓ→∞.
We present some of our solutions in figure 3.2, where we show the rescaled
density profiles (defined as ̺= 4πr2ρ such that M =
∫
̺dr) associated with some
5We note that in all the solutions we have found α′(r) > 0 for r > 0, such that the lapse is
monotonously increasing.
48 3.3 Extreme ℓ-boson stars
0
0.5
1
1.5
2
2.5
3
3.5
0 50 100 150 200
unstable
0
0.1
0.2
0.3
0 5 10
4π
r2
ρ
µ r
ℓ C99 Cm
0 0.081 0.120
1 0.116 0.146
5 0.168 0.182
25 0.385 0.383
25 0.319 0.318
25 0.209 0.209
25 0.131 0.131
25 0.057 0.057
50 0.218 0.216
100 0.225 0.223
Figure 3.2: Rescaled density profiles, 4πr2ρ(r), of solutions with maximum MT
for various values of ℓ (solid lines), and of varying compactness for fixed ℓ = 25,
both in the stable and unstable region (red lines). The inner panel shows a zoom
into the small r region for a better reference of the cases with ℓ= 0 and 1.
of our configurations.6 Since one needs some criterion in order to compare solu-
tions through different values of ℓ, in this case we chose to display configurations
that, for each ℓ, have the maximum total mass, which are also the most compact
stable solutions. This is a criterion we will adopt in most of this work. In the same
figure we also show some solutions for given ℓ (= 25) and varying compactness,
the more compact ones being unstable. The solutions clearly exhibit a shell-like
morphology, at least for ℓ> 1. For bigger ℓ the stars are larger both in size and in
total mass. We will see that the compactness also increases with ℓ. In contrast, if
one considers stars with fixed ℓ and increasing size, the compactness decreases.
We also note that, as is the case for the traditional ℓ = 0 boson stars, the most
compact solutions belong to the unstable branch.
Figure 3.3 shows the dependence of the total mass on the frequency and on the
R99 radius for ℓ = 0, 1, 5, 25, 50 and 100. For each ℓ we indicate the maximum
of MT (squares), which we denote Mmax, and the first appearance of a light rings
pair (circles) and of an ISCO-OSCO pair (triangles). We have seen in previous
works [101, 102] that the state of maximum mass marks the transition from the
stable solutions (to the right in these figures) to the unstable ones (to the left) for
ℓ in the interval from 0 to 5. We also corroborated in the present work that this
fact is still true for larger values of ℓ.
In the following subsections we analyze various properties of these solutions,
including their compactness, anisotropy and causal circular orbits.
6Throughout this section we alternate between showing results in terms of ρ and ̺, depending on
what we find more illustrative.
Extreme ℓ-boson stars 49
0.1
1
10
0.7 0.75 0.8 0.85 0.9 0.95 1
Mmax
← UOs
LRs →
µ
M
T
ω/µ
ℓ 100
50
25
5
1
0
0.1
1
10
10 100
Mmax
← UOs
←
LR
sµ
M
T
µR99
ℓ 100
50
25
5
1
0
Figure 3.3: MT vs. ω and vs. R99 for ℓ = 0, 1, 5, 25, 50 and 100. Each point
on these curves corresponds to a different solution, including for instance those
shown in figure 3.2. The squares denote the maximum of the total mass, which
separates the stable and unstable regions. The circles denote the first appearance
of light rings, while the triangles denote the first appearance of an ISCO-OSCO
pair and, hence, the existence of unstable orbits (UOs).
3.3.1 Compactness
In this section we explore the compactness of our solutions using the definitions of
equations (3.7). As can be seen from figure 3.3, larger values of ℓ lead to solutions
with higher total mass MT . On the other hand, considering for instance the solu-
tions of maximum mass, the radius also increases with ℓ, as can be inferred from
that same figure and figure 3.2. However, the increase in mass tends to “win” over
the increase in radius in such a way that their ratio, the compactness, increases
with ℓ. Note that said solutions are the most compact stable ones for each ℓ.
After inspection of our solutions we note that the two mass definitions MT and
Mm from equations (3.7), as well as their associated radii R99 and Rm, seem to
both show a linear relation with ℓ, at least at large ℓ (ℓ& 10). This can be seen
in the first two panels of figure 3.4. Once again, in order to compare configura-
tion with different ℓ’s between each other, we have chosen those solutions with
maximum total mass Mmax for each ℓ.
The apparent linear dependence in ℓ suggests that simple expressions can be
obtained by performing linear fits. We show the results of said fits in the figure
(continuous lines), together with the fit coefficients (a to d) and their respective
errors. Keeping only two significant figures and omitting the errors we can write:
50 3.3 Extreme ℓ-boson stars
0
10
20
30
40
50
0 20 40 60 80 100
a =0.500741 ± 0.000924
b =0.818344 ± 0.025114
c =0.498183 ± 0.000747
d =0.553337 ± 0.020308
µ
M
ℓ
µMT
aℓ+ b
µMm
cℓ+ d
0
50
100
150
200
0 20 40 60 80 100
a =2.197661 ± 0.004953
b =8.689029 ± 0.134570
c =2.214881 ± 0.004211
d =6.710842 ± 0.160246
0 5 10
µ
R
ℓ
µR99
aℓ+ b
µRm
cℓ+ d (ℓ ≥ 10)
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0 20 40 60 80 100
C
ℓ
C99
C99 fit
Cm
Cm fit (ℓ ≥ 10)
Figure 3.4: We show the dependence on ℓ of the stars’ mass, radius and compact-
ness. All quantities shown here correspond to the solution of maximum MT for
each value of ℓ. Left panel: MT and Mm, together with the corresponding linear
fits. Center panel: R99 and Rm, together with their linear fit. In the case of Rm
we do the linear fit only to the points with ℓ ≥ 10, which is the region where we
actually see a linear dependence. Right panel: Compactness Cm and C99, and,
in each case, the compactness calculated from the fits of the previous panels. We
also indicate the asymptotic value as a dotted line (see section 3.4).
µMT ≈ 0.50ℓ+0.82, (3.14a)
µMm ≈ 0.50ℓ+0.55, (3.14b)
µR99 ≈ 2.2ℓ+8.7, (3.14c)
µRm ≈ 2.2ℓ+6.7. (3.14d)
From here, expressions for our two definitions of compactness can be found by
taking the quotient of each M vs. R pair. Said quotients, i.e. C99 and Cm, are
shown in the last panel of figure 3.4. The point values shown in that panel are
obtained by taking individually the quotient of the corresponding data pairs that
appear in the first panels, while the continuous line represent the quotient of the
linear fit’s expressions.
The almost linear relations shown in the first two panels of figure 3.4 suggest
that solutions might have simple rescaling properties with ℓ, at least at large
enough ℓ. A more detailed analysis of such scaling properties will be given in
section 3.4, where we will see that an asymptotic value can be obtained for the
compactness at large ℓ. That value is indicated in the right panel of figure 3.4
as a dotted line. Note that initially the compactness increases rapidly with ℓ,
and continues to rise monotonically, remaining close to and below the asymptotic
value derived in section 3.4.
Extreme ℓ-boson stars 51
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0 1 2 3 4 5 6 7 8
ℓ = 0
µ r
pr/µ
2
pT/µ
2
−0.0001
−5× 10−5
0
5× 10−5
0.0001
0.00015
0.0002
0 2 4 6 8 10 12 14
ℓ = 1
µ r
pr/µ
2
pT/µ
2
0
5× 10−6
1× 10−5
1.5× 10−5
2× 10−5
2.5× 10−5
3× 10−5
ℓ = 5
−3× 10−7
0
3× 10−7
0 5 10 15 20 25
pr/µ
2
pT/µ
2
µ r
0
2× 10−6
4× 10−6
6× 10−6
8× 10−6
1× 10−5
1.2× 10−5
ℓ = 25
−1× 10−9
0
1× 10−9
0 5 10 15 20 25 30 35 40 45
pr/µ
2
pT/µ
2
µ r
Figure 3.5: Radial and tangential pressures vs. radial coordinate for the solutions
of maximum MT in the cases of ℓ= 0, 1, 5 and 25. We see that the tangential pres-
sure becomes larger and larger relative to the radial pressure when ℓ increases.
3.3.2 Anisotropy
We move now to the description of the stars’ anisotropy. In figure 3.5 we show the
pressure profiles for ℓ = 0, 1, 5 and 25, in all cases for the solution of maximum
mass Mmax. Notice how different the profiles are for ℓ= 0, ℓ= 1, and ℓ> 1. The
typical ℓ = 0 “solid-sphere” star has pr > pT , while for larger ℓ’s the “shell-like”
stars have mostly pr < pT , with this difference becoming more pronounced the
higher the value of ℓ. This behavior seems intuitively natural given the stars’
morphology. As ℓ increases, the "shells" become larger, as well as thinner relative
to their radius, in such a way that the tangential pressure has to become larger
relative to the radial one in order to support the configuration.
In figure 3.6 we show parametric plots of [pr,pT ] vs. r, in which the larger
the deviation from the identity pr = pT (shown as a dotted line of unit slope), the
larger the anisotropy. Additionally, we indicate the density as a color map, as well
as the compactness in each case. The differences at the starting points of these
curves, which correspond to the pressure values at the origin r = 0, are consistent
with the stars’ shape as seen in our previous work: while they are “empty” at
the center when ℓ > 1, they have maximum density there when ℓ = 0, as is a
well known property of standard boson stars. In the intermediate case, ℓ = 1,
the density is greater than zero at the center, but it does not reach its maximum
value at that point. It is also clear from these plots that the anisotropy, as well as
52 3.3 Extreme ℓ-boson stars
the compactness, grow with ℓ, the tangential pressure becoming larger and larger
compared to the radial pressure. In section 3.4 we will see that the limiting case
ℓ→∞ would display a vertical line in this type of plot. On the other hand, we see
little differences in anisotropy when transitioning between stable and unstable
solutions for any given value of ℓ.
3.3.3 Geodesic motion
Given the large compactness that ℓ-boson stars may achieve, one may wonder
whether they admit light rings and/or ISCOs/OSCOs. In fact, it is known that
even traditional ℓ = 0 boson stars can have light rings and ISCOs/OSCOs, al-
though this is true only in the case of solutions located very deep into the unsta-
ble region7. In the remainder of this section we will analyze the appearance of
light rings and ISCOs/OSCOs, paying particular attention to their relation with
compactness and stability of the underlying spacetime solutions.
In figure 3.3 we indicated with a circle the point corresponding to the first, or
less compact, solutions containing a pair of light rings. In all the cases we studied,
such solutions are always in the unstable region, although they get closer to the
stable region as ℓ increases. It is unclear, however, whether light rings may be
found in the stable region for large enough ℓ, although one would expect that this
is not the case given that the maximum compactness that a stable ℓ-boson star is
able to achieve, C ≈ 0.235, is far from the expected one for the appearance of light
rings, C = 1/3. The results presented in [109] seem to indicate that light rings
can only exist for unstable solutions. Another matter of astrophysical interest is
whether such unstable solutions have a relatively short or a rather long life-time.
However, this question goes beyond the scope of the present chapter, so we leave
it for future work.
Regarding the existence of ISCOs, we also indicated in figure 3.3 the first ap-
pearance of an ISCO-OSCO pair (triangles). We see that for large enough ℓ these
pairs can also exist in the case of stable spacetime solutions. In fact, we have
found that the smallest ℓ for which stable ℓ-boson stars with ISCO-OSCO pairs
exist is ℓ= 9.
We now go into more detail and analyze the different stability regions in fig-
ure 3.7, where we show plots of radius vs. compactness for ℓ= 0, 1, 5 and 25. Each
vertical line in these plots corresponds to a solution, and we can see the transi-
tions through different stability regions as r varies along said line. The green
regions are those where the timelike circular orbits are stable (SCOs). The red
region is where the circular orbits are unstable (UCOs), and it is delimited by an
ISCO at the top and by an OSCO at the bottom. Similarly, the dark gray region
is that for which no circular orbitss exist, and it is delimited by a pair of light
7Note that the situation may change when non-canonical kinetic terms are considered [108].
Extreme ℓ-boson stars 53
0
2
4
6
8
10
12
0 2 4 6 8 10 12
ℓ
=
0
←
r
C99 = 5.46 × 10−3
Cm = 7.65 × 10−3
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16
←
r
C99 = 8.06 × 10−2
Cm = 12.0 × 10−2
0
2
4
6
8
10
12
14
16
0 2 4 6 8 10 12 14 16
←
r
C99 = 7.42 × 10−2
Cm = 25.8 × 10−2
p
T
/µ
2
[×
10
−
8
]
pr/µ
2 [×10−8]
0
2
4
6
8
10
12
14
ρ
/µ
2
[×
10
−
6
]
STABLE
p
T
/µ
2
[×
10
−
4
]
pr/µ
2 [×10−4]
0
1
2
3
4
5
6
7
8
ρ
/µ
2
[×
10
−
3
]
MARGINALLY STABLE
p
T
/µ
2
[×
10
3
]
pr/µ
2 [×103]
0
2
4
6
8
10
12
14
16
ρ
/µ
2
[×
10
3
]
UNSTABLE
−3
−2
−1
0
1
2
3
4
5
−3 −2 −1 0 1 2 3 4 5
ℓ
=
1
r
→ ←
r
C99 = 1.02 × 10−2
Cm = 1.26 × 10−2
−15
−10
−5
0
5
10
15
20
−15 −10 −5 0 5 10 15 20
r
→ ←
r
C99 = 1.16 × 10−1
Cm = 1.46 × 10−1
−6
−4
−2
0
2
4
6
8
10
−6 −4 −2 0 2 4 6 8 10
r
→ ←
r
C99 = 9.32× 10−2
Cm = 36.9× 10−2
p
T
/µ
2
[×
10
−
8
]
pr/µ
2 [×10−8]
0
1
2
3
4
5
6
ρ
/µ
2
[×
10
−
6
]
p
T
/µ
2
[×
10
−
5
]
pr/µ
2 [×10−5]
0
2
4
6
8
10
12
14
16
ρ
/µ
2
[×
10
−
4
]
p
T
/µ
2
[×
10
0
]
pr/µ
2 [×100]
0
2
4
6
8
10
12
14
16
18
20
ρ
/µ
2
[×
10
0
]
0
5
10
15
20
25
30
0 5 10 15 20 25 30
ℓ
=
5 r
→ ←
r
C99 = 2.32 × 10−2
Cm = 2.50 × 10−2
0
5
10
15
20
25
30
0 5 10 15 20 25 30
r
→ ←
r
C99 = 1.69× 10−1
Cm = 1.82× 10−1
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7
r
→ ←
r
C99 = 3.25× 10−1
Cm = 3.65× 10−1
p
T
/µ
2
[×
10
−
9
]
pr [×10−9]
0
5
10
15
20
25
30
ρ
/µ
2
[×
10
−
7
]
p
T
/µ
2
[×
10
−
6
]
pr/µ
2 [×10−6]
0
5
10
15
20
25
30
ρ
/µ
2
[×
10
−
5
]
p
T
/µ
2
[×
10
−
3
]
pr/µ
2 [×10−3]
0
2
4
6
8
10
12
14
ρ
/µ
2
[×
10
−
3
]
0
10
20
30
40
50
0 10 20 30 40 50
ℓ
=
25
r
→ ←
r
C99 = 1.31 × 10−1
Cm = 1.31 × 10−1
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35 40
r
→ ←
r
C99 = 2.09× 10−1
Cm = 2.09× 10−1
0
5
10
15
20
25
30
0 5 10 15 20 25 30
r
→ ←
r
C99 = 3.85× 10−1
Cm = 3.83× 10−1
p
T
/µ
2
[×
10
−
8
]
pr [×10−8]
0
2
4
6
8
10
12
ρ
/µ
2
[×
10
−
6
]
p
T
/µ
2
[×
10
−
7
]
pr/µ
2 [×10−7]
0
5
10
15
20
25
30
35
40
ρ
/µ
2
[×
10
−
6
]
p
T
/µ
2
[×
10
−
5
]
pr/µ
2 [×10−5]
0
1
2
3
4
5
6
7
8
ρ
/µ
2
[×
10
−
4
]
Figure 3.6: Parametric plots of [pr , pT ](r). Each row corresponds to a value of ℓ,
while each column corresponds to a stability type. These figures are particularly
well suited for analyzing the pressure anisotropies, measured by the deviation
from the identity pr = pT . The value of the density ρ along the curves is indicated
with a color map. The direction of growing r is also indicated. To help guide the
eye, the curves pr =±pT are shown as dotted lines, and the same scale was used
in each pair of axes.
54 3.4 Scaling properties for large ℓ
rings, indicated with a red line. Since ℓ-boson stars are shell like for ℓ> 1, with
density much smaller than its maximum value and falling quickly towards the
center in the interior region –where the spacetime is very close to Minkowski–
the circular orbits are almost non-existent there, having speed v ≪ 1. To make
this more apparent we shaded in a darker green the regions in which v < 10−5,
noting that the rotation curves of these configurations have a maximum in the
interval 0.4. v < 1 (see figure 3.8).
Figure 3.7 also indicates the stars’ radii, R99 and Rm, and also, as a guide, the
limit of spacetime stability (vertical dotted line) and the locations of the Schwarzschild
ISCO and light ring, given by r = 6M and r = 3M, respectively, where for the value
of M we used both MT and Mm. We see that, as the compactness increases, the
light rings first appear at or very close to Rm, and soon they move to each side
of that location. For small ℓ the definition Rm seems more meaningful than R99
when comparing to the location of light rings. On the other hand, both definitions
tend to coincide at large ℓ.
Although all the solutions with light rings found in this work are unstable, we
see that, for larger values of ℓ, solutions with light rings exist closer and closer to
the stable region. However, as mentioned earlier, it is unlikely that stable solu-
tions with light rings exist, even for extremely large ℓ.
Interestingly, the unstable circular orbit regions for large ℓ tend to be delim-
ited almost exactly by the star’s radius (from below) and the Schwarzschild ISCO
(from above). We can see again in the last panel of figure 3.7 (ℓ = 25) that re-
gions of instability can exist even for stable ℓ-boson star spacetimes. As already
mentioned, this happens for solutions starting at ℓ = 9. This could constitute
an observable feature that might help distinguish some ℓ-boson stars from other
dark compact objects.
In figure 3.8 we show the rotation curves for the solutions shown in figure 3.2,
that is: solutions of maximum MT for ℓ= 0, 1, 5, 25, 50 and 100; and for ℓ= 25,
also some solutions with varying compactness, both in the stable and unstable
spacetime branch. The curves have been extended beyond the domain of numer-
ical integration using the Schwarzschild expressions with mass MT . We can see
that this gives and excellent match. The points where the curves reach v = 1 cor-
respond to light rings, and no circular orbits exist in the region in between those
points (red line and dark gray region of figure 3.7). We also indicate the regions
where the circular orbits are unstable (thick gray line).
3.4 Scaling properties for large ℓ
In this section we discuss the scaling properties of the fields in the asymptotic
limit ℓ → ∞. This is achieved by rescaling the fields (M,α,γ,ψ) and by shift-
Extreme ℓ-boson stars 55
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.225 0.23 0.235 0.24 0.245 0.25 0.255 0.26
µ
r
Cm
LRs
Rm
6Mm
3Mm
R99
6MT
3MT
← unstable ℓ-BSs →
SCOs
UCOs
no COs
ℓ = 0
0
1
2
3
4
5
6
0.15 0.2 0.25 0.3 0.35
µ
r
Cm
LRs
Rm
6Mm
3Mm
R99
6MT
3MT
↑
s
t
a
b
le
ℓ-
B
S
s
unstable ℓ-BSs →
SCOs
U
C
O
s
no COs
ℓ = 1
0
5
10
15
20
0.1 0.15 0.2 0.25 0.3 0.35
µ
r
Cm
LRs
Rm
6Mm
3Mm
R99
6MT
3MT
← stable ℓ-BSs unstable ℓ-BSs →
SCOs
U
C
O
s
v < 10−5 SCOs
no COs
ℓ = 5
0
10
20
30
40
50
60
70
80
90
0.1 0.15 0.2 0.25 0.3 0.35
µ
r
Cm
LRs
Rm
6Mm
3Mm
R99
6MT
3MT
← stable ℓ-BSs unstable ℓ-BSs →
SC
O
s
U
C
O
s
v < 10−5
SCOs
no COs
ℓ = 25
Figure 3.7: For solutions of high compactness we indicate the regions of existence
and stability of causal circular orbits (COs). In these plots, each vertical line of
constant Cm corresponds to a different solution. The green regions indicate the
radii with stable (timelike) circular orbits (SCOs), while the red region indicates
those with unstable orbits (UCOs). On the other hand, no COs exist in the dark
gray region, which is limited by a pair of light rings (LRs), red line. Finally, the
dark green region indicates the “almost empty, almost flat” central region of the
ℓ> 1 “shells”, where the circular orbits have speed v < 10−5. We also include as a
guide R99, Rm, and the corresponding locations of a Schwarzschild LR and ISCO.
56 3.4 Scaling properties for large ℓ
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
v
:=
r
d
φ
/d
t
µr
ℓ C99 Cm
0 0.081 0.120
1 0.116 0.146
5 0.168 0.182
25 0.385 0.383
25 0.319 0.318
25 0.209 0.209
25 0.131 0.131
25 0.057 0.057
50 0.218 0.216
100 0.225 0.223
Figure 3.8: Rotation curves as defined by equation (3.13). The solid lines corre-
spond to solutions with maximum MT for ℓ ranging from 0 to 100. For ℓ= 25 we
also show, in red, some cases with varying compactness around the maximum MT
solution. Beyond the numerical integration region, we extended the curves using
the Schwarzschild spacetime with mass MT . Those parts of the curves are shown
in a lighter color. Thick gray lines indicate regions where the circular orbits are
unstable. The solutions represented here are the same as those in figure 3.2.
ing and rescaling the radial coordinate r in an appropriate way (which is largely
motivated by the empirical numerical data and trial-and-error) such that, when
taking the limit ℓ→∞, one obtains a set of effective field equations which can be
solved separately. As we show, combining the solution of these effective equations
with the aforementioned rescaling, one obtains the correct asymptotic behavior
for the fields and related quantities for large values of ℓ. For clarity, we include a
summary of these results in the final paragraph of this section.
To describe our scaling method, we consider a family of configurations with
increasing value of ℓ and fixed ω. As ℓ becomes large, the numerical data (see
figure 3.9) suggests that the fields’ profiles depend only on the variable
y :=
r−ℓx0
ℓa
, (3.15)
with x0 a positive constant that depends on ω but not ℓ, and a a parameter within
the range 0 < a < 1 that will be determined later. This means that the profiles
have their center shifted outwards by ℓx0 and stretched by the factor ℓa as ℓ→∞.
The fields’ amplitudes are rescaled as follows:
M∗(y) :=
M(r)
ℓ
, α∗(y) :=α(r) γ∗(y) := γ(r) ψ∗(y) := ℓ1+ a
2 ψℓ(r), (3.16)
the data suggesting that the quantities with a star have finite limits when ℓ→∞.
Note that equations (3.15,3.16) and the definition of γ in equation (3.4) imply that
Extreme ℓ-boson stars 57
−10 −5 0 5 10
µ (r − ℓx0)/ 3
√
ℓ
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
ℓ7
/
6
ψ
ℓ
−10 −5 0 5 10
µ (r − ℓx0)/ 3
√
ℓ
0.0
0.1
0.2
0.3
0.4
0.5
µ
M
/ℓ
ℓ = 25
ℓ = 50
ℓ = 100
ℓ = 200
ℓ = 400
−10 −5 0 5 10
µ (r − ℓx0)/ 3
√
ℓ
1.00
1.05
1.10
1.15
1.20
γ
−10 −5 0 5 10
µ (r − ℓx0)/ 3
√
ℓ
0.80
0.81
0.82
0.83
0.84
0.85
0.86
0.87
α
Figure 3.9: Scaling of the solutions with ω= 0.8612, for the quantities ψℓ, M(r)/ℓ,
γ and α. In these plots, the constant x0 appearing on the horizontal axis has been
estimated using as reference the ℓ = 1600 configuration with the same ω, using
the formula x0 = r1600/1600, with r1600 defined as the location of the maximum
of ψ1600(r). The procedure to determine x0 without resorting to any particular
finite ℓ solution is explained in the text.
58 3.4 Scaling properties for large ℓ
γ−2
∗ (y)= 1−
2M∗(y)
x0
1
1+ℓa−1 y
x0
, (3.17)
such that in the limit ℓ→∞ (with fixed y) it follows that γ−2
∗ (y) = 1−2M∗(y)/x0.
In terms of the rescaled quantities M∗, α∗ and ψ∗ equations (3.5) can be written
as
dM∗
dy
= x2
0
(
1+ℓa−1 y
x0
)2
ρ∗, (3.18a)
1
γ2
∗α∗
dα∗
dy
= x0
(
1+ℓa−1 y
x0
)
pr∗+
M∗
x2
0
ℓa−1
(
1+ℓa−1 y
x0
)2
, (3.18b)
1
α∗γ∗
d
dy
(
α∗
γ∗
dψ∗
dy
)
(3.18c)
+
2
x0
ℓa−1
1+ℓa−1 y
x0
1
γ2
∗
dψ∗
dy
=−ℓ2a




ω2
α2
∗
−µ2 −
1
x2
0
1+ 1
ℓ
(
1+ℓa−1 y
x0
)2




ψ∗,
where we have introduced the rescaled energy density and radial pressure
ρ∗(y) := 4πℓ1+aρ(r) (3.19a)
=
(
1+
1
2ℓ
)





ℓ−2a 1
γ2
∗
(
dψ∗
dy
)2
+




ω2
α2
∗
+µ2 +
1
x2
0
1+ 1
ℓ
(
1+ℓa−1 y
x0
)2




ψ2
∗





,
pr∗(y) := 4πℓ1+a pr(r) (3.19b)
=
(
1+
1
2ℓ
)





ℓ−2a 1
γ2
∗
(
dψ∗
dy
)2
+




ω2
α2
∗
−µ2 −
1
x2
0
1+ 1
ℓ
(
1+ℓa−1 y
x0
)2




ψ2
∗





.
Let us consider the limiting case a = 0 first and take the limit ℓ→∞ in these
equations (with y held fixed). In this case, one obtains the effective equations
dM∞
dy
= x2
0ρ∞, ρ∞ =
1
γ2
∞
(
dψ∞
dy
)2
+
(
ω2
α2
∞
+µ2
0
)
ψ2
∞, (3.20a)
1
γ2
∞α∞
dα∞
dy
= x0 pr∞, pr∞ =
1
γ2
∞
(
dψ∞
dy
)2
+
(
ω2
α2
∞
−µ2
0
)
ψ2
∞,(3.20b)
1
α∞γ∞
d
dy
(
α∞
γ∞
dψ∞
dy
)
=−
(
ω2
α2
∞
−µ2
0
)
ψ∞, (3.20c)
where the index∞ refers to the (pointwise) limit for ℓ→∞, i.e. M∞(y)= limℓ→∞ M∗(y)
and similarly for α∞, γ∞ and ψ∞. We have also introduced the shorthand nota-
tion µ0 :=
√
µ2 +1/x2
0
in order to abbreviate the notation. Equations (3.20) look
Extreme ℓ-boson stars 59
like a nice system of differential equations for (M∞,α∞,ψ∞) which could be inte-
grated numerically and whose solution with the appropriate boundary conditions
should approximate the solution of the full system when ℓ is large and |y| . ℓ.
However, it is not difficult to show that these equations imply that
α∞γ∞pr∞ = const, (3.21)
and by virtue of the boundary conditions this constant must be zero. Therefore,
pr∞ = 0 which implies that α∞ is constant and ω2/α2
∞−µ2
0
< 0. Then, multiplying
both sides of equation (3.20c) with ψ∞, integrating over y and using integration
by parts reveals that ψ∞ = 0 is the only solution which decays to zero as y→±∞.
This indicates that the choice a = 0 in the rescaling (3.15) is not the correct one.
Therefore, let us assume that 0 < a < 1 is strictly positive and take again the
pointwise limit ℓ→∞ in equations (3.18). This yields
dM∞
dy
= x2
0ρ∞, ρ∞ =
(
ω2
α2
∞
+µ2
0
)
ψ2
∞, (3.22a)
1
γ2
∞α∞
dα∞
dy
= x0 pr∞, pr∞ =
(
ω2
α2
∞
−µ2
0
)
ψ2
∞, (3.22b)
while the rescaled Klein-Gordon equation (3.18c) implies that pr∞ must vanish
in order for the right-hand side to be finite. It follows that
α∞ =
ω
µ0
(3.23)
is constant and that
dM∞
dy
= 2x2
0µ
2
0ψ
2
∞ = 2(1+µ2x2
0)ψ2
∞. (3.24)
The problem is that (so far) we have no differential equation for ψ∞. However,
a differential equation for ψ∞ can be obtained by expanding the rescaled fields:
ψ∗(y)=ψ∞(y)+εψ1(y)+O (ε2), (3.25)
and similarly for M∗ and α∗ in powers of ε = ε(ℓ) and looking at the next-order
contributions from equations (3.18). For the following, we choose ε(ℓ)= ℓa−1 since
most of these corrections terms are of this order, and we expand the rescaled lapse
in the form
α∗(y)=α∞
[
1+ℓa−1δ(y)+O (ℓ−1)
]
, (3.26)
with the function δ(y) describing the first-order correction. Using equation (3.23)
the right-hand side of equation (3.18c) gives, to leading order in 1/ℓ,
2ℓ3a−1

µ2
0δ−
y
x3
0

ψ∞, (3.27)
60 3.4 Scaling properties for large ℓ
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
ℓ7
/
6
ψ
ℓ
−8 −6 −4 −2 0 2 4 6 8
µ y
−0.004
−0.002
0.000
0.002
ℓ7
/
6
ψ
ℓ
−
ψ
∞
0.0
0.1
0.2
0.3
0.4
0.5
µ
M
/ℓ
ℓ = 25
ℓ = 200
ℓ = 1600
ℓ→∞
−8 −6 −4 −2 0 2 4 6 8
µ y
−0.02
0.00
0.02
0.04
µ
(M
/ℓ
−
M
∞
)
Figure 3.10: Solutions with ℓ≫ 1 compared to the same ω = 0.8612 solution ob-
tained from the effective (ℓ→∞ limit) equations (3.24,3.28). The asymptotic solu-
tion ℓ→∞ yields the value x0 = 2.73. In the bottom panels we show the difference
between the finite ℓ configurations and the ℓ→∞ case, which converges to zero.
which yields a finite contribution if a = 1/3. Choosing a = 1/3 in the expan-
sion (3.26), equations (3.18b,3.18c) yield the two differential equations
1
γ2
∞
dδ
dy
= x0



1
γ2
∞
(
dψ∞
dy
)2
−2

µ2
0δ−
y
x3
0

ψ2
∞



+
M∞
x2
0
,(3.28a)
1
γ∞
d
dy
(
1
γ∞
dψ∞
dy
)
= 2

µ2
0δ−
y
x3
0

ψ∞, (3.28b)
which can be integrated along with equation (3.24) and γ−2
∞ = 1−2M∞/x0 in order
to find (M∞,δ,ψ∞). Note that the expression inside the square parenthesis on
the right-hand side of equation (3.28a) is the ℓ−2/3-contribution to pr∗.
The rescaled equations (3.24,3.28) are solved on a finite interval [yL, yR] with
yL < 0< yR , fixing the left boundary conditions M∗ = 0, δ= 0, ψ∗ =ψ∗L at y= yL
and the right boundary condition ψ∗ ∼ 0 at y = yR . The integration is carried
out by means of a shooting method from left to right, where ψ∗ is fixed at yL
and the value of x0 for which the field matches the boundary condition at yR is
searched for. In this procedure, it is necessary to provide the value of dψ∗/d y at
yL given ψ∗L; this can be done by studying the asymptotic behavior y →−∞ of
the rescaled equations. It is obtained that the scalar field takes the form ψ∗ ∝
Ai(z) ≈ exp(− 2
3
z3/2)/z1/4 with z = − 3
p
2y/x0 and Ai the Airy function of the first
kind, obtaining dψ∗/dy≈ (
√
−2y/x3
0
−1/(4y))ψ∗.
After x0 is found, the total mass of the solution is obtained by evaluating
Extreme ℓ-boson stars 61
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
µ
M
T
/ℓ
ℓ = 2
ℓ = 50
ℓ = 100
ℓ→∞, (M∞T )
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
µ x0
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
C
ℓ = 2 (Cm)
ℓ = 50 (Cm)
ℓ = 100 (Cm)
ℓ→∞ (Cx0
)
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.4
0.5
0.6
0.7
0.8
0.9
1.0
α
0
0.80 0.81 0.82
0.70
0.72
0.74
ℓ = 2
ℓ = 50
ℓ = 100
ℓ→∞, (α∞)
Figure 3.11: Equilibrium ℓ ≫ 1 configurations. Left panel: Total mass vs. the
frequency ω/µ. Center panel: Compactness Cx0 for the ℓ→∞ limit. Right panel:
Minimum of the lapse α0 = α(r = 0) vs. ω/µ. The green square represents the
maximum (ℓ→∞) configuration and the red circle denotes the first appearance
of light rings, at Cx0 = 1/3.
M∞T := M∞(y = yR). Outside the spherical shell, at y = yR , we evaluate α∞ =
1/γ∞(yR) and calculate ω from equation (3.23). Once the pair (ω, x0) is obtained
from the effective equations, we can compare the fields with those corresponding
to the same ω finite ℓ solutions. Notice that there is no loss in generality in choos-
ing δ(yL) = 0 since the system (3.24,3.28) is invariant under the transformation
(y,δ) 7→ (y−ζ,δ−ζ/(µ2
0
x3
0
)) with M∞ and ψ∞ unchanged. In fact, as the lower right
panel of figure 3.9 shows, the δ correction to α is not zero in the inner shell re-
gion. In turn, the previous transformation will translate horizontally the scalar
field profile. So there are two ways to calculate ζ, the first is to take a solution
with large ℓ and find the value of the δ correction within the shell, the second is
to use the scalar field profile and make the maxima of ψ∗ of the large ℓ solution
and the effective ℓ→∞ solution to overlap; these two forms are equivalent.
We illustrate the fields’ rescaling in figure 3.10, showing convergence to the
limiting ℓ→∞ case, as expected. We have found that the value of x0 that corre-
sponds to a solution with the frequency of the previous (stable branch) figure 3.9
configurations, ω= 0.8612, is x0 = 2.73. The estimated value for the α correction
in this case is ζ= 2.17. It is found that for this ζ value, the maxima of ψ∗ overlap,
as expected. Figure 3.11 shows a plot for certain global quantities of the equilib-
rium solutions of the rescaled ℓ→∞ limit. The left panel shows that a critical
mass solution, M∞T = 0.49031, is obtained at ω = 0.8077 (marked with a green
square). This solution is obtained for the values x0 = 2.0902 and ζ= 1.58.
Next, we evaluate the anisotropy and compactness of this particular configu-
ration. In contrast to the rescaled radial pressure (3.19b), the rescaled tangential
pressure
pT∗(y) := 4πℓ1+a pT =
(
1+
1
2ℓ
)

−ℓ−2a 1
γ2
∗
(
dψ∗
dy
)2
+
(
ω2
α2
∗
−µ2
)
ψ2
∗

 , (3.29)
62 3.4 Scaling properties for large ℓ
−6 −4 −2 0 2 4 6
µ y
0.000
0.005
0.010
0.015
0.020
0.025
0.030
ρ∞
pT∞
Figure 3.12: Rescaled energy density and tangential pressure for the ℓ→∞ max-
imum mass solution. Notice that pT is always positive, while pr is strictly zero
in this limit.
does not vanish in the pointwise ℓ→∞ limit:
pT∞ =
(
ω2
α2
∞
−µ2
)
ψ2
∞ =
ψ2
∞
x2
0
, (3.30)
which is consistent with the observations made in section 3.3.2.
We show in figure 3.12 a plot for the tangential pressure as well as the rescaled
energy density, equation (3.22a), for the solution of maximum mass. Now, to de-
termine the compactness of these solutions, the easiest way is to note that the
quotient M(r)/r in terms of the rescaled quantities in equations (3.15,3.16), re-
duces to M∞(y)/x0 in the ℓ→∞ limit, allowing us to define,
Cx0
:=
M∞T
x0
. (3.31)
In the central panel of figure 3.11 we show the x0 value of the solution as
a function of the compactness Cx0
(and Cm for the finite ℓ solutions). For the
maximum mass solution, the compactness obtained is Cx0
= 0.234554. Like the
finite ℓ solutions, the compactness increases as the value of the boson star radius
decreases, approaching the limit value of 0.5. However, the ℓ=∞ solutions with
compactness exceeding ≃ 0.433 have frequencies ω larger than µ, and thus they
do not correspond to a limit of solutions with finite ℓ which must have ω/µ< 1 due
to the exponential decay of the scalar field at spatial infinity.
Next, we wonder about the presence of light rings for the ℓ≫ 1 configurations.
As stated above, the existence of these rings is given by the existence of local
extrema of Veff. For large ℓ the effective potential for null geodesics is
Veff(r)= L2 α
2
r2
=
L2
ℓ2
α2
∞
x2
0
[
1+
2
ℓ2/3
(
δ(y)−
y
x0
)
+O
(
1
ℓ
)
]
. (3.32)
Extreme ℓ-boson stars 63
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
y
−1.4
−1.2
−1.0
−0.8
−0.6
−0.4
V
1
Cx0
> 1/3 : ω = 0.71
ℓ = 50
ℓ = 100
ℓ→∞
rin, ℓ→∞
Figure 3.13: Effective rescaled potential V1 for circular null geodesics for ℓ =
50,100 solutions together with the ℓ→∞ case. The circles indicate the position
of the light rings.
For the following it is convenient to introduce the rescaled potential V1, defined
as
V1(y) := ℓ2/3
(
x2
0
α2
∞
ℓ2
L2
Veff(r)−1
)
= 2
(
δ(y)−
y
x0
)
+O
(
1
ℓ1/3
)
. (3.33)
Figure 3.13 shows the function V1(y) for ℓ= 50 and ℓ= 100 along with the ℓ→∞
limit. Starting from the knowledge of δ(y) we can obtain the approximate location
of rin, the inner light ring for ℓ≫ 1 solutions. However, as shown in figure 3.13 at
this order one is unable to determine the position of the outer ring which moves
away from y = 0 as ℓ increases; in fact, it seems that the location of this second
light ring diverges to y →∞ when ℓ→∞. Evaluating the condition dV1/d y = 0
along the ℓ → ∞ family of configurations we obtain that the solution closest to
the critical mass point satisfying this condition for some value of y, is the one
that has compactness Cx0
= 1/3 (red dot in figure 3.11). This correspond to the
solution with x0 = 1.32 and ω= 0.7251.
To close the discussion of this section on the rescaling properties for large ℓ,
we list the transformations involving certain relevant quantities mentioned in the
previous paragraphs. To do this, suppose the situation in which an ℓ-boson star
solution has been obtained for a certain value of ω and sufficiently large ℓ = ℓ1;
then starting from it we can obtain an approximate solution with arbitrarily large
ℓ= ℓ2 > ℓ1 for the same ω in the following way: first, identify the position r of the
maximum of ψℓ and estimate8 x0 = r/ℓ1. Then, apply the transformation r 7→
3
√
ℓ2/ℓ1(r−ℓ1x0)+ℓ2x0. As function of this redefined coordinate r, the amplitude
of the scalar field becomes smaller according to ψℓ1
(r) 7→ψℓ2
(r)= (ℓ1/ℓ2)7/6ψℓ1
(r)
while the mass function grows as M(r) 7→ (ℓ2/ℓ1)M(r). The energy density and the
8The error induced by this estimation as well as the following ones presented in this paragraph is
of the order ℓ−2/3
1
.
64 3.5 Conclusions
tangential pressure both decrease according to ρ(r) 7→ (ℓ1/ℓ2)4/3ρ(r) and pT (r) 7→
(ℓ1/ℓ2)4/3 pT (r). On the other hand, the mass, radius and compactness parameters
rescale as follows: (MT ,R99,C99) 7→ (ℓ2/ℓ1MT ,ℓ2/ℓ1R99,C99) and (Mm,Rm,Cm) 7→
(ℓ2/ℓ1Mm,ℓ2/ℓ1Rm,Cm). Let this example above serve as an illustration of the
rescaling properties; certainly a better way to obtain a solution with ℓ = ℓ2 is to
solve the effective equations and then obtain the quantities with finite ℓ by in-
verting the definitions of the rescaled variables. In this case the error would be
of order ℓ−2/3
2
or even smaller for some of the quantities.
3.5 Conclusions
We have studied various properties of the recently introduced ℓ-boson stars [99],
analyzing in each case the role played by the angular momentum parameter ℓ,
and paying particular attention to the large ℓ regime. These objects are com-
posed of 2ℓ+1 massive complex scalar fields and present notable characteristics
which single them out from the standard ℓ = 0 boson stars, while still sharing
with them several common features [99,101–104]. Among these features are the
fact that both are formed with complex scalar fields on a spherically symmetric
spacetime; they both admit diluted and compact solutions; and they possess sta-
ble and unstable branches separated by the solution of maximum mass for a given
ℓ. On the other hand, we had previously [99, 101, 102] observed some character-
istics related to the ℓ parameter: an increase in the compactness and size of the
maximum mass configurations and the fact that their morphology tends to form
a hollow-like central region (even in the ℓ= 1 boson star case), with the position
of the maximum of density moving away from r = 0. The purpose of the present
work was to take a step forward and perform a thorough examination of how these
features change with ℓ. In particular, using different numerical methods, we were
able to increase notably the magnitude of the parameter ℓ and finally, with the
information of ℓ≫ 1 solutions and a careful analysis of the system of equations,
we were able to study the limiting case when the parameter ℓ goes to infinity.
One of the interesting features that can be observed is the fact that for ℓ> 1
the density in the central region is much smaller than in the shell region. We
have shown in this work that, as ℓ grows, so does the object and also the almost
empty central region, tending to form shells of scalar fields where the size of the
almost empty central region is much larger than the size of the region where the
scalar field is mainly distributed. This tendency in the behavior goes all the way
to infinity, making the objects look like larger and larger shells. We have shown
that, when ℓ ≫ 1 the scalar field profile is shifted outwards proportionally to ℓ
while its width grows as 3
p
ℓ. Furthermore, the spatial components of the stress
energy-momentum tensor tend to be highly anisotropic as ℓ increases. Indeed,
Extreme ℓ-boson stars 65
as ℓ grows the radial pressure tends to zero, while the tangential one remains
finite. In this way, for large values of ℓ, the shells tend to have no radial pressure
and are supported solely by the tangential ones, analogous to the way in which a
Roman arch supports its own weight. This increase in the anisotropy seems re-
lated to an increase in the compactness [107] of the ℓ-boson star. The mass of the
solutions that divide the stable and the unstable branches, as well as their size,
grows with ℓ, but in such a way that the compactness tends to a finite value. We
have proven that in the ℓ→∞ limit the compactness tends to about 0.23 for the
maximum mass configuration; that is, about half the Buchdahl limit. However,
unstable configurations may be much more compact, reaching a compactness of
about 0.433 in the large ℓ limit. In this regard, it is interesting to point out that
(single and multiple) shell-type configurations have also been found when analyz-
ing the spherically symmetric steady-state solutions of the Einstein-Vlasov sys-
tem [110]. In particular, it has been proven that such shells satisfy the Buchdahl
inequality and that static shells of Vlasov matter can have M(r)/r arbitrarily close
to 4/9 [111].
Regarding orbiting particles, the high compactness that ℓ-boson stars can
achieve while remaining stable gives rise to new features, which differentiate
them from standard boson stars and also from black holes. Schwarzschild black
holes have an ISCO located at 6M, with no stable circular orbits below that value.
Consequently, accretion disks around non-rotating black holes typically have an
inner boundary and “end” at r = 6M. Stable standard boson stars do not have
ISCOs, meaning that they could in principle possess an accretion disk extending
all the way to the star’s center. On the other hand, stable ℓ-boson stars exist with
an ISCO-OSCO pair. In this case, accretion disks could show a “gap” between the
ISCO and OSCO, to then again extend all the way to the center. These differences
could constitute an important observable feature.
Besides causal circular orbits, we studied null ones, also known as light rings.
We found that, for each ℓ, a pair of light rings appears at high enough compact-
ness, the exterior one beeing unstable and the interior one stable. These light
rings are always in the unstable spacetime regions, although they begin appear-
ing closer and closer to the stable region as ℓ increases, which seems reasonable
given that more compact stable solutions exist for larger ℓ. Our findings are con-
sistent with the results of [109]: if a regular compact object has a light ring, it
must have at least two9, one of them being stable; and the presence of the stable
light ring is expected to lead to nonlinear spacetime instabilities.
In our vast parameter exploration we have not included excited modes (higher
frequency solutions containing one or more nodes of ψℓ), which would be unsta-
ble if results for standard boson stars also hold here [112]. However, solutions
9Except, of course, for the degenerate case in which the two light rings coincide.
66 3.6 Appendix
Table 3.1: Summary of the main definitions used in this chapter.
Symbol Definition Depends on
M Mass function, also M(r) ℓ, u0, r
MT Total mass (or mass function at outer boundary) ℓ, u0
R99 Areal radius containing 99% of the total mass ℓ, u0
C99 MT /R99 ℓ, u0
Cm Maximum of M(r)/r over r > 0 ℓ, u0
Rm Location of maximum M(r)/r ℓ, u0
Mm Mass function evaluated at Rm ℓ, u0
rin Location of the inner light ring ℓ, u0
rout Location of the outer light ring ℓ, u0
rosco Location of the OSCO ℓ, u0
risco Location of the ISCO ℓ, u0
Mmax Maximum of MT (for a given ℓ) ℓ
that combine a stable ground state solution with excited ones might again be sta-
ble [113]. We expect to address these questions in future works.
Apart from the properties discussed in this chapter, ℓ-boson stars open up
the possibility to consider a larger landscape of solutions such as the ones de-
scribed in [104]. These results along with the existence of a stable branch for the
ℓ-boson stars [101–103] make us conclude that such localized bosonic systems
may play an important role in modeling astrophysical objects, such as galactic
halos or black hole mimickers with potential observable consequences. Further
work along these lines is underway and will be presented in the near future.
3.6 Appendices of Chap. 3
Appendix: Definitions and rescaling in µ
We include tables that provide summarized information in a single place, aiding
in the reading of this chapter. A summary of the main definitions used in this work
is shown in Table 3.1. Rescaling rules in µ, which allows one to obtain solutions
for arbitrary µ from the solution of any given µ0, are shown in Table 3.2.
Appendix 3.B: Numerical methods
We obtain solutions of the eigenvalue problem in equations (3.5) numerically using
two different methods, implemented in independent codes. For ℓ. 25 we use a
shooting method similar to the one described in our previous work [99], but with
some improvements. For larger ℓ it becomes more and more difficult for this code
Extreme ℓ-boson stars 67
Table 3.2: Solutions for arbitrary values of µ can be obtained from those of a given
value by performing a rescaling as shown in this table.
µ 7→ λµ
(α, γ, ψℓ) 7→ (α, γ, ψℓ)
u0 7→ λℓ u0
ω 7→ λω
(r, M) 7→ λ−1 (r, M)
(ρ, pr, pT ) 7→ λ2 (ρ, pr, pT )
to converge to a given mode. In those cases we switch instead to a spectral method.
These methods, which are described in the following subsections, give the same
results in the parameter region where both are able to obtain solutions.
Shooting Method To obtain solutions for ℓ. 25 we use a “shooting to a fitting
point method” based on [114], implemented in a code which is described in [115].
It consist of doing a direct numerical integration of the ordinary differential equa-
tions starting both from the left and right boundaries, at which one imposes either
appropriate physical conditions or guesses when those are undetermined, with
the goal of matching both the fields and their first derivatives at some interme-
diate point. This defines a function of the mentioned guesses, plus an additional
guess, the eigenvalue ω2, whose roots correspond to the fitting condition being
satisfied. In order to find such roots, a Newton-Raphson method is used. The
fitting point method is particularly useful when one has a system with a pair of
solutions, one rapidly growing and the other rapidly decreasing at each bound-
ary, and one wants to obtain the (physical) solution that decays to zero at both
boundaries, as in the large ℓ cases. For the numerical integration, instead of the
algorithm described in [114], we use a more sophisticated step adaptive method
provided by the LSODE routines. For the particular applications of this work, it
was also helpful in a few cases to modify the left boundary conditions in order to
be able to set them at locations quite a bit to the right of r = 0. This is due to
the shell-like shape of the stars for large enough ℓ. The details are given below.
Finally, even though the solutions for a given value of µ can be trivially obtained
from a rescaling of the µ= 1 case (see appendix 3.6), which is the value we fixed
in most situations, sometimes it helped the numerical code to easily find solu-
tions to vary µ depending on the particular region of the parameter space. This
is because some fields may become many orders of magnitude different when one
restricts oneself to the µ= 1 case. Nevertheless, we present all our results in a µ
independent form.
68 3.6 Appendix
Approximate solutions for low density As mentioned throughout the chap-
ter, for large values of ℓ the scalar field distribution is shell-like, with very low
density (as compared to its maximum value) in an interior region with r < r1 and
in an exterior region with r2 < r for certain values r1 < r2. In the interior region
the solutions can be approximated by those of a scalar field on a flat spacetime,
while in the exterior region they can be approximated by solutions of a scalar field
on a Schwarzschild spacetime with mass MT .
In the interior region (r < r1) we can assume α = γ = 1. Then, from equa-
tion (3.5c), we get
1
r2
(
r2ψ′
in
)′
=
(
µ2 −ω2 +
ℓ(ℓ+1)
r2
)
ψin, (3.34)
with solutions
ψin(r)= C1
Jℓ+ 1
2
(
√
ω2 −µ2 r
)
p
r
+C2
Yℓ+ 1
2
(
√
ω2 −µ2 r
)
p
r
, (3.35)
where Jν(x) and Yν(x) are the Bessel functions of the first and second kind, re-
spectively. Keeping only the solution with the proper behavior at r = 0 and writing
the arbitrary amplitude in terms of u0 we obtain
ψin(r)= u0
2
(
ℓ+ 1
2
)
Γ
(
ℓ+ 3
2
)
(
√
ω2 −µ2
)ℓ+ 1
2
Jℓ+ 1
2
(
√
ω2 −µ2 r
)
p
r
. (3.36)
In order to transform to the gauge used in the remainder of this work, in which
α= 1 at r =∞, rather than at r = 0, one just needs to replace ω with γ∞α∞ω in
equation (3.36). We show an example of this approximation in figure 3.14. We see
a very good agreement between the scalar field and its approximation even well
beyond µ r =µ r1 ≈ 40.
Finally, we note that in the exterior region one can assume the metric is given
by a Schwarzschild solution with mass MT . Then, the scalar field can be expressed
in terms of the confluent Heun functions. However, we did not use the external
region approximations in this chapter, hence we will not present any details here.
Spectral Method An independent code was built based on a multidomain spec-
tral method. Specifically, a collocation method has been used with Chebyshev
polynomials as the basis functions. Details of the code described in the following
paragraphs were essentially implemented based on [116] which is a review on
spectral methods in numerical relativity. The Einstein-Klein-Gordon equations
were solved in isotropic coordinates, where the differential operators in the re-
sulting equations in the system are similar to each other and therefore easier to
implement in this particular method.
Extreme ℓ-boson stars 69
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
1× 10−12
1× 10−10
1× 10−8
1× 10−6
0.0001
0.01
0
0.4
0.8
1.2
0 10 20 30 40 50 60 70 80
ψ
i
ψℓ
ψin
ψℓ−ψin
ψℓ
4π
r2
ρ
µ r
4πr2ρ
Figure 3.14: Solution ψℓ and approximation ψin for the case with ℓ = 25 and
maximum mass MT = 13.45 (top panel). The middle panel shows the relative
error. For reference we also include the density profile in the bottom panel.
The physical domain, parameterized by the radial coordinate is decomposed
into 5 carefully placed domains depending on what range of solutions we want to
obtain in a single run, given ℓ. For the outer domain a compactification is carried
out so that the external boundary conditions can be imposed at spatial infinity. On
the other hand, in the domain that contains the origin, an even base of Chebyshev
polynomials is used for the lapse and the conformal factor Ψ, while an even (odd)
base is used for the field if ℓ is even (odd), which guarantee the solution is regular
at the origin. The non-linear system of equations that results for the coefficients
of the expansion is solved iteratively using a Newton scheme, the extra variable ω
is compensated with an extra equation α(r = 0) = α0 > 0, which ensures that the
code does not converge to the trivial solution.
An initial guess is required in the Newton scheme for the coefficients of the
expansion in all the functions (as well as the frequency), this is equivalent to pro-
vide an initial guess for the functions. Given certain value of ℓ, the first solution
is obtained from reasonable choices for the three parameters, σ, r0 and φ0, which
control the properties of the following simple initial guess
ψℓ =
(
r̄
r0
)ℓ
φ0 exp
(
−
r̄2 − r2
0
σ2
)
, (3.37)
α=−(1−α0)exp(−r̄2)+1, (3.38)
Ψ= 1. (3.39)
Here r̄ refers to the isotropic radial coordinate. The solutions are easier to find
in the Newtonian regime where the frequency is close to one. For example, in the
ℓ = 50, 100 cases presented here we start with an initial guess of ω = 0.95 and
once the first solution is obtained we slightly decrease the value of α0 and take as
70 3.6 Appendix
the new initial guess the previous solution.
We have checked that in the spectral code, as in the convergence test performed
in [117] for the ℓ= 0 case, the error indicators, as for example the frequency and
the difference of the ADM and Komar masses converge exponentially to a fixed
value and to zero respectively, as we increase the number of Chebyshev basis
polynomials, as expected for a spectral method.
Appendix 3.C: Summary of numerical data
Table 3.3 shows information regarding most of the solutions analyzed in this chap-
ter.
Extreme ℓ-boson stars 71
T
ab
le
3.
3:
P
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4

CHAPTER 4
CONFINED GHOST1
Contents
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Theoretical setting . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Action and equations of motion . . . . . . . . . . . . . . 75
4.2.2 Field equations . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.3 Quantities of interest . . . . . . . . . . . . . . . . . . . . 77
4.3 Equilibrium solutions . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Boson stars and phantom solitons . . . . . . . . . . . . . 79
4.3.2 E -boson stars . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
W
e present the E -boson star: A novel configuration of a boson star with
an exotic matter nucleus; the exotic matter being described by a real
massive scalar field with self-interaction term and kinetic term of the
opposite sign. The other scalar field is canonical, so that the system is similar to
the material component of the quintom cosmological scenario. Considering the
static spherical symmetric case, we obtain cases where both fields are distributed
as concentric spheres, and others with the remarkable feature that the canonical
matter is pushed outwards and obtain a shell like distribution of the canonical
field, with a nucleus of exotic matter at the center. We present global properties
of such E -boson stars and stress the differences that these configurations have
with respect to the usual boson stars. In particular, we obtain cases where the
compactness goes beyond the Buchdahl limit.
1This chapter is extracted with minor revisions from Ref. [118], which was written in collaboration
with E. Jiménez-Vázquez and D. Núñez. © American Physical Society. Reproduced with permission.
All rights reserved.
73
74 4.1 Introduction
4.1 Introduction
The study of the dynamics of matter different to the usual gas or fluid, is an aca-
demic subject by itself. Such studies acquire more relevance once that matter is
related to possible models describing the dark components of the Universe. In-
deed, several models proposed for describing the dark matter, and all the ones
proposed to describe the dark energy, even including the cosmological constant
one, violate the strong energy condition. Some particular models, specially those
concerning dark energy fail to satisfy the null energy condition (NEC) which in
turn implies the violation of the weak energy condition, meaning that at least
some time-like observer with four-velocity uµ measures negative energy densities
ρ = Tµν uµ uν. An example of this are the phantom models [34], whose simplest
cases consist of a scalar field with a negative kinetic term in the Lagrangian.
On the other hand, quintessence and scalar field dark matter models, which en-
compass scalar fields with a canonical kinetic term, always satisfy the NEC [35],
nevertheless typical cosmological solutions with this type of matter within Gen-
eral Relativity, lead to periods in time with negative pressures T i
j, as measured
by an static observer, in an analogy with the fluid description [30].
In the Cosmological models the matter is characterized by their equation of
state2 , the ratio of the pressure to the density, p/ρ = w. For instance in quintessence,
w stays above the cosmological constant value w ≥−1, while for the phantom field
w ≤−1. In describing the dark energy with a dynamical equation of state, cosmo-
logical data analyses interestingly suggest that the boundary w =−1 is crossed in
the evolution of the Universe [36, 37]. Models that can achieve this, require two
scalar field (or fluid) components [36,38] and are called quintom (see Ref. [39] for
a review).
Self-gravitating scalar fields that violate the NEC, such as the phantom field,
are the constituent material from which hypothetical objects called wormholes are
constructed [119–123]. On the other hand, canonical (complex) scalar fields build
up dynamically robust gravitational solitons called boson stars [44], which are
regular solutions in Euclidean topology and exist in a variety of different models,
reaching compactness values comparable to those of the neutron stars. Also so-
lutions to the Einstein-Klein-Gordon equations with phantom fields exist in Eu-
clidean topology [124] which could well be thought of as phantom boson stars.
Along the lines of the latter solutions and boson stars in this work we construct
objects that combine properties of both. Indeed, in this manuscript, we present
solutions to a boson star model, consisting of a canonical complex scalar field and
2Perhaps this is not a fortunate definition, as long as it applies directly to fluids and as mentioned
above, for some models of matter the concepts of density or pressure are very peculiar and do not fit
with the usual ones related to the fluid. In any case, the matter is characterized in such a way and
several models have a negative w.
Confined ghost 75
a phantom real scalar field, both minimally coupled to Einstein gravity, where the
ghost matter is confined within a bounded object made of a canonical scalar field.
We call these objects E -boson stars.
We are able to capture ghost matter in two different ways: with a small dis-
tribution inside a bigger soliton of canonical scalar field; and we have also found
distribution where the ghost field is present and the canonical field profile tends
to zero, with a maximum located away from the center, forming a shell surround-
ing the ghost matter. In both cases, these solutions can be thought of as boson
stars with a phantom matter core that repels (via gravity) the positive density
scalar field, allowing more massive and compact solutions compared to standard
boson stars.
As mentioned in several works, for instance [125], the scalar field is not a kind
of fluid and care should be made when trying to understand it as a kind of fluid.
It is interesting that the dynamics described in this manuscript due to the pres-
ence of both types of fields, involves in an effective manner a kind of gravitational
repulsive interaction which has no analogy to the gradients of pressure of the
Archimedes’ principle regarding a body submerge in a liquid.
The paper is organized as follows. In Sec. 4.2 we present the action, specify
the metric and scalar fields ansatz, derive the static spherically symmetric field
equations and define various physical quantities that will be used in the analysis.
Numerical solutions of are presented in Sec. 4.3, starting from the known boson
star and ghost field soliton solutions and then constructing composite solutions,
describing global properties of the configuartions, as well as concrete individual
examples. Finally we present a review of our findings and give some final remarks
in Sec. 4.4. We use geometrized units, in which G = 1 = c, and the convention
(−,+,+,+) for the metric signature.
4.2 Theoretical setting
4.2.1 Action and equations of motion
We consider a canonical complex scalar field ϕ and a ghost real scalar field χ,
minimally coupled to Einstein gravity. The system is represented by the action,
S =
∫
d4x
p
−g
(
1
16π
R+Lϕ+Lχ
)
, (4.1)
which contains the Einstein-Hilbert action, R is the Ricci scalar, g the determi-
nant of the metric, and the canonical scalar field ϕ contribution with Lagrangian,
Lϕ =−
1
2
∇µϕ∇µϕ∗−
1
2
Vϕ (4.2)
76 4.2 Theoretical setting
as well as the phantom field contribution whose Lagrangian has the following
form
Lχ =
1
2
∇µχ∇µχ−
1
2
Vχ. (4.3)
The functions Vϕ and Vχ denote scalar potentials given by
Vϕ(|ϕ|)=µϕ|ϕ|2; Vχ(χ)=−µχχ
2 +λχ4 . (4.4)
µϕ and µχ are the mass parameters and λ is a (positive) self-interaction param-
eter. When χ= 0 the system described by Eq. (4.1) contains the necessary ingre-
dients to construct a mini-boson star [44] and on the other hand, when ϕ= 0 the
system reduce to the action used for the phantom non-singular spherical solu-
tion [124]
Variation of the action with respect to gµν, ϕ and χ lead to the Einstein equa-
tions
Rµν−
1
2
gµνR = 8πTµν, (4.5a)
Tµν = gµν(Lϕ+Lχ)−2
∂(Lϕ+Lϕ)
∂gµν
, (4.5b)
and the Klein-Gordon equations
∇µ∇µϕ=µ2
ϕϕ ; (4.6a)
∇µ∇µχ=µ2
χχ−2λχ3 . (4.6b)
We look for static spherically symmetric solutions and thus employ the following
line element,
ds2 =−N2 dt2 +Ψ
4
(
dr2 + r2 dΩ2
)
, (4.7)
with N,Ψ functions of r only and 0≤ r <∞. For the scalar fields we assume that
χ= χ(r) and for the canonical one we take the anzatz:
ϕ =φ(r)eiωt. (4.8)
4.2.2 Field equations
Substitution of the metric form (4.7) and the ansatz of the field (4.8), into Einstein
Equations (4.5) yield
∆3Ψ+πΨ5
[
(
ωφ
N
)2
+
∂φ2
Ψ4
+µ2
ϕφ
2 −
∂χ2
Ψ4
−
(
µ2
χ−λχ2
)
χ2
]
= 0,
∆3N +
2∂N∂Ψ
Ψ
−4πNΨ
4
[
2
(
ωφ
N
)2
−µ2
ϕφ
2 +
(
µ2
χ−λχ2
)
χ2
]
= 0,
(4.9)
Confined ghost 77
where ∂ f := d f
dr
and ∆3 f := ∂2 f + 2
r
∂ f . The Klein Gordon equation for the canonical
field is
∆3φ+
∂φ∂N
N
+2
∂φ∂Ψ
Ψ
−Ψ
4
[
µ2
ϕ−
(
ω
N
)2
]
φ= 0 , (4.10)
and the equation for the ghost field is
∆3χ+
∂χ∂N
N
+2
∂χ∂Ψ
Ψ
−Ψ
4
(
µ2
χ−2λχ2
)
χ= 0 , (4.11)
To solve the Einstein-Klein-Gordon system of equations (4.5)-(4.11), we de-
mand regularity at the origin r = 0 and asymptotical flatness at r →∞. We apply
these boundary condition to φ, χ, N, Ψ:
∂φ|r=0 = ∂χ|r=0 = ∂N|r=0 = ∂Ψ|r=0 = 0 ; (4.12)
φ|r→∞ = χ|r→∞ = 0, N|r→∞ =Ψ|r→∞ = 1 . (4.13)
Also asymptotically flat equilibrium solutions exists only if ω < µϕ. Both ω
and λ are eigenvalues to be determined iteratively together with the integration
of the differential equations.
4.2.3 Quantities of interest
The Komar expression which gives the gravitational mass of the star is given by
the following expression
M =
1
4π
∫
Σt
RµνnµξνdV , (4.14)
where Σt is an hypersurface of constant t, ξ = ∂/∂t is the Killing vector associ-
ated with stationarity, nµ the future-directed unit vector normal to Σt. Using the
ansatz (4.7) we arrive at the formula M = r2 limr→∞∂N, for the Komar mass of
the present model.
We are also interested in evaluating the compactness of the obtained solutions
and for this reason we use the quantity R99, defined as the areal radius (Ψ2r)
that encompasses 99 percent of the total mass of the star. We then define the
compactness as
C =
M
R99
, (4.15)
which is an important quantity in describing the properties of several compact
objects and will be dealt with bellow.
The presence of light rings in horizonless compact objects (ultracompact ob-
jects) has been deeply discussed in [109, 126, 127] and are a clear signal of the
(in)stability of the space time under study. These results and theorems have the
NEC as a requirement for the implications of the existence of such light rings on
78 4.2 Theoretical setting
the (in)stability of the spacetime. We will see that E -boson stars not always sat-
isfy this energy conditions. With these facts in mind, we have searched for the
presence of light-rings in the solutions that we construct, by means of the deter-
mination of the light-ring radius rlr using the procedure described in [85], which
impose a condition on the lapse-function:
N(rlr)− rlr
(
1+2rlr
dlnΨ
dr
∣
∣
∣
∣
rlr
)−1
dN
dr
∣
∣
∣
∣
rlr
= 0. (4.16)
In this way, the energy density as measured by the Eulerian static observer
with four-velocity, uµ, which coincides with the vector normal to the hypersur-
faces, nµ, is an important feature of the solutions and the corresponding expres-
sion for both fields are obtained directly from the projection of the stress energy
tensors on such velocity:
ρ = Tµν nµ nν = ρϕ+ρχ, (4.17)
with
ρϕ =
1
2
[
ω2φ2
N2
+
∂φ2
Ψ4
+µ2
ϕφ
2
]
, (4.18)
ρχ =−
1
2
[
∂χ2
Ψ4
+
(
µ2
χ−λχ2
)
χ2
]
, (4.19)
Finally, the geometric scalars are also useful for characterizing the solutions,
analyzing the curvature of the spacetime and asses its regularity. From the line
element Eq. (4.7) we obtain the following expressions for the 4D-scalar of curva-
ture:
R =−
2
Ψ4
(
∆3 N
N
+4
∆3Ψ
Ψ
+2
∂N
N
∂Ψ
Ψ
)
, (4.20)
for the Weyl scalar
W =
4
3Ψ8



∂2 N − ∂N
r
N
−
2
Ψ

∂2
Ψ−
∂Ψ
r
−3
(
∂Ψ
)2
Ψ

−4
∂N
N
∂Ψ
Ψ



2
, (4.21)
and for the Kretschmann scalar:
K =−
4
Ψ8




(
∂2N
)2
−4∂2 N ∂N ∂ lnΨ
N2
+
8
Ψ2
(
(
∂2
Ψ
)2
+2∂2
Ψ∂Ψ
(
1
r
−
∂Ψ
Ψ
)
)
+
2
(
∂N
)2
N2


6
(
∂Ψ
)2
Ψ2
+
4∂ lnΨ+1/r
r

+
8
(
∂Ψ
)2
Ψ2


3
(
∂Ψ
)2
Ψ2
+
2∂Ψ
rΨ
+
3
r2





.
(4.22)
Confined ghost 79
101 102
λ
−10
−8
−6
−4
−2
0
µ
M
Figure 4.1: Non-singular solutions to Einstein-Klein-Gordon equations with a
phantom scalar field [124]. Plot of the (dimensionless) total mass of the exotic
configuration, as a function of the self interaction parameter. Notice that this
mass is negative for all values of λ. Such plot was originally presented in [124].
As mentioned in [122], the canonical matter generates wells in the geometry,
whereas the exotic matter produces bumps, so that the plot of the geometric scalar
quantities is also useful to characterize the effects on the different types of matter
on the geometry.
4.3 Equilibrium solutions
Before reporting numerical results for the full system (4.1) we first discuss and
review, in Sec. 4.3.1, some of the general properties of the cases with χ = 0 and
then with ϕ= 0. Next, in Sec. 4.3.2 we take into account both canonical and exotic
fields to construct a new type of configuration.
In all the configurations presented in this section, we have taken µϕ =µχ and
named this quantity simply µ. In the Appendix we present some cases where the
masses are different µϕ 6=µχ.
4.3.1 Boson stars and phantom solitons
Boson stars and the non-singular solutions with a phantom scalar field presented
in [124] are very similar solutions in the sense that they both are static spherical
and regular solutions of the Einstein-Klein-Gordon system in Euclidean topology.
However, the latter only exists for λ > 0 and have a total mass that is always
negative as can be seen in Fig. 4.1. Another difference is that these ghost solitons
are known to be unstable.
Boson stars on the other hand are stable in a region of the parameter space.
Given certain model, they are parameterized by the value of the scalar field at
80 4.3 Equilibrium solutions
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
φ0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
µ
M
boson star
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
sph. shells
 
0.4 0.5 0.6 0.7 0.8 0.9 1.0
ω/µ
0.2
0.4
0.6
0.8
1.0
1.2
1.4
µ
M
boson star
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
sph. shells
 
Figure 4.2: Left panel: Value of the canonical scalar field φ at r = 0 vs. total
mass. Right panel: Total mass vs. frequency. Triangles locate the transition point
between rmax(φ) = 0 (concentric spheres) and rmax(φ) 6= 0 (shell distribution of the
canonical field). The regions that connect the triangles with the corresponding
φ0 = 0 solutions (ω/µ→ 1) correspond to the sell-like E -boson stars.
the center of the star, φ0 := φ|r=0, the family of solutions is separated by certain
configuration that maximizes the value of the mass as can be seen from the thick
red line at the left panel of Fig. 4.2. In the right panel of this same figure it can
be seen how mini-boson stars, which are recovered when χ= 0 in (4.1), spiral to a
limit solution when mass is plotted versus frequency.
It is worth mentioning that interesting multi-field solutions for boson stars
have been reported in the literature using a complex and a real scalar field (see
also [99, 104, 128] for other boson star solutions with multiple complex scalar
fields). Such is the case of Boson Stars in the Friedberg-Lee-Sirlin model [129],
in which both fields are coupled (the mass of the complex field is given by its
interaction with the real field) and possess a canonical kinetic term. In this FLS-
model it has even been possible to construct rotating and black hole solutions with
hair. Another example are the boson stars in non-trivial topology [130], in which a
massless real field is included, giving rise to solitonic configurations with a worm-
hole in the core of the star. This solutions happen to inherit the Bronnikov-Ellis
wormhole instability inside the star, although the authors show that in a region
of parameter space the instability can be very weak.
4.3.2 E -boson stars
In generating the solutions presented in this paper, we consider dimensionless
quantities constructed from re-scaling with the mass of the canonical scalar field,
µϕ. For example r → µϕr, ω → ω/µϕ, µχ → µχ/µϕ, M → µϕM, etc. so that the
solutions are obtained for arbitrary µϕ. Then the set of four coupled non-linear
ordinary differential equations for φ, χ, N and Ψ (4.5)-(4.11) is solved numerically
Confined ghost 81
0 10 20 30 40 50 60 70
µR99
0.2
0.4
0.6
0.8
1.0
1.2
1.4
µ
M
boson star
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
sph. shells
 
0.5 0.6 0.7 0.8 0.9 1.0
ω/µ
0.00
0.05
0.10
0.15
0.20
C
boson star
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
sph. shells
 
Figure 4.3: First panel: Radius R99 vs. total mass. Second panel: Frequency vs.
compactness.
subject to the boundary conditions at (4.12) and (4.13).
We have used a Chebyshev spectral method with the collocation approach us-
ing 24 spectral coefficients in all solutions as well as 5 radial domains with bound-
aries at µr = {0.1,2.5,5,25}. The last domain is compactified and comprises µr
from 25 to ∞. The resulting nonlinear algebraic system of equations is solved
using a Newton-Raphson iteration.
The composite solutions (E -boson stars), consisting of non-zero φ and χ fields,
are uniquely determined by the value φ0 (defined above) and χ0 := χ|r=0, as long
as we restrict ourselves to the solutions in the base state (no nodes), for all the
configurations presented in the manuscript. A first solution for the composite
system has been obtained by fixing small values for φ0 and χ0 and taking as initial
guess a superposition of a boson star solution {φBS, NBS,ΨBS;ω} and a phantom
soliton solution {χPh, NPh,ΨPh;λ} in the following form;
φ=φBS, χ= χPh ; (4.23)
N2 = N2
BS
+N2
Ph
−1, Ψ
4 =Ψ
4
BS
+Ψ
4
Ph
−1 . (4.24)
Clearly this superposition introduce constraint violations, however the initial
guess leads to configurations close to the true solution of Einstein equations, to
which the code converges after a small number of steps. Then, sequence of con-
stant χ0 solutions are obtained by slowly varying φ0. Different global quantities
obtained for this families of boson star solutions are displayed in Fig. 4.2 and 4.3,
where we present the total mass of the stars vs φ0, ω and R99. Also in the right
panel of Fig. 4.3 we show the compactness as function of the frequency ω. We ob-
serve that the mass and even the compactness grow notably by including a larger
component of ghost matter, which, as we will see below, is stored in the core of the
star.
The first and second column of Fig. 4.4 show solutions with χ0 = 0.5 and three
different values for φ0. We observe an interesting feature: As the value of φ0/χ0
82 4.3 Equilibrium solutions
10−2 10−1 100 101 102
µr
10−6
10−5
10−4
10−3
10−2
10−1
100
φ0 = 0.35
φ
χ
10−2 10−1 100 101 102
µr
0.25
0.50
0.75
1.00
1.25
1.50
1.75 N
Ψ
10−2 10−1 100 101 102
µr
0
1
2
3
4
5 ρ
ρϕ
ρχ
10−2 10−1 100 101 102
µr
10−6
10−5
10−4
10−3
10−2
10−1
100
φ0 = 0.1
φ
χ
10−1 100 101 102 103
µr
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3 N
Ψ
10−2 10−1 100 101 102
µr
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6 ρ
ρϕ
ρχ
10−2 10−1 100 101 102
µr
10−6
10−5
10−4
10−3
10−2
10−1
100
φ0 = 0.001
φ
χ
10−1 100 101 102 103
µr
1.0
1.5
2.0
2.5
3.0
N
Ψ
10−2 10−1 100 101 102
µr
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6 ρ
ρϕ
ρχ
Figure 4.4: E -boson star solutions with χ0 = 0.5 and φ0 = 0.35 (upper row), 0.1
(middle row) and 0.001 (bottom row). In the first column the fields are plotted; in
the second column, the metric coefficients and in the third column the individual
and total densities, all as a function of the distance to center. We notice that the
energy density in the lower panel is difficult to be appreciate for the canonical
field, since it is four orders of magnitude smaller.
decreases (in fact, for a whole region near ω/µ = 1) the φ field profile shifts its
maximum away from the origin. It is also observed that for small values of φ0, the
metric factors undergo qualitative changes. They go from having a monotonically
increasing (decreasing) behavior for N (Ψ) of the standard boson star, to having
local maxima and/or minima. In the third column, we present the densities, the
individual corresponding to each field, and the total one, Eqs. (4.19, 4.18, 4.17); as
expected [122], the exotic density has regions with negative values and we see that
there are configuration such that the total density is positive, while others where
the total density has regions with negative values; this features are also related
to the distribution being concentric spheres or shells with a nucleus. Fig. 4.5
illustrate the different morphologies of the E -boson stars as 3D plots.
The left panel of Fig. 4.6 shows the outward shift of the maximum of φ for a
set of solution families. The right panel illustrates how the minimum of N ceases
to be located at r = 0, as happens in the case of the standard boson stars, χ0 = 0.
In Figs. 4.2, 4.3 we have indicated with a triangle the transition point from which
Confined ghost 83
µr
φ
χ
µr
φ
χ
Figure 4.5: Left figure: Concentric solid spheres configuration (χ0 = 0.1, φ0 =
4×10−2). Left figure: Shell-like configuration (χ0 = 0.6, φ0 = 3×10−4). The field
profiles in the lower plots have been normalized and reflected for −r. Notice that
the scalar fields, do not interact with one another except gravitationally and yet
give rise to this type of morphologies.
10−3 10−2 10−1
max(φ)
10−1
100
101
102
r m
a
x
(φ
)
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
N(r = 0)
0.2
0.4
0.6
0.8
1.0
m
in
(N
)
boson star
χ0 = 0.1
χ0 = 0.2
χ0 = 0.3
χ0 = 0.4
χ0 = 0.5
χ0 = 0.6
Figure 4.6: Left panel: Maximum of the canonical scalar field φ vs. position at
which this maximum is found. Right panel: Value of the metric function
p−gtt
at r = 0 vs. its global minimum.
we have solutions with the maximum of φ at r 6= 0, i.e., where the canonical field
changes its morphology, from a shell distribution to a solid sphere.
The transition to the solutions in the limit χ0 → 0 (standard boson star) occurs
in a simple way: the ghost field χ, vanishes keeping its center at r = 0 while the
associated back-reaction of the spacetime also vanishes. However, the limitφ0 → 0
is not equally straightforward as the reader can probably already anticipate from
the presented results. As already mentioned, the maximum of φ moves away from
the origin and the value of the absolute maximum of φ decreases (Fig. 4.6). Also,
as seen in Fig. 4.7, when ω/µ → 1 the λ eigenvalue tends asymptotically to the
value that the Dzhunushaliev et al. solution would have with the corresponding
χ0. Although the canonical field energy density decreases as it approaches to
84 4.3 Equilibrium solutions
0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
103
2× 102
3× 102
4× 102
6× 102
λ
χ0 = 0.2 E-boson star
χ0 = 0.2 phantom soliton
Figure 4.7: Eigenvalues ω and λ for a sequence of solutions with χ0 constant.
this limit, the shell is of larger size and therefore contributes enough to obtain
solutions whose total mass is positive. With the numerical resolution to which we
have access we have not been able to find solutions with small φ0 6= 0 transiting
to the negative mass region, which we believe should be the case for ω/µ→ 1 in
order to have a continuous transition to the phantom solitonic solution.
With respect to the energy conditions, the E -boson stars have a remarkable be-
havior as well. Indeed, the null energy condition states: Tµνkµkν ≥ 0, for all null
vectors kµ. If the energy momentum tensor is of the form Tµ̂ν̂ = diag(τ, p1, p2, p3)
with respect to an orthonormal basis (first Segrè type), then the NEC is satisfied
if and only if τ+ pi ≥ 0 (i = 1,2,3) [35, 131]. For the orthonormal components of
the energy-momentum tensor of the present model (4.1) and spacetime (4.7) we
have
τ+ p1 =
(
ωφ
N
)2
+
(
1
Ψ2
dφ
dr
)2
−
(
1
Ψ2
dχ
dr
)2
; (4.25)
τ+ p2 = τ+ p3 =
(
ωφ
N
)2
. (4.26)
From these relations it can be seen that, for the Dzhunushaliev et al. soliton
[124]. when φ= 0, the NEC is always violated (in particular Eq. (4.25)), however
with respect to the E -boson stars, it can be seen that there are configurations
which abide the energy conditions at all points. In Table 4.1 we include data for a
selection of the solutions obtained. We present two cases corresponding to boson
stars, another two corresponding to the Phantom configuration and six cases of
the E -boson star. Fig. 4.8 evaluates the NEC by plotting the quantity τ+p1 for the
E -boson star configuration in Table 4.1. We note that among the configurations
with χ0 = 0.1 (A, B, C) all of them violate the energy condition; the configuration
A even for all r. On the other hand, for the configurations with χ0 = 0.01 (D, E, F)
only D violates the NEC in a small part of the domain while E and F satisfy it at
Confined ghost 85
10−2 10−1 100 101
µr
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
0.25
τ
+
p 1
×10−2
E-BS-A
E-BS-B
E-BS-C
E-BS-D
E-BS-E
E-BS-F
10−1 100
0.0
0.5
1.0
×10−4
Figure 4.8: Null energy condition for E -boson star configurations described in
table 4.1.
all points.
Related to the NEC violation, the high compactness obtained by the solutions
(as will be seen below) and the recent results on compact objects and circular pho-
ton orbits, we wonder if the appearance of light rings happens before the critical
value of the mass, in the branch that in standard boson stars corresponds to the
family of stable solutions. However, what we obtain when we evaluate the condi-
tion in Eq. (4.16) across fixed χ0 families of solutions, is that although the light
ring "climbs" to the critical mass configuration as we increase the value of χ0, it
never passes to the so-called stable branch.
The E -boson stars have also the remarkable property that the compactness of
the configurations can easily rise above the Buchdahl limit as seen in Fig. 4.9. In-
deed, the interplay of both scalars field, the exotic pushing away the canonical one,
and the canonical exercising a larger pressure, provokes that the matter could be
compressed to very high values3 As shown in Fig. 4.9 and in the Appendix, in-
creasing values of χ0 or µχ/µϕ points to the fact that the black hole limit could
be reached. There was already experience with the ℓ-boson stars, [85] where the
maximum compactness tends to the Buchdahl limit as the parameter ℓ grows,
but they never rose above such limit. The C -stars, presented in [133] are solu-
tions which describe anisotropic fluid stars with compactness above the Buchdahl
limit, due to anisotropic pressures. Gravastars are also an example of compact
object solutions within general relativity with compactness above the Buchdahl
limit [134]. See [135] for a discussion on the conditions needed to rise above such
limit, and it is presented the case for a collection of non-interactive particles, the
3This is similar to the effect on the compactness and total mass of self-interacting boson stars [132],
where the repulsive (positive) self-interaction allows to accumulate more scalar field within the star,
except that in the case presented in this manuscript, the repulsion is driven by the gravitational
interaction.
86 4.3 Equilibrium solutions
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75
χ0
0.0
0.1
0.2
0.3
0.4
0.5
C
m
a
x
Buchdahl limit
Black hole
Mini-boson stars
Figure 4.9: Maximum compactness for E -boson stars with fixed values of χ0.
10−2 10−1 100 101
µr
0
1
2
3
4
R
E-BS-A
E-BS-B
E-BS-C
10−2 10−1 100 101
µr
0.00
0.02
0.04
0.06
0.08
R
E-BS-D
E-BS-E
E-BS-F
Figure 4.10: 4D curvature scalar for E -boson star configurations described in Ta-
ble 4.1.
Vlasov gas, where the Buchdahl limit is not surpassed. See Ref. [136] for inter-
esting discussions on the maximum compactness of horizonless compact objects
and [137] for case of bosonic fields.
Regarding the geometric scalars, we present their behavior for the six cases of
the Table 4.1 labeled as cases E -BS. Several of them describe concentric spheres
and one of them (solution E -BS-A) describe a shell-like configuration. In all cases
the scalar of curvature (which follows the total density), the Kretschmann and
Weyl scalars are well-behaved Fig. 4.10, 4.11, 4.12. For cases A-C, all curvature
scalars behave in a similar manner given that we have similar quantities of ex-
otic and canonical matter, not so for the D-F cases that shown differences in their
maximum (or minimum) values. Central values φ0 and χ0 are similar in mag-
nitude, but the curvature behavior is affected by the content of canonical scalar
field, smaller values for φ0 results in flatter spaces.
Confined ghost 87
10−2 10−1 100 101
µr
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
K
E-BS-A
E-BS-B
E-BS-C
10−2 10−1 100 101
µr
−1.75
−1.50
−1.25
−1.00
−0.75
−0.50
−0.25
0.00
K
×10−2
E-BS-D
E-BS-E
E-BS-F
Figure 4.11: Kretschmann scalar for E -boson star configurations described in ta-
ble 4.1.
10−2 10−1 100 101
µr
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
W
×10−2
E-BS-A
E-BS-B
E-BS-C
10−2 10−1 100 101
µr
0
1
2
3
4
5
W
×10−4
E-BS-D
E-BS-E
E-BS-F
Figure 4.12: Weyl scalar for E -boson star configurations described in table 4.1.
µϕ =µχ =µ
Solution φ0 χ0 ω/µ λ µM µR99 C
BS-A 1.057×10−2 0 0.975 - 0.3696 24.27 1.523×10−2
BS-B 4.650×10−2 0 0.901 - 0.6027 10.68 5.645×10−2
Ph-A 0 0.1 - 925.8 −0.0102 2.903 −3.519×10−3
Ph-B 0 1.0 - 2.559 −11.44 6.665 −1.743
E -BS-A 9.951×10−3 0.1 0.975 925.6 0.3793 24.80 1.530×10−2
E -BS-B 4.512×10−2 0.1 0.901 925.6 0.6148 10.81 5.688×10−2
E -BS-C 5.409×10−2 0.1 0.885 927.2 0.6322 9.687 6.526×10−2
E -BS-D 1.033×10−2 0.01 0.975 9.403×104 0.3662 24.57 1.491×10−2
E -BS-E 4.698×10−2 0.01 0.901 9.411×104 0.6040 10.62 5.689×10−2
E -BS-F 5.539×10−2 0.01 0.886 9.430×104 0.6196 9.614 6.445×10−2
Table 4.1
88 4.4 Final remarks
4.4 Final remarks
We have designed a configuration such that the exotic matter, described by a real
massive scalar field with self-interaction and such that the corresponding stress
energy tensor has a global sign opposite to the one corresponding to a canoni-
cal scalar field, is distributed inside a usual, canonical boson star configuration
described by a massive complex scalar field.
We have been able to solve the corresponding differential equations consider-
ing that the configuration is static and has spherical symmetry, using an inte-
gration method based on the spectral solver procedure and demanding regularity
at the origin and asymptotic flatness. We obtain several examples of configura-
tions based on three free parameters, namely the central amplitudes of both scalar
fields and the ratio between the exotic scalar mass µχ and the canonical one µϕ.
From the configurations obtained, several interesting features were deduced.
First of all, the static spherical configurations inherit a part of the properties of
boson stars and phantom solitons, however the gravitational (repulsive) interac-
tion between the fields creates new global features of the configurations, such as
an appreciable increase in compactness and also different morphologies, with the
phantom field always within the canonical field. As in boson stars, it is obtained
that in the plots of the total mass of the configuration versus the frequency, have
the usual snail-like plot, with the possible implication that the maximum of each
plot separates a region with stable total configurations from the unstable ones.
We also found another interesting feature regarding the ratio of the central
amplitudes of the exotic scalar field, χ0, to the corresponding of the canonical one,
φ0; when the ratio is much smaller than one, both distributions, the exotic and the
canonical one, are concentric spheres, however, as the ratio grows, the canonical
distribution seems to be pushed outwards form the center, forming a shell-like
distribution containing in the center the exotic field.
We obtained the scalars of curvature which are regular at all points and de-
scribe the characteristic features of the different types of matter on the geometry.
An eloquent feature of E -boson stars is that it is possible to tune the amplitudes
of the fields in such a way that the NEC is always satisfied and stable numeri-
cal implementation could be obtained even with a phantom scalar field. Further
information will be given in the forthcoming part II of our work.
We have thus designed and obtained interesting configurations with two types
of scalar fields. Clearly, the dynamics of such configurations, with some possible
stable or stationary states, is a most pressing question which will be dealt with in
a forthcoming work. Preliminary results show that some of these configurations
remain bounded.
Confined ghost 89
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
φ0
0.2
0.4
0.6
0.8
1.0
µ
M
µχ/µϕ = 1
µχ/µϕ = 0.5
µχ/µϕ = 2
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.2
0.4
0.6
0.8
1.0
µ
M
µχ/µϕ = 1
µχ/µϕ = 0.5
µχ/µϕ = 2
Figure 4.13: χ0 = 0.4 configurations with three different values for the ratioµχ/µϕ.
Left panel: Value of the canonical scalar field φ at r = 0 vs. total mass. Right
panel: Total mass vs. frequency.
4.5 Appendix of Chap. 4. Solutions with µϕ 6=µχ
The ratio between the mass parameter of χ and ϕ is also a free parameter in the
model and cannot be absorbed by any simple rescaling. Nevertheless the implica-
tion of modifying this parameter are relatively simpler than the effects of the χ0
parameter.
In Figs. 4.13 and 4.14 we show some global quantities of solutions for fixed
value χ0 = 0.4 and three different values of the quotient µχ/µϕ. The conclusions
regarding the mass and compactness are straightforward: as µχ/µϕ increases, the
effects of the phantom field are more present and bigger values for the total mass
of the distribution and for the compactness are obtained. We have verified that
similar effects are obtained for other values of χ0, in addition to those presented
in the above figures. For example, we have configurations with µχ/µϕ = 1 which
do not exceed the Buchdahl limit, but it is possible to modulate the value of the
quotient to obtain configurations with compactnesses as close to 0.5 as desired.
90 4.5 Appendix
0 10 20 30 40 50 60 70
µR99
0.2
0.4
0.6
0.8
1.0
µ
M
µχ/µϕ = 1
µχ/µϕ = 0.5
µχ/µϕ = 2
0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
C
µχ/µϕ = 1
µχ/µϕ = 0.5
µχ/µϕ = 2
Figure 4.14: χ0 = 0.4 configurations with three different values for the ratioµχ/µϕ.
First panel: Radius R99 vs. total mass. Second panel: Frequency vs. compact-
ness.
CHAPTER 5
TRAVERSABLE ℓ-WORMHOLES1
Contents
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.2.1 Metric ansatz . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 Matter content . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3 Stationary Wormhole equations . . . . . . . . . . . . . . . . . . 98
5.3.1 Qualitative analysis of the solutions . . . . . . . . . . . 102
5.3.2 Numerical shooting algorithm . . . . . . . . . . . . . . . 104
5.3.3 Energy density, mass and curvature scalars . . . . . . 105
5.4 Numerical wormhole solutions . . . . . . . . . . . . . . . . . . . 107
5.4.1 0-wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.2 ℓ-wormhole . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.5 Embedding diagrams and geodesic motion . . . . . . . . . . . . 112
5.5.1 Geodesic motion . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 Discussion and concluding remarks . . . . . . . . . . . . . . . . 116
W
e present new, asymptotically flat, static, spherically symmetric and
traversable wormhole solutions in General Relativity which are sup-
ported by a family of ghost scalar fields with quartic potential. This
family consists of a particular composition of the scalar field modes, in which each
mode is characterized by the same value of the angular momentum number ℓ,
yet the composition yields a spherically symmetric stress-energy-momentum and
metric tensor. For ℓ = 0 our solutions reduce to wormhole configurations which
had been reported previously in the literature. We discuss the effects of the new
parameter ℓ on the wormhole geometry including the motion of free-falling test
particles.
1This chapter is extracted with minor revisions from Ref. [122], which was written in collaboration
with B. Carvente, J. C. Degollado, D. Núñez and O. Sarbach. © IOP Publishing. Reproduced with
permission. All rights reserved
91
92 5.1 Introduction
5.1 Introduction
The essential property of General Relativity, namely, that matter determines the
geometry of the spacetime, acquires a new light when the matter is such that
it violates the energy conditions [138–142]. In particular, the violation of the
null energy condition opens the possibility for the existence of globally hyperbolic,
asymptotically flat spacetimes with non-trivial topological structures [143]. Such
matter, usually referred to as exotic in the literature, generates peculiar responses
in the properties of the spacetime curvature with important consequences on the
effective gravitational potential, producing potential “bumps" instead of the usual
potential wells. To provide an explicit example, in Fig. 5.1, we present the grav-
itational effective potentials for a massive, radially infalling test particle for two
cases: the first one is due to the presence of a point mass which generates the
usual gravitational well, whereas the second one is generated by a distribution of
exotic matter (as the one discussed later in this work) in which case the potential
exhibits a different type of convexity corresponding to a gravitational potential
bump.
Figure 5.1: Gravitational potential produced by a point mass ∼ −1/r (left panel)
and by the exotic distribution presented in this work (right panel), see section
5.5.1. In both cases a test particle with vanishing angular momentum is consid-
ered.
Moreover, under specific circumstances, the bump could be such that it con-
nects two separated regions of spacetime. The resulting configuration is dubbed
wormhole, and it offers challenges and opportunities to better understand the re-
lation between matter and geometry, aside from the fact that, being bona fide
solutions to Einstein’s equations, it could potentially describe an astrophysical
scenario if exotic matter turns out to actually being present in our Universe.
In cosmology, matter with negative pressure can be used to describe the ob-
served accelerated expansion of the Universe [144–147] and seems to be favored by
several observational constraints [29,148,149]. Additionally, modeling the dark
energy with an equation of state of the form p = ωρ, the observations suggest a
Traversable ℓ-wormholes 93
value of ω close to −1 or even smaller, in which case the existence of astrophysical
or cosmological wormholes becomes plausible.
The studies of traversable wormholes have their origin with Ellis’ work [150],
where the author presented a black hole like solution to Einstein’s equations,
and in order to remove the singularity, used a scalar field and drain the hole.
Actually, the same solution, based on a different approach was obtained almost at
the same time by Bronnikov [119]. It turned out that the solution represented a
bridge between two regions of the spacetime. Over the years, the idea was further
developed, and the best known example of a traversable wormhole appeared in
1988, in the work of Morris and Thorne [151]. Since then, a plethora of literature
has arisen and the complexity of the models has increased, see for example [152,
153] and references therein.
In order to obtain a wormhole solution to the Einstein equations, some works
use generation procedures, such as the Newman-Janis algorithm, which allows
to obtain a Kerr black hole solution starting from a Schwarzschild one; in this
way, a rotating (although not asymptotically flat) solution was obtained start-
ing from one of the original Ellis models [120]. There is also a technique which
uses the thin shell approach, which assumes that the matter is concentrated in
a three-dimensional submanifold. However, a common practice is to analyze the
geometry describing a putative wormhole without mentioning the possible mat-
ter that could generate it, artificially producing general forms of such wormholes
which might even include rotation; in the words of Morris and Thorne: fixed the
geometry... “and let the builders of a wormhole synthesize, or search throughout
the universe for, materials or fields with whatever stress-energy tensor might be
required” [151].
In the present work, we prefer to avoid this “reversed engineering approach”
and assume the specification of a suitable matter model which allows for a large
class of static, spherically symmetric and traversable wormhole solutions. In par-
ticular, following the recent approach in [99] to construct a generalized class
of static and spherically symmetric boson stars, we consider a family of mas-
sive, complex and self-interacting ghost scalar fields similar to the one consid-
ered in Dzhunushaliev et al. [121, 124], but which includes an extra parameter
ℓ mimicking the effects of the angular momentum. In this way, new spherical
and traversable wormhole solutions can be constructed which generalize those of
Refs. [121,124] to ℓ> 0. Accordingly, and following the terminology of the ℓ-boson
stars, we dub these solutions ℓ-wormholes.
While Ellis’ original solutions use a massless, time-independent real scalar
field without self-interaction, in this work we consider massive, complex and self-
interacting scalar fields with a harmonic time-dependency à la Dzhunushaliev
et al. [121, 124], but instead of considering just a single field we consider a fam-
94 5.2 Foundations
ily of fields with angular momentum number ℓm with ℓ fixed and m = −ℓ, . . . ,ℓ.
Assuming like in [99] that each of these fields has exactly the same radial depen-
dency, we obtain static, spherically symmetric wormhole solutions. When ℓ = 0,
the mass of the scalar field, its self-interaction and the time-frequency vanish one
recovers Ellis’ wormhole solutions.
The new wormholes have several interesting characteristics, such as curva-
ture scalars and effective potentials which smooth out the features of the cor-
responding 0-wormholes. The geodesic motion helps us to understand the role
played by the ℓ–parameter in the spacetime configuration. Finally, the presence
of a new parameter gives rise to the possibility that this wormhole might be stable,
a feature that will be discussed in a followup work.
The paper is organized as follows. In Section 5.2, we specify our metric ansatz
describing static, spherical symmetric and traversable wormhole spacetimes and
introduce the matter model. In Section 5.3, we derive the static field equations
in spherically symmetry, discuss some qualitative properties of the wormhole so-
lutions and then construct numerical solutions to the field equations whose main
properties are discussed next in Section 5.4. In Section 5.5, we discuss the em-
bedding diagrams visualizing the spatial geometry of the solutions, derive the
geodesic equations for massive or massless test particles propagating in the worm-
hole metric and analyze the motion under several conditions determined by the
wormhole parameters. Finally, we discuss and summarize our results in Sec-
tion 5.6.
5.2 Foundations
The determination of the stress-energy-momentum tensor that supports a worm-
hole geometry is of the utmost importance to understand its physical properties
and structure. As already mentioned in the introduction, an asymptotically flat
wormhole geometry in general relativity requires the matter to be exotic, that is,
matter that does not fulfill the regular properties of the usual matter we deal
with everyday.2 More specifically, the matter must violate the null energy condi-
tion, Tµνkµkν ≥ 0, where Tµν is the stress-energy-momentum tensor and kµ any
null vector [151,153]. Incidentally, this is also the fundamental ingredient of the
so-called ghost energy, a model not excluded by observations to be a candidate
for dark energy. For instance, constraints from the Supernovae Ia Hubble dia-
gram [156] favor the existence of an equation of state for such dark fluid, p =ωρ
with ω<−1, a model consistent with ghost energy [141].
2However, it should be mentioned that there are examples of traversable wormholes without ex-
otic matter in modified theories of gravity [154] or in general relativity when the asymptotic flatness
condition is replaced by adS-asymptotics [155].
Traversable ℓ-wormholes 95
In practice, violation of the null energy condition is accomplished by changing
the global sign in the stress-energy-momentum tensor in Einstein’s equations.
Ellis called this the other polarity of the equations [150]. This change in sign in
the equations is attributed to the type of matter, and has multiple implications
which might lead to misunderstandings. A global change in sign to the stress-
energy-momentum tensor implies that the usual definition of density also has the
opposite sign and is thus negative.
In this work we interpret the physical properties of the wormhole directly in
terms of the theory of general relativity and Einstein’s field equations, so that
the exotic matter produces a different reaction in the curvature of the spacetime,
particularly in the effective potential in which the test particles move, generating
bumps instead of wells, so that a particle has to spend potential energy in order to
get closer to the source, while it gains kinetic energy and accelerates when getting
away from it; like when climbing a mountain to reach the summit and then going
down.
In particular, we stress that when talking about test particles we assume the
validity of the weak equivalence principle, which assumes that the inertial and
gravitational masses are equal to each other. Therefore, free-falling test particles
or photons always follow causal geodesics of the underlying spacetime, regardless
of the sign of their mass. The “bump interpretation" mentioned so far will become
evident when analyzing the geodesic motion of test particles in Section 5.5.
5.2.1 Metric ansatz
We will consider a static spherically symmetric spacetime with a line element of
the form:
ds2 =−a(r) c2 dt2 +a(r)−1 dr2 +R2(r) dΩ2, (5.1)
where R and a are positive functions only of the radial coordinate r, and dΩ2 =
dθ2+sin2θ dϕ2. Notice that for R2 = r2+b2, with b a positive constant, and a = 1,
the reflection-symmetric Ellis wormhole metric is recovered [150] and, from it,
with a suitable redefinition of the radial coordinate, one obtains the usual form of
the Morris–Thorne like wormhole [151]. Also note that the coordinate r our work
is based on extends from −∞ to +∞, and we will demand that R be regular at the
throat r = 0, which corresponds to a minimum of the area 4πR2 of the invariant
two-spheres.
5.2.2 Matter content
In the present work, we consider a set of several massive scalar fields with a self-
interaction term. Our configurations are constructed in such a way that the sum
of the fields preserves the spherical symmetry of the stress-energy-momentum
96 5.2 Foundations
tensor and includes an extra parameter associated with the angular momentum
number ℓ. This approach was introduced in [100] in the context of critical collapse,
and recently used in [99] to construct ℓ-boson stars.
We start with the Lagrangian density for N complex massive scalar fields
LΦ =−
1
2κ
[
N
∑
i=1
η∇µΦi∇µ
Φ
∗
i +V (
∣
∣Φ
∣
∣
2
)
]
, (5.2)
with a quartic potential
V (
∣
∣Φ
∣
∣
2
)=
N
∑
i=1
V (i) =
N
∑
i=1

ηµ
m2
Φ
c2
~2
|Φi|2 +ηλ
λ
2~2
∣
∣Φi
∣
∣
2
N
∑
j=1
∣
∣
∣Φ j
∣
∣
∣
2

 (5.3)
where κ= 8πG/c4, ~ is the reduced Planck constant, mΦ is the mass of the scalar
field particle and λ is the parameter measuring the strength of the quartic inter-
action term. The values η= ηµ = ηλ = 1 represent the canonical scalar fields while
η= ηµ =−ηλ =−1 describe the type of ghost fields in which we will be interested
in, and from now on we fix the latter choice. In the following, for convenience, we
will work with the rescaled quantities µ = mΦc/~ and Λ = λ/2~2 instead of mΦ
and λ.
The stress-energy-momentum tensor associated with the scalar fieldΦi is thus
given by
T(i)
µν =
c4
16πG
[
−
(
∇νΦi ∇µΦ
∗
i +∇νΦi ∇µΦ
∗
i
)
− gµν
(
−∇αΦi∇α
Φ
∗
i +V (i)
)
]
, (5.4)
while the total stress-energy-momentum tensor that we plug into Einstein’s field
equations is
Tµν =
N
∑
i=1
T(i)
µν. (5.5)
In [100], for the case of real scalar fields, and in the appendix of [99], for com-
plex ones, it was shown that for an appropriate superposition, a stress-energy-
momentum tensor of the form (5.5) with Λ= 0 may be spherically symmetric, even
though the individual fields Φi have non-vanishing angular momentum. Here, we
generalize this result further to include the self-interaction of the field. The pro-
cedure is as follows.
Each scalar field Φi has the form
Φi(t, r,θ,ϕ)= φℓ(t, r)Y ℓm(θ,ϕ), i = ℓ+1+m, (5.6)
where m varies over −ℓ,−ℓ+ 1, . . . ,ℓ (such that i varies from 1 to N = 2ℓ+ 1)
and Y ℓm(θ,ϕ) denote the standard spherical harmonics. Here, the parameter ℓ
is kept fixed and the amplitudes φℓ(t, r) are equal to each other for all m. Using
the addition theorem for the spherical harmonics, one can show that the resulting
Traversable ℓ-wormholes 97
stress-energy-momentum tensor in Eq. (5.5) for the N = 2ℓ+1 fields is spherically
symmetric.
Next, one considers a stationary state with harmonic time dependence for the
scalar field
φℓ(t, r)= eiωtχℓ(r), (5.7)
where χℓ is function of r and ω is a real constant. Once such procedure is carried
out, the following non-trivial components of the stress-energy-momentum tensor
are obtained:
T t
t =
c4
8πG
2ℓ+1
8π



a
(
dχℓ
dr
)2
+
[
ℓ
(
ℓ+1
)
R2
+µ2 −
2ℓ+1
4π
Λχ2
ℓ+
ω2
a
]
χℓ
2



,(5.8)
Tr
r =
c4
8πG
2ℓ+1
8π



−a
(
dχℓ
dr
)2
+
[
ℓ
(
ℓ+1
)
R2
+µ2 −
2ℓ+1
4π
Λχ2
ℓ−
ω2
a
]
χℓ
2



,(5.9)
Tθ
θ = Tϕ
ϕ =
c4
8πG
2ℓ+1
8π



a
(
dχℓ
dr
)2
+
[
µ2 −
2ℓ+1
4π
Λχ2
ℓ−
ω2
a
]
χℓ
2



. (5.10)
Notice how the procedure of adding individual stress-energy-momentum ten-
sors maintains the spherical symmetry and yields a result that depends on the
angular momentum number ℓ through the centrifugal-like terms ℓ(ℓ+1)/R2. As
expected and shown below, this dependency plays a nontrivial role in the solutions
of Einstein’s equations. The mixed components T t
r, T t
θ and T t
ϕ vanish; indicat-
ing that there are no fluxes of matter in this case, which is compatible with the
assumption of staticity of the metric.
Notice also that the stress-energy-momentum tensor (5.8–5.10) violates the
null energy condition everywhere; for instance, the null vector field k = a−1/2c−1∂t+
a1/2∂r gives
Tµνkµkν =−T t
t +Tr
r =−
c4
8πG
2ℓ+1
4π
[
a
(
dχℓ
dr
)2
+
ω2
a
χ2
ℓ
]
, (5.11)
which is negative unless the scalar field vanishes.
Now, one can compute the equation of motion for each individual field, i. e.
the Klein-Gordon equation, using the fact that the divergence of the total stress-
energy-momentum tensor is zero. Each amplitude obeys the identical equation:
d
d r
[
aR2 dχℓ
dr
]
+R2
(
ω2
a
−
ℓ
(
ℓ+1
)
R2
−µ2 +
2ℓ+1
2π
Λχ2
ℓ
)
χℓ = 0, (5.12)
where we have used the fact that spherical harmonics are eigenfunctions of the
Laplace-Beltrami operator
∆S2 Y ℓm =
(
∂2
∂θ2
+cotθ
∂
∂θ
+
1
sin2θ
∂2
∂ϕ2
)
Y ℓm =−ℓ
(
ℓ+1
)
Y ℓm . (5.13)
98 5.3 Stationary Wormhole equations
Figure 5.2: Normalized density at the throat for a ℓ = 1 wormhole. The first
sphere represents the m = ±1 contribution while the center sphere corresponds
to the m = 0 contribution.
As an example of the construction of the ℓ−wormhole by the contribution of
individual non-spherical scalar fields, in Fig. 5.2 we show the distribution of the
density at the throat for the fields (φ1Y 1−1,φ1Y 10,φ1Y 11). The values are given
by Eq. (5.5) for the T t
t ∝ ρ component. This is the case for ℓ = 1 wormhole, so
that there are three values for m. The first sphere represents the sum of the
m = −1 and m = 1 contributions, the second one represents the m = 0 field, and
the combination is given in such a way that the total density (the third sphere) is
spherically symmetric.
5.3 Stationary Wormhole equations
In order to obtain the remaining field equations, it is helpful to notice that with
the stress-energy-momentum tensor components given by Eqs. (5.8, 5.9, 5.10) the
following equation is satisfied
T t
t −Tr
r
2
−Tθ
θ =−
c4
8πG
2ℓ+1
8π
(
µ2 −
2ℓ+1
4π
Λχ2
ℓ−
2
a
ω2
)
χ2
ℓ. (5.14)
On the other hand, from the line element Eq. (5.1), we obtain the Bianchi-Einstein
tensor, Gµ
ν, and the same linear combination of the components gives:
G t
t −Gr
r
2
−Gθ
θ =−
1
2R2
d
dr
[
R2 da
dr
]
. (5.15)
Thus, with the aid of Einstein’s equations:
Gµ
ν =
8πG
c4
Tµ
ν, (5.16)
we obtain our next field equation:
1
2R2
d
dr
[
R2 da
dr
]
=
2ℓ+1
8π
(
µ2 −
2ℓ+1
4π
Λχ2
ℓ−
2
a
ω2
)
χ2
ℓ. (5.17)
Notice that when a static, massless scalar field (regular or exotic) without inter-
action is considered, then a particular solution is obtained in which the metric
function a is constant.
Traversable ℓ-wormholes 99
A further field equation comes from the combination of the G t
t component plus
the Gr
r one, and the corresponding Tµ
ν components:
dR2
dr
da
dr
+a
d2 R2
dr2
−2=
2ℓ+1
4π
[
R2
(
µ2 −
2ℓ+1
4π
Λχ2
ℓ
)
+ℓ(ℓ+1)
]
χ2
ℓ. (5.18)
As mentioned above, for the first independent field equation, we consider the
Klein-Gordon equation, Eq. (5.12). In this way we obtain a system of equations in
which each function χℓ(r), a(r) and R2(r) appears as the only second derivative:
χ′′
ℓ = −
(
R2′
R2
+
a′
a
)
χ′
ℓ+
1
a
[
µ2 −
ω2
a
+
ℓ(ℓ+1)
R2
−
2ℓ+1
2π
Λχ2
ℓ
]
χℓ, (5.19)
a′′ = −
R2′
R2
a′+
2ℓ+1
4π
(
µ2 −
2
a
ω2 −
2ℓ+1
4π
Λχ2
ℓ
)
χ2
ℓ, (5.20)
R2′′ =
1
a



−a′ R2′+2+
2ℓ+1
4π
[
(
µ2 −
2ℓ+1
4π
Λχ2
ℓ
)
R2 +ℓ (ℓ+1)
]
χ2
ℓ



(5.21)
where a prime denotes derivative with respect to r. The remaining field equation
is the rr-component of Eq. (5.16) which yields
R2′
2R2
(
a′+
aR2′
2R2
)
−
1
R2
=
2ℓ+1
8π

−aχℓ
2′+
(
µ2 −
2ℓ+1
4π
Λχℓ
2 −
ω2
a
+
ℓ
(
ℓ+1
)
R2
)
χℓ
2

 ,
(5.22)
which can be interpreted as a constraint since it only involves zeroth and first-
order derivatives of the fields. Provided the second-order field equations (5.19, 5.20, 5.21)
are satisfied, the twice contracted Bianchi identity ∇µGµ
r = 0 and ∇µTµ
r = 0 im-
ply that
d
dr
[
aR4
(
Gr
r −
8πG
c4
Tr
r
)
]
= 0, (5.23)
such that it is sufficient to solve Eq. (5.22) at one point (the throat, say).
A particular simple solution arises when a static, spherical, massless scalar
field (regular or exotic) without interaction is considered. In this case, the param-
eters ω, µ, and Λ vanish and considering ℓ= 0, the field equations (5.19–5.22) can
be integrated explicitly [119, 150], see also [157]. The simplest (but not unique)
solution is obtained assuming that the metric function a is constant. This yields
the solution
a = 1, R2 = b2 + r2, χEllis (r)=
p
8πarctan
(
r
b
)
, (5.24)
which has the property that the metric functions a, R2 and the gradient of χEllis
are reflection symmetric about the throat r = 0. In Fig. 5.3, we present the plot
of Ellis’ ghost field and the corresponding energy density. Although the scalar
100 5.3 Stationary Wormhole equations
Figure 5.3: Ghost field and the corresponding density for the original Ellis worm-
hole.
Figure 5.4: Kretschmann scalar and effective potential, Veff = L2/R2 + κ for a
massive particle propagating on the reflection-symmetric Ellis wormhole.
field itself does not decay to zero simultaneously at both asymptotic ends r →
±∞, its gradient does. Since in the massless case the stress-energy-momentum
tensor and equations of motion only depend on the gradient of the scalar field, the
configuration is localized from a physical point of view. Furthermore, we observe
that the density is negative everywhere. The curvature and Kretschmann scalars
are given by Rs,Ellis =− 2b2
(
r2+b2
)2 and KEllis = R2
s,Ellis
(see Fig. 5.4), respectively, and
like the density, they have a fixed sign.
In the following, we consider much more general wormhole solutions in which
the parameters ω, ℓ, µ and Λ do not necessarily vanish. These solutions are ob-
tained by numerically integrating the field equations (5.19, 5.20, 5.21) and tak-
ing into account the constraint (5.22). For simplicity, in this article, we restrict
ourselves to the reflection-symmetric case (although more general wormhole so-
lutions which are asymmetric about the throat could also be considered). These
solutions satisfy the following boundary conditions at the throat, r = 0:
χ′
ℓ(0)= 0, (5.25)
a′(0)= 0, (5.26)
R2′(0)= 0. (5.27)
Traversable ℓ-wormholes 101
Denoting by b := R(0) the areal radius of the throat, the constraint (5.22) yields
the following condition at r = 0:
[
1+
(2ℓ+1)ℓ(ℓ+1)
8π
χℓ(0)2
]
b−2 =
2ℓ+1
8π
(
ω2
a(0)
−µ2 +
2ℓ+1
4π
Λχℓ(0)2
)
χℓ(0)2,
(5.28)
which fixes the radius b of the throat and requires a(0) and χℓ(0) 6= 0 to be chosen
such that
ω2
a(0)
+
2ℓ+1
4π
Λχℓ(0)2 >µ2. (5.29)
Note that this inequality and Eq. (5.20) also imply that a has a local maximum at
the throat. Next, Einstein’s equation (5.21) together with the conditions (5.25, 5.26, 5.27),
implies the relation
1
2
a(0)(R2)′′(0)=
2ℓ+1
8π
[
b2
(
µ2 −
2ℓ+1
4π
Λχℓ(0)2
)
+ℓ(ℓ+1)
]
χ2
ℓ(0)+1. (5.30)
Using Eq. (5.28) this can be simplified considerably,
1
2
a(0)(R2)′′(0)=
2ℓ+1
8π
b2ω2
a(0)
χℓ(0)2, (5.31)
which shows that the throat is indeed a local minimum3 of R2. For ℓ = 0 the
Eqs. (5.28, 5.31) reduce to the corresponding equations in Ref. [121] (see their
equation (18) and the unnumbered equation below it.) Due to Eq. (5.28), one
has two free parameters at the throat, given by χℓ(0) 6= 0 and a(0), say. As can be
checked, the field equations (5.19,5.20, 5.21, 5.22) as well as the conditions (5.28, 5.31)
are invariant with respect to the transformations
(ω, t, r,a,R2,χℓ) 7→
(
ω
p
B
,
p
Bt,
r
p
B
,
a
B
,R2,±χℓ
)
, (5.32)
with B > 0 a real parameter. Therefore, one can fix the value of a(0) to one, say,
and adjust the value of B such that a(r) → 1 for r →∞. In this way, one is left
with just one shooting parameter (χℓ(0)> 0, say) at the throat r = 0.
At r →±∞, we require asymptotic flatness,
χℓ(r)→ 0, (5.33)
a(r)→ 1, (5.34)
R(r)
r
→ 1. (5.35)
3For ω 6= 0 the right-hand side of Eq. (5.31) is positive since χℓ and χ′
ℓ
cannot both vanish at r = 0;
otherwise it would follow from Eq. (5.19) that χℓ vanishes identically. For the special case ω = 0 see
the proof of Theorem 2 below.
102 5.3 Stationary Wormhole equations
Under these assumptions, the field equation (5.19) for the scalar field reduces to
χ′′
ℓ(r)≈ (µ2 −ω2)χℓ(r), (5.36)
which shows that4
ω2 <µ2 (5.37)
is required to have the exponentially decaying solution χℓ(r) ≈ e−
p
µ2−ω2r. Ap-
proximating the (exponentially decaying) right-hand sides of Eqs. (5.17, 5.18) to
zero, one obtains the following behavior of the metric coefficients in the asymptotic
region:
a ≈ e
− c0
r+c1 , R2 ≈ (r+ c1)2e
c0
r+c1 , r →∞, (5.38)
for some constants c0 and c1.
5.3.1 Qualitative analysis of the solutions
Before numerically constructing the wormhole solutions, we make a few general
remarks regarding the restrictions on the parameters ω, µ and Λ and the initial
condition χℓ(0) and regarding the qualitative properties of the solutions. We as-
sume in the following that (χℓ(r),a(r),R2(r)) is a smooth solution of Eqs. (5.12),
(5.17), (5.18) (or, equivalently, of Eqs. (5.19, 5.20, 5.21)) on the interval [0,∞)
which satisfies a > 0, R2 > 0, the boundary conditions (5.25, 5.26, 5.27) at r = 0
and (5.33, 5.34, 5.35) at r → ∞, and is subject to the conditions (5.28, 5.31) at
the throat. We had already observed that an exponentially decaying solution at
infinity requires ω2 ≤µ2. Furthermore, at the throat, the inequality (5.29) needs
to be satisfied. A first immediate consequence of this last inequality is that the
parameters ω and Λ cannot be both zero. In fact, one has the following stronger
result which shows that the self-interaction term is needed.
Theorem 1 There are no reflection-symmetric solutions with the above properties
if Λ= 0.
Proof. We prove the theorem by contradiction. If Λ = 0, the inequality (5.29)
implies that
µ2 <
ω2
a(0)
≤
µ2
a(0)
, (5.39)
which requires a(0) < 1 and ω2 > 0. However, Eq. (5.20) with Λ = 0 implies that
at any point r = rc where the derivative of a vanishes, the equality
a′′(rc)=
2ℓ+1
4π
(
µ2 −
2ω2
a(rc)
)
χℓ(rc)2 (5.40)
holds. Since µ2 −2ω2/a(0) < 0, a has a local maximum at the throat, as already
remarked above, such that a(r) decreases for r > 0 small enough. Since a(r)→ 1 as
4The limiting value ω2 =µ2 is discussed in Ref. [121].
Traversable ℓ-wormholes 103
r →∞ there must be a point r = rc for which a ceases to decrease, corresponding
to a (local) minimum of a. At this point, we must have a(rc)< a(0), a′(rc)= 0 and
a′′(rc)≥ 0. On the other hand, since
µ2 −
2ω2
a(rc)
<µ2 −
2ω2
a(0)
< 0, (5.41)
Eq. (5.40) implies a′′(rc)< 0, provided that χℓ(rc) 6= 0, which leads to a contradic-
tion. If χℓ(rc)= 0, we do not obtain an immediate contradiction since in this case
it follows that a′′(rc)= 0. However, in this case, we must have χ′
ℓ
(rc) 6= 0 since oth-
erwise χℓ (as a solution of the second-order equation (5.19)) would be identically
zero. By differentiating Eq. (5.20) twice with respect to r and evaluating at r = rc
one obtains a′′′(rc)= 0 and
a′′′′(rc)=
2ℓ+1
2π
(
µ2 −
2ω2
a(rc)
)
χ′(rc)2 < 0, (5.42)
which shows that r = rc is a local maximum of a and yields again a contradiction.
This concludes the proof of the theorem.
The next result implies that there cannot be more than one throat.
Theorem 2 Under the assumptions stated at the beginning of this subsection,
the function R2(r) is strictly monotonously increasing and strictly convex on the
interval [0,∞).
Proof. By combining Eqs. (5.21,5.22) one obtains the simple equation
(R2)′′
R2
=
1
2
[
(R2)′
R2
]2
+
2ℓ+1
4π
(
χ′2
ℓ +
ω2
a2
χ2
ℓ
)
(5.43)
for R2, which shows that (R2)′′ ≥ 0 and hence that R2 is convex. We show further
that the right-hand side of Eq. (5.43) cannot vanish at any point. This is clearly the
case if ω 6= 0 since χ′
ℓ
and χℓ cannot vanish at the same point (otherwise it would
follow from Eq. (5.19) that χℓ is identically zero). Next, we rule out the exceptional
case in which ω= 0 and there existed a point r0 ≥ 0 where (R2)′(r0) = χ′
ℓ
(r0) = 0.
If this case occurred, successive differentiation of Eq. (5.43) would yield
(R2)′′′(r0)= 0,
(R2)′′′′(r0)
R2(r0)
=
2ℓ+1
2π
[χ′′
ℓ(r0)]2. (5.44)
Further, evaluating Eq. (5.22) at r = r0 one would obtain
−
1
R2(r0)
=
2ℓ+1
8π
(
µ2 −
2ℓ+1
4π
Λχℓ(r0)2 +
ℓ(ℓ+1)
R2(r0)
)
χℓ(r0)2, (5.45)
implying that χℓ(r0) 6= 0 and that the expression inside the parenthesis on the
right-hand side must be negative. Eq. (5.19) would then imply that
a(r0)
χ′′
ℓ
(r0)
χℓ(r0)
<−
2ℓ+1
4π
Λχℓ(r0)2 < 0, (5.46)
104 5.3 Stationary Wormhole equations
and hence χ′′
ℓ
(r0) 6= 0 and (R2)′′′′(r0) > 0. It follows that any critical point of R2
must be a strict minimum of R2. However, since R2 is convex there can be only
one such critical point which is the one at the throat. Therefore, it follows from
Eq. (5.43) that (R2)′′(r)> 0 for all r > 0 and the theorem is proven.
5.3.2 Numerical shooting algorithm
Next, we describe a shooting algorithm which allows us to find asymptotically flat
wormhole solutions from a given set of initial conditions at the throat by numer-
ically integrating the equations outwards. As discussed above, there is only one
free parameter to start the shooting procedure. Such parameter is the value of
the scalar field at the throat, χℓ(0).
We are looking for the desired solutions in the same spirit as the boson stars
(see for instance [46]), in which the solutions are parametrized by the value of
the scalar field at the center of the configuration so that for each solution a set of
discrete values for the frequency is found to satisfy the asymptotic flatness con-
ditions, each with different number of nodes for the scalar field profile. Qualita-
tively, the same happens with the ℓ-wormhole solutions discussed here. All the
solutions reported in this article are those corresponding to the ground state, in
which the scalar field χℓ has no nodes.
So for given values of a(0), Λ, ℓ, ω, only one particular value of χℓ(0) picks
the χℓ → 0 solution at infinity. We can see this in the approximation of the Klein-
Gordon equation for large r. If we assume that a(r) tends to unity and R(r) to
the coordinate r fast enough, then Eq. (5.36) is satisfied, which is consistent with
exponential decay of χℓ for large r as long as µ2 −ω2 > 0. In a similar way we see
that if χℓ is exponentially decaying at both infinities then, from (5.20) we obtain
d2a
dr2 + 2
r
da
dr
≈ 0, which has solutions:
a ≈ B+
A
r
, (5.47)
where A and B are constants. This is a particular simplification over Eq. (5.38)
that is useful in the numerical procedure. In particular B will enter as a normal-
ization factor, since we will ask for B = 1, as required by the asymptotic condi-
tion (5.34).
As mentioned previously, χℓ(0) is used as the shooting parameter so the re-
quirements needed to find a solution are those described in the previous para-
graphs. Using the Lsoda Fortran solver for initial value problems of ordinary
differential equations, we perform the integration of the system (5.19–5.21) start-
ing at the value r = 0 using steps of ∆r = 1×10−6 until a final value is reached.
This finite value of the asymptotic boundary needs to be sufficiently large for the
functions to reach their asymptotic behavior. Once the desired behavior of χℓ is
Traversable ℓ-wormholes 105
ω= 0 ℓ= 0 ℓ= 1 ℓ= 2
χℓ(0) R(0) a(0) χℓ(0) R(0) a(0) χℓ(0) R(0) a(0)
Λ= 0.5 6.26 1.07 1.40 3.94 1.72 7.43 3.08 2.71 20.21
Λ= 0.7 4.91 1.75 0.80 3.21 1.97 2.59 2.58 2.81 7.04
Λ= 1.0 3.90 2.79 0.58 2.48 2.70 1.03 2.08 3.14 2.51
Λ= 4.0 1.81 13.21 0.41 1.05 13.14 0.42 0.82 13.00 0.44
Table 5.1: Central values of the field χℓ and metric functions R and a for several
values of Λ= 0.5,0.7,1.0,4.0 and ℓ= 0,1,2 with ω= 0.
Λ= 0.7 ℓ= 0 ℓ= 1 ℓ= 2
χℓ(0) R(0) a(0) χℓ(0) R(0) a(0) χℓ(0) R(0) a(0)
ω= 0.1 4.88 1.77 0.83 3.18 2.01 2.60 2.56 2.85 7.03
ω= 0.2 4.64 2.01 1.07 2.95 2.34 2.82 2.36 3.28 7.13
ω= 0.3 4.06 3.01 2.25 2.45 3.67 4.49 1.94 5.01 9.77
Table 5.2: Central values of the field χℓ and metric functions R and a for several
values of ω and ℓ= 0,1,2 with Λ= 0.7.
obtained up to a precision of order ∆r, the asymptotic values of R and a are ad-
justed by means of the transformation (5.32) which leaves the system of equations
invariant, where the parameter B is chosen equal to the corresponding coefficient
in Eq. (5.47).
Examples are shown in Tables 5.1 and 5.2. Their physical implications are
shown in section 5.4. The 0-wormhole recovers the wormhole studied by Dzhunushaliev
et al. in [121] for complex, massive and self-interacting ghost scalar fields. Our
results match those of them as can be seen in the Λ= 4.0 row in Table 5.1 when
the following change of variables is performed:
r 7→
∫r
0
dr
p
a(r)
, Λ 7→
Λ
4
, χℓ 7→
√
8π
2ℓ+1
χℓ, (5.48)
which takes into account the differences in the definitions, nondimensionalization
and the coordinate election. These authors also studied the case for a real scalar
field in a previous work [124], which in fact corresponds to the ω = 0 results in
this paper.
5.3.3 Energy density, mass and curvature scalars
In order to help interpreting the solutions presented in the next section, we dis-
cuss several scalar quantities, like the energy density ρ of the ghost field measured
by static observers, the Misner-Sharp mass function and the scalars related with
106 5.3 Stationary Wormhole equations
the curvature of the spacetime, such as the Ricci scalar Rs and the Kretschmann
scalar K . These quantities will turn out to be helpful for understanding the fea-
tures of the ghost field and its action on the geometry.
Explicitly, the function ρ, associated with the density of the ghost field, is given
by
ρ =−
T t
t
c2
=
(
c2
8πG
)
ρ̂, (5.49)
where ρ̂ is defined by
ρ̂ =−(2ℓ+1)

aχ′2
ℓ +
(
ℓ(ℓ+1)
R2
+µ2 −
2ℓ+1
4π
Λχ2
ℓ+
ω2
a
)
χ2
ℓ

 . (5.50)
A striking feature of the wormhole solutions is that despite the presence of the
exotic matter which violates the null energy condition everywhere, the density
may still be positive at the throat,5 as will be shown in the numerical examples
discussed in the next section. In fact, using Eq. (5.22) one can obtain the following
simple expression for ρ̂ at the throat:
ρ̂(0)=
8π
b2
−2(2ℓ+1)
ω2
a(0)
χℓ(0)2, (5.51)
which shows explicitly that for those solutions with ω= 0 the energy density is in-
deed positive near the throat. The plots in the next section show that this behavior
also holds for other solutions with small enough values of ω2.
The total (ADM) mass of the wormhole configurations can be computed from
the asymptotic limit M∞ = lim
r→∞
M(r) of the Misner-Sharp mass function M(r),
defined by
2GM
c2
= R
[
1− gµν(∇µR)(∇νR)
]
= R
(
1−aR′2
)
. (5.52)
From Eqs. (5.38, 5.47) one obtains GM∞/c2 = c0/2 = −A/2. Alternatively, using
Eqs. (5.21, 5.22) one also obtains M′ = 4πρR2R′ which can be integrated to
M(r)=
c2
2G



b+
r
∫
0
ρ̂(r̄)R2(r̄)R′(r̄)dr̄



, (5.53)
with b = R(0) the throat’s areal radius. As long as ρ̂ is positive near the throat,
the mass function increases as one moves away from the throat. However, M
decreases as soon as ρ̂ becomes negative, so that solutions which have either sign
of the total mass are possible. This is shown in Table 5.3, where values of the
total mass M∞ for our wormhole were computed taking several values of ℓ and
fixing the values of all other parameters.
5Note that the violation of the null energy condition implies the violation of the weak energy con-
dition, which means that there exists at least one observer which measures negative energy density.
Our example shows that this observer does not necessarily need to be a static one.
Traversable ℓ-wormholes 107
ω= 0, Λ= 1 M∞ (m2
pl/mΦ)
ℓ= 0 0.677
ℓ= 1 0.224
ℓ= 2 −1.25
ℓ= 3 −4.69
Table 5.3: Total mass values for ℓ-wormholes with Λ= 1, ω= 0 and ℓ= 0,1,2,3.
The Ricci scalar, Rs = Rµ
µ, associated with the geometry given by Eq. (5.1)
has the form
Rs =−a′′−2a
R2′′
R2
+a
(
R2′
R2
)2
+
2
R2
(
1−a′ R2′
)
. (5.54)
A further commonly used curvature measure is the Kretschmann scalar, defined
by K = Rµνστ Rµνστ. For the metric under consideration, Eq. (5.1), the Kretschmann
scalar has the following explicit form:
K = a′′+2
(
aR2′′
R2
)2
+2a
R2′′ R2′
R4
(
a′−
aR2′
R2
)
+
4
R4
+


3
4
(
aR2′
R2
)2
−
a2′ R2′
2R2
+a′2 −2
a
R2


(
R2′
R2
)2
.
(5.55)
All these quantities will turn out to be helpful when understanding the role
played by the several parameters of the solution in the geometry and in the dy-
namics of the bodies moving on it.
From Einstein’s equations, Eq. (5.16), we have that Rs = − 8πG
c4 T with T the
trace of the stress-energy-momentum tensor. Numerical experiments show that
the behavior of the stress-energy-momentum tensor components in the throat re-
gion are similar to each other, and thus Rs ≈ 8πG
c2 ρ, as seen in the actual solutions.
That is, the Ricci scalar goes as the density, irrespective of its character, exotic or
usual matter. We will discuss this fact in more detail in the explicit cases that we
present below.
5.4 Numerical wormhole solutions
Following the procedure described above, we are able to obtain several solutions to
the Einstein-Klein-Gordon system, given the four parameters, namely µ,ω,Λ, and
ℓ. We will present the solutions first for trivial values of the angular momentum
parameter, ℓ = 0, and vary the self-interaction parameter Λ, while keeping the
oscillation frequency ω fixed and then we explore the properties of the solution
108 5.4 Numerical wormhole solutions
Figure 5.5: Solitonic profile for the 0-wormhole, for Λ ∈ [0.5,4.0], and ω= 0.
for some values of ω maintaining Λ fixed, as was done in [121, 124]. Next, we
repeat the study for different values of ℓ. In all our solutions presented in this
work, we keep the mass of the scalar field µ fixed. These experiment allow us to
have a better understanding on the role that each parameter plays in determining
the geometry of the solutions.
All the solutions presented are asymptotically flat, and are generated by look-
ing for a solution of the ghost scalar field, once the parameters ℓ,Λ and ω are
chosen. We fix the value of mass parameter µ to one, and the distance scale of the
solution is given by the dimensionless parameter r̂ =µr. Also, from Eq. (5.28) we
see that the size of the wormhole throat R(0) is given by
R(0)=



ℓ(ℓ+1)+ 8π
(2ℓ+1)χ2
ℓ
(0)
2ℓ+1
4π
Λχ2
ℓ
(0)+ ω2
a(0)
−µ2



1/2
. (5.56)
In Fig. 5.5, we present this localized solution, for the case ℓ = 0, ω = 0, for
several values of Λ. All the other solutions with ℓ > 0 are localized as well. No-
tice how the amplitude of the pulse decreases as the value of the self-interaction
parameter Λ increases.
In our experiments, we see that the ghost density, in order to form a worm-
hole, is distributed in such a way that it has a positive value in the region of the
throat, and then it starts to have larger concentrations of negative ghost density
on both sides of the throat, as shown in Fig. 5.6. From the geometric perspective,
as suggested above, the profile of the Ricci scalar follows the density one and has
a convex region at the throat, surrounded by concave zones, see Fig. 5.8.
Also we will show that, in general, as can be seen in Fig. 5.6, the action of the
self-interaction parameter, Λ, smooths out this behavior of the exotic density and
spacetime interaction. Indeed, the scalar field, at least the massive ghost field,
possess a radial pressure that creates the throat and then the spacetime strongly
reacts generating regions of negative density; it is the role of the self-interaction
term to smooth down such reaction and allows to keep the wormhole throat open
Traversable ℓ-wormholes 109
Figure 5.6: Density profile for the 0-wormhole, for Λ ∈ [0.5,4.0] and ω= 0.
Figure 5.7: Metric coefficients for Λ ∈ [0.5,4.0] and ω= 0.
with smaller amount of ghost density. Conversely, as the self-interaction parame-
ter Λ becomes smaller, the metric coefficient a, the curvature scalars and density
at the throat become more and more localized, an observation which is compati-
ble with the result in Theorem 1 where we have shown that the solutions cease to
exist for Λ= 0.
5.4.1 0-wormhole
We start our discussion for the case with vanishing angular momentum, i. e.
ℓ= 0. Setting also ω equal to zero for the moment, we start by sweeping a range
of values for the self-interaction parameter, Λ. The corresponding results for the
scalar field and the density profile are shown in Figs. 5.5 and 5.6, respectively. As
mentioned above, the ghost density has regions of positive magnitude near the
throat, and regions with negative density which tend to zero from below in the
asymptotic region.
The corresponding metric coefficients, a(r) and R(r) are shown in Fig. 5.7. No-
tice how the metric coefficient a(r) shows concave regions which will determine a
similar behavior in the effective potential of the spacetime, which in turn will im-
ply the existence of particles moving on bound trajectories. Again, the effect of the
self-interaction parameter is to smooth out the concavity of the metric functions.
Regarding the curvature scalars, as expected, the Ricci scalar Rs has a be-
110 5.4 Numerical wormhole solutions
Figure 5.8: Ricci and Kretschmann scalars for Λ ∈ [0.5,4.0] and ω= 0.
Figure 5.9: Density profile, curvature scalar and metric function a for the 0-
wormhole, Λ= 0.7 and ω= 0.1,0.2,0.3.
havior which follows the one of the density, with regions of positive values and
then valleys with negative values of the curvature as we can see in Fig. 5.8. The
Kretschmann scalar, however, is very different and shows two peaks of positive
values and they decrease as the self-interaction parameter grows, and the central
one is negative in the region of the throat, surrounded by bumps.
The next step is to increase the parameter ω keeping ℓ = 0 and the self-
interaction parameter Λ= 0.7 fixed. We show in Fig. 5.9 the corresponding den-
sity and Kretschmann scalar for three non-zero values, ω = 0.1,0.3,0.5, of the
frequency. Notice the difference between the behavior of the Kretschmann scalar,
in which a larger value of ω gives the effect of increasing the central value, acting
in the same way as the parameter Λ discussed above.
Traversable ℓ-wormholes 111
Figure 5.10: Metric function a(r), density profile ρ̂ and Rs, K scalars for ℓ ≥ 0,
Λ= 1.0 and ω= 0.
5.4.2 ℓ-wormhole
In this section we present the behavior of the ℓ parameter and the effect on the
metric functions and the curvature scalars. For the latest we can see in Fig. 5.10
that an increment on the ℓ parameter increases the central peak for the Ricci
scalar and decreases the central bump for the Kretschmann scalar. The case for
a(r) is quite different: while for ℓ= 1 the two minima are still present, for larger
values of the ℓ parameter the central peak is increased and the minima disappear.
The presence of a minimum (or two in this case) also corresponds to positive total
masses as can be verified in Table 5.3 and Fig. 5.11, consequently, its absence
corresponds to negative masses. This is a general property of all solutions given
the asymptotic behavior of a (see Eq. 5.47).
On the other hand, as is shown in Fig. 5.12, the increment of the ω parameter
plays a role quite similar to the one made by the ℓ parameter: an increase on the
former elevates the central peak on the metric function a(r).
Moreover, as can be seen from a comparison of Figs. 5.7 and 5.10, the effect
of the Λ parameter on the metric coefficient a is opposite to the one generated
by the ℓ parameter on that metric coefficient. Indeed, for small values of Λ, the
metric coefficient a has a global maximum at the throat, while for large values of
this Λ, the metric coefficient a only has a local maximum. Thus, for small values
of Λ, the ℓ parameter is not able to change the qualitative behavior of the metric
coefficient, while for larger values of Λ, the appearance of the local maximum is
112 5.5 Embedding diagrams and geodesic motion
Figure 5.11: Mass function M(r) for the parameters ℓ≥ 0, Λ= 1 andω= 0.
Figure 5.12: Metric function a for different values of the parameters ℓ, ω andΛ.
recovered or enhanced with the parameter ℓ. This fact will have consequences on
the effective potential and the geodesic motion of particles, as discussed below.
5.5 Embedding diagrams and geodesic motion
In order to gain a better understanding of the configurations described by the
scalar field and the geometry in the vicinity of the throat, in this section we discuss
the embedding procedure and geodesic motion. Because the metric (5.1) is static
and spherically symmetric, it is sufficient to analyze the induced geometry on a
t = constant and θ =π/2 slice, described by the two-metric
dΣ2 = a−1 dr2 +R2 dϕ2 . (5.57)
In order to visualize this geometry as a two-dimensional surface embedded in
three-dimensional flat space we shall employ cylindrical coordinates (ρ, ϕ, z). The
metric for a flat space in these coordinates is
dS2 = dρ2 +ρ2 dϕ2 +dz2 . (5.58)
We seek for the functions ρ(r) and z(r), specifying a surface with the same geom-
etry as the one described by the metric (5.57).
Traversable ℓ-wormholes 113
Figure 5.13: Embedding of the different ℓ-wormholes, for Λ = 1.5,ω = 0. The
complete embedding diagram is obtained by rotating this figure about the z axis.
In the enlarged picture we underline the change of the value of the throat radius
for the given values of ℓ.
The line element for the embedding surface will be
dΣ2 =
[
(
dz
dr
)2
+
(
dρ
dr
)2
]
dr2 +ρ2dϕ2 , (5.59)
if the following conditions are satisfied:
ρ = R , (5.60)
and
(
dz
dr
)2
+
(
dρ
dr
)2
=
1
a
. (5.61)
Using the expression (5.60) to calculate dρ
dr
, Eq. (5.61) gives the following dif-
ferential equation for z(r):
dz
dr
=
[
1
a
−
1
4
(R2′)2
R2
]1/2
. (5.62)
Integrating this equation gives the function z = z(r); in order to plot it in an
Euclidean space, we need to find r as a function of ρ. However, it is not possible
to express this function r = r(ρ) in closed form because R was found numerically.
Nevertheless, one can obtain r = r(ρ) numerically from (5.60) and finally get z =
z(ρ).
In Fig. 5.13 we show the visualization of this embedding. It is seen that as ℓ
increases from 0 to 2, the profile of of the embedding representing the wormhole’s
geometry becomes more and more curved (which is analogous to the increase of
|Rs| and |K | shown in Fig. 10) , with a slight decrease in the throat’s radius.
114 5.5 Embedding diagrams and geodesic motion
Figure 5.14: Effective potential for time-like (left panel) and null geodesics (right
panel) for L =
√
1
10 .
5.5.1 Geodesic motion
In order to describe the motion of the particles in the spacetimes described above,
we start from the Lagrangian for the metric (5.1),
L = gµνuµuν+κ c2 =−ac2(u0)
2+a−1(ur)
2+R2[(uθ)
2+sin2θ(uϕ)
2
]+κ c2, (5.63)
where uµ = ẋµ is the four velocity and the parameter κ assumes the values 1 or
0, depending on whether the particle is massive or massless. The line element
is spherically symmetric and static, so that the energy, E =− ∂L
∂u0 , the azimuthal
momentum, Lϕ = ∂L
∂uϕ , and the total angular momentum, L2 =
(
∂L
∂uθ
)2
+ Lϕ
2
sin2 θ
, are
conserved quantities. Explicitly, they have the form:
E = −
∂L
∂u0
= ac2u0, (5.64)
Lϕ =
∂L
∂uϕ
= R2 sin2θuϕ. (5.65)
Since we are only interested in the motion of a single particle (as opposed to a
swarm of particles) we can choose the angles such that the orbital plane coin-
cides with the equatorial plane θ = π/2, in which case Lϕ = L. In this way, we
can express the components of the four-velocity in terms of the conserved quanti-
ties L and E, and the normalization condition gµν uµ uν =−κ c2 yields the radial
equation of motion:
(ur)2 +Veff =
E2
c2
, (5.66)
with the effective potential
Veff =
a L2
R2
+aκ c2. (5.67)
In Fig. 5.14 we plot Veff for the parameter choices L =
√
1
10
, ω= 0 and Λ= 1.5
for time-like and a null geodesics. As expected, the term involving L generates
Traversable ℓ-wormholes 115
Figure 5.15: Different ℓ= 0 (top), ℓ= 1 (middle) and ℓ= 2 (bottom) geodesics for
κ= 1, ω= 0, Λ= 1.5, L =
p
1/10. On the left we plot the motion of the particle in
the embedding surface of the wormhole and on the right the value of E and Veff.
an angular momentum barrier, corresponding to a local maximum of the effective
potential located at the throat r = 0. (Recall from Section 5.3 that a(r) has a local
maximum while R2(r) has a local minimum at r = 0.) This maximum corresponds
to an unstable equilibrium point giving rise to circular unstable particle orbits.
For the ℓ = 0 case, and for this value of L and with κ = 1, the effective potential
also has a minimum at r ≈ ±1.34, which means that bound orbits also exist for
this value of L.
In Fig. 5.15 we plot different geodesics for massive particle with L =
p
1/10 and
Λ = 1.5 in the ℓ = 0,1,2 wormholes. Here we picked the same initial conditions
in terms of the initial radial velocity ur(0) = 0 and initial position r(0) = 0.65,
ϕ(0)=π ending up with particles with different energies and qualitatively differ-
ent motion. As stated above, one can assume without loss of generality that the
motion is confined to the equatorial plane θ = π
2
, so that it can be plotted in the
embedding surface. As can be noticed from the plots, the motion is quite interest-
ing and can be understood based on the behavior of the effective potential and the
energy level of the test particle.
In order to further clarify the behavior of the geodesics, in Fig. 5.16 we plot
the trajectory in the embedding diagram and the radial velocity ur for two dif-
116 5.6 Discussion and concluding remarks
(a) A particle having insufficient energy to pass through the wormhole. The motion starts
at r = 2.6, the velocity decreases until the particle arrives near the throat after which it
returns, moving away with increasing speed. In this case the particle stays on the same
side of the wormhole and does not cross the throat.
(b) A particle starts its motion at r = 2.6, traverses the throat, and continues its motion with
a growing absolute value of the velocity.
Figure 5.16: Two different ℓ = 1 geodesics for κ = 0, L =
p
1/10, ω = 0, Λ = 1.5.
We plot the motion with their corresponding E (right y-axis) and ur (left y-axis)
values.
ferent null geodesics with angular momentumL =
p
1/10 propagating in the ℓ= 1
wormhole. In the first case, shown in Fig. 5.16a, the particle does not have suf-
ficient energy to traverse the throat so it starts approaching the throat with a
decrement of the velocity, reaches a zero radial velocity and resumes its motion
going away from the throat. On the other hand, the second example in Fig. 5.16b
shows that, for a particle that has enough energy to pass through the throat, the
absolute value of its velocity decreases as it moves towards the throat (from right
to left) until it traverses the throat, after which the absolute value of the velocity
increases again as the particle moves away from the throat on the other side of
the wormhole.
5.6 Discussion and concluding remarks
We have described how to construct new families of traversable wormhole solu-
tions which are parametrized by a parameter ℓ, related to the angular momentum
of the ghost fields supporting the throat, and discussed its effects on the shape of
Traversable ℓ-wormholes 117
the geometric functions characterizing the solution, and on the geodesic motion of
the corresponding spacetime. These families generalize previous wormhole space-
times discussed in the literature [121, 124] which are recovered from our models
by setting ℓ= 0.
Indeed, we have obtained bona fide solutions to the Einstein-Klein-Gordon
system and performed a detailed analysis of such solutions, which allowed us to
gain a better understanding on the effects of the new parameter ℓ. We have been
able to establish that its role on the geometric function a (determining the redshift
factor), on the curvature scalars and on the density is quite similar to the role
played by the frequency ω characterizing the time-dependency of the field, while
its effect on these quantities is opposite to the one generated by the parameter
of self-interaction Λ. Moreover, as can be clearly seen in the plot of the effective
potential for time-like geodesics shown in Fig. 5.14, as the value of ℓ grows, the
positions of the local minima move farther away from the throat, which is similar
to the effect of increasing the angular momentum L of the test particles. In this
sense, from the point of view of the test particle, the parameter ℓ plays a similar
role than its conserved total angular momentum L.
It is interesting to point out that the energy density of some of our solutions –
despite of the fact that the stress-energy-momentum violates the null energy con-
dition – is actually positive close to the throat (but changes its sign as one moves
away from it and then converges to zero which is consistent with our asymptotic
flatness asymptions). In fact, the construction of a wormhole does not necessar-
ily require measurements of a negative energy density made by static observers,
as already indicated in [140]. However, the fact that the null energy condition
is violated at the throat implies that such observers also measure a “superlu-
minal" energy flux. In general, the wormhole solutions discussed in this article
possess a much richer structure than the simple, reflection-symmetric Bronnikov-
Ellis wormholes, whose energy density is everywhere negative. In particular, the
spacetimes discussed here exhibit a rich profile of bumps and wells in their cur-
vature scalars whose precise shape depends on the values of ℓ as much as it does
on the other parameters.
Indeed, we presented a detailed analysis of the role played by the several pa-
rameters in our wormhole solutions, namely the self-interaction term Λ, the oscil-
lating frequency, ω, and the angular momentum parameter, ℓ of the scalar fields.
Moreover, we have proved that there are no solutions for which the metric and
scalar fields are reflection-symmetric about the throat if Λ = 0 (see Theorem 1).
In this sense, the self-interaction term needs to be included in the action in order
to extend the solution space. Actually, we have seen that it plays a smoothing role
in the geometric reaction to the ghost matter. Also, we have seen that the effects
on the geometry of the self-interaction parameter is opposite to the effects due
118 5.6 Discussion and concluding remarks
to the frequency ω. As mentioned previously, the role of the ℓ parameter on the
geometry is similar to the one generated by the frequency. This fact can be used
to obtain real scalar field wormholes, with a new degree of freedom analogous
to the case in which the solution space is extended by permitting the scalar field
to be complex and harmonic in time. As shown in Theorem 2, all our wormhole
solutions are characterized by a single throat whose areal radius is fixed by the
parameters and the value of the scalar field at the throat, see Eq. (5.56).
We also provided a study of the effects of the parameters on the embedding dia-
grams visualizing the spatial geometry of the solutions, including the shape of the
throat for several relevant cases. Finally, we presented a detailed analysis of the
effective potential describing the motion of free-falling test particles as a function
of the parameters, and we showed how the potential may present a local maximum
at the throat which is surrounded by regions with local minima. Accordingly, we
obtained several interesting types of trajectories. Depending on the values of the
parameters and on those of the constants of motion (namely, the energy and angu-
lar momentum of the particle), we displayed trajectories approaching the throat
until they reach a turning point and go back, other trajectories which describe
bound motion on either side of the throat, and then we even obtained orbits that
are bound but cross the throat repeatedly and keep passing from one side of the
Universe to the other; a nice property for a space station!
In the plots of Fig. 5.16 we have shown the behavior of the geodesics passing
through the throat, we presented the absolute value of the particle’s radial ve-
locity and showed that it decreases as the particle approaches the throat until it
crosses it after which it increases again as the particle gets further away from
the throat. Such behavior is consistent with the interpretation that the reaction
of the geometry to the ghost matter is to create bumps in the effective potential,
instead of the wells generated by the usual matter. As mentioned at the begin-
ning of Section 5.2, there is no need to invoke negative masses to explain such
behavior; it is simpler to imagine that the reaction of the geometry to the ghost
matter is to create bumps that the particle have to surmount, consistent with the
fact that the absolute value of the velocity decreases as it approaches the throat,
and then, goes down the hill.
The new configurations we have found and discussed in this article consider-
ably extend the parameter space describing wormhole solutions of the Einstein-
scalar field equations, and they provide a large arena that offers the possibility
to further study the intriguing properties of wormhole spacetimes, including the
relation between the properties of exotic matter and their geometry. While it has
been shown that the solutions ℓ= 0 are linearly unstable [121,157,158], there is
hope that such a large arena may contain a set of parameter values with ℓ > 0
describing stable wormholes or unstable wormholes.
Part II
Axisymmetric solutions

CHAPTER 6
MAGNETOSTATIC BOSON STAR1
Contents
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2.1 Field equations . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.2 Global quantities . . . . . . . . . . . . . . . . . . . . . . . 125
6.2.3 Static axisymmetric spacetime and Ansätze for the fields126
6.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3.1 Boundary conditions and numerical method . . . . . . 127
6.3.2 Structure of the stars . . . . . . . . . . . . . . . . . . . . 129
6.3.3 Sequence of magnetic boson stars . . . . . . . . . . . . . 131
6.4 Magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
W
e solve the Einstein-Maxwell-Klein-Gordon system of equations and de-
rive a compact, static axially symmetric magnetized object which is
electrically neutral and made of two complex massive charged scalar
fields. We describe several properties of such solution, including the torus form
of the matter density and the expected dipolar distribution of the magnetic field,
with some peculiar features in the central regions. The solution shows no diver-
gences in any of the field and metric functions. A discussion is presented on a case
where the gravitational and magnetic fields in the external region are similar to
those of neutron stars.
6.1 Introduction
Boson stars are self-gravitating solitons made up of complex scalar field. This ob-
jects are interesting for various reasons. They serve as simple models for compact
1This chapter is extracted with minor revisions from Ref. [128], which was written in collaboration
with D. Núñez. © American Physical Society. Reproduced with permission. All rights reserved.
121
122 6.1 Introduction
objects in the interplay of Field Theory and General Relativity and are also inter-
esting on their own since they possess important dynamical properties that allow
to study them in hypothetical strong gravity astrophysical scenarios, for example,
in the gravitational waveform research; see e.g. [95]. Bosonic stars have applica-
tions also as black hole mimickers [159,160] and, more generally, scalar fields are
relevant in cosmology as quintessence [161], ultralight dark matter [25–27] and
as a source of inflation [162].
Static and stationary single field configurations have been presented in the
literature in the past years, for instance, the free and massive bosonic field so-
lution of the Einstein-Klein-Gordon equations in spherical symmetry [44] can
be generalized gauging the global U(1) symmetry, which leads to the charged
version of boson stars [163]. Rotating generalizations of boson stars were ob-
tained in [164] within the Einstein-Klein-Gordon setup and in the Einstein-Klein-
Gordon-Maxwell extension, the charged rotating one [165]. In these models the
coupling constant parameter is freely specifiable, however, in order to obtain equi-
librium solutions, the value of the coupling parameter ranges from zero up to a
critical value [163,165].
Staying within the complex, massive, free scalar field case there are two more
characterizations that can be found in the literature up to the present time, these
are the multipolar boson stars [166], which are static nonspherical configurations
with similar morphologies to the probability density of atomic orbitals and the
multifield (ℓ-) boson stars [99, 103, 104], in which the U(1) symmetry is general-
ized by considering a U(N) symmetry. Of particular interest in this paper is the
toroidal static boson star of [104], which can be understood as the superposition
of two contrarotating solutions that give rise to a static equilibrium configuration.
The rotating charged boson stars share some properties with the uncharged
rotating case, such as the toroidal shape and with the charged (electrostatic)
case, such as the critical value of the coupling constant. As one would expect
the charged and rotating general solutions include electric charge and magnetic
dipole moment with the particularity that the magnetic moment is nonzero only
if the electric charge is nonzero, therefore obtaining solutions where both elec-
tric and magnetic fields are present in the local inertial frame of zero angular
momentum. Until now, no electrically neutral and magnetized self-gravitating
bosonic stars have been constructed, which might be relevant models in the study
of strong gravity and magnetic fields phenomena.
Magnetic fields play an important role in many astrophysical scenarios. Some
of the relativistic applications involve compact, electrically neutral objects and
strong magnetic fields where their self-gravitation must be taken into account.
Fully relativistic and self-consistent models of neutron stars with magnetic fields
where first presented in [167] (see also e.g. [168,169] for poloidal and [170–172] for
Magnetostatic boson star 123
toroidal magnetic fields). These are numerical solutions of the Einstein-Maxwell-
Euler system in axial symmetry which can possess angular momentum and total
electric charge. Even the globally neutral and static cases are deformed by the
effect of the magnetic field, changing in consequence the global properties of the
equilibrium configurations.
The purpose of this paper is to construct and study magnetostatic solutions of
boson stars with zero total electric charge. To do so we obtain a generalization of
the toroidal static boson stars by coupling the scalar fields to the electromagnetic
field, we shall show that the gauge coupling parameter can exceed the critical
value obtained for the electrostatic boson stars. In Sec. 6.2 we present the action
for the model, the ansätze for the fields and the conserved quantities. Then, in
Sec. 6.3 we describe the numerical procedure and the construct sequence of solu-
tions with different values of the coupling parameter, discussing the physical prop-
erties of the configurations. The magnetic field for the constructed solutions and
a comparison with strongly magnetized neutron stars is presented Sec. 6.4. We
conclude our manuscript and give some perspectives of future works in Sec. 6.5.
In Appendix 6.6 we give the complete set of elliptic partial differential equations
of our model together with explicit expressions for the 3+1 decomposition of the
energy momentum tensor. We work with c = G = 1 and the metric signature is
taken to be (−,+,+,+).
6.2 Model
A single electrically charged scalar field in spherical symmetry allows to construct
charged boson stars, which are static solutions that give rise to an electric field
as measured by an observer at rest [163]. Even the rotating generalization of this
configurations, which also generate a magnetic field, possess an electric field that
does not vanish [165]. There are no immediate simple models consisting of one
scalar field that give rise to magnetostatic self-gravitating solutions, however the
multifield approach leads to a natural way of obtaining such boson stars.
On the other hand, although fully relativistic neutron stars with magnetic
fields have been constructed, all the solutions (to the best of our knowledge) have
been obtained using the free current assumption, i.e., electromagnetic sources Jµ
independent to the fluid movement. This means in particular that in the (mag-
neto)static solutions the fluid is at rest while the spatial components of the electric
current are nonzero. This is a limiting assumption, and as pointed out in [169], in
principle the currents should be derived from a microscopic model which“would
require a multifluid approach to model the movements of free protons and elec-
trons”.
For boson stars, in the Einstein-Klein-Gordon-Maxwell framework, the free
124 6.2 Model
current assumption cannot be even made since the electric current is determined
by the scalar fields, however the multifield approach is feasible. In our approach
globally neutral configurations are constructed by superposition of two contraro-
tating “thick current loops” made of charged scalar fields.
The general framework in which magnetized boson stars are constructed con-
sists of two self-gravitating complex scalar fields minimally coupled to the electro-
magnetic four-potential with coupling constants of opposite sign. In this section
we summarize the basic equations needed to construct the solutions and the con-
served quantities that will be useful in the analysis.
6.2.1 Field equations
We consider two massive complex scalar fields, Φ(1) and Φ(2), minimally coupled
to the Einsteinian gravity and to the electromagnetic field,
S =
∫
d4x
p
−g


R
16π
−
1
2
2
∑
j=1
(
gµν(D
( j)
µ Φ( j))(D
( j)
ν Φ( j))
∗+µ2|Φ( j)|2
)
−
FµνFµν
4µ0

 ,
(6.1)
where Fµν = ∂µAν−∂νAµ is the Faraday tensor and the covariant derivative op-
erators, D
(1)
µ =∇µ+ iqAµ and D
(2)
µ =∇µ− iqAµ couple both scalar fields with Aµ.
Notice that we have chosen equal mass terms for both scalar fields and opposite
signs for the electromagnetic coupling constants (boson charges). The scalar fields
interact with each other indirectly, through gravity and the electromagnetic field.
Variation of Eq. (6.1), with respect to the different fields leads to the Euler-
Lagrange equations of the model (see e.g. [173]). Variation with respect to gµν
leads to
Rµν−
1
2
R gµν = 8πTµν; (6.2a)
Tµν = T(1)
µν +T(2)
µν + (TEM)µν, (6.2b)
T
( j)
µν :=
1
2
(D
( j)
µ Φ( j))(D
( j)
ν Φ( j))
∗+
1
2
(D
( j)
ν Φ( j))(D
( j)
µ Φ( j))
∗ (6.2c)
−
1
2
gµν
(
gαβ(D
( j)
α Φ( j))(D
( j)
β
Φ( j))
∗+µ2|Φ( j)|2
)
,
(TEM)µν :=
1
µ0
FµσFνλgσλ−
1
4µ0
gµνFαβFαβ. (6.2d)
The equation for the fields Φ(1) and Φ(2) are the Klein-Gordon equations,
gµνD
( j)
ν D
( j)
µ Φ( j) =µ2
Φ( j). (6.3)
Variation with respect to Aµ leads to the Maxwell equations with source the
Magnetostatic boson star 125
charged scalar fields which define a current four-vector Jµ,
∇νFµν =µ0Jµ :=µ0(q j
µ
1
− q j
µ
2
); (6.4a)
j
µ
i
:=
igµν
2
(Φ(i)
∗D(i)
ν Φ(i) −Φ(i)(D
(i)
ν Φ(i))
∗), (6.4b)
here Jµ is the (total) electromagnetic current.
6.2.2 Global quantities
The spacetime we will consider in this work is stationary (static) and axisym-
metric. Komar expressions allow to calculate global quantities for each of this
isometries; if ξ is the Killing vector associated with stationarity, Σt is a spacelike
surface and nµ the unit vector normal to this hypersurface, then the quantity,
M =
1
4π
∫
Σt
RµνnµξνdV , (6.5)
defines the Komar mass. Similarly, if χ is the Killing vector associated with the
axial symmetry, the quantity
J =
1
8π
∫
Σt
Rµνnµχν. (6.6)
gives the angular momentum of the spacetime.
The quantities j
µ
1
and j
µ
2
defined in Eq. (6.4) are Noether density currents
(∇µ jµ = 0) which arise from the invariance of Eq. (6.1) under the U(1) gauge trans-
formation of Φ(1) and Φ(2). It follows that integration of the projection onto nµ of
this currents over Σt leads to the conserved particle numbers
N1 =
∫
Σt
j
µ
1
nµdV , N2 =
∫
Σt
j
µ
2
nµdV ; N :=N1 +N2. (6.7)
In the rotating boson stars, it was shown [117, 164, 174] that the angular
momentum J takes values that are integer multiples of the particle number,
J = mN , with m the winding number (defined below) of the scalar field ansatz,
this result is also valid in the charged rotating case [165]. However, in the mag-
netized solutions obtained in this work this relation does not hold since they are
by construction J = 0 static, as will be argued in the next section.
The associated total electric charge, related to the sources at the Maxwell
equation is given by Q =
∫
Σt
JµnµdV = q(N1 −N2). In the single field static and
rotating charged boson stars the total charge of the system is related to the par-
ticle number by Q = qN and it was obtained [163] that Q coincides with the
asymptotic value extracted from the electric potential and matches the exterior
Reissner-Nordström solution. Again, the relation Q = qN is not valid for our case
because, as we will see bellow, the obtained solutions satisfy Q = 0.
126 6.2 Model
6.2.3 Static axisymmetric spacetime and Ansätze for the fields
In coordinates adapted to the Killing fields, where ξ= ∂/∂t and χ= ∂/∂ϕ, the gen-
eral static and axially symmetric line element we will consider is in the Lewis-
Papetrou form,
gµνdxµdxν =−e2F0 dt2 + e2F1 (dr2 + r2dθ2)+ e2F2 r2 sin2θdϕ2, (6.8)
where the metric functions F0, F1 and F2 depend only on the coordinates r and θ.
We have used the same line element as the one in Ref. [104], where the toroidal
static boson star is constructed, however, the gtϕ term usually written as the
function w(r,θ) or w(r,θ)/r, is not included in (6.8) because we are looking for
static configurations with zero total angular momentum J, and in this case it can
be seen [40,175] that w = 0 if and only if the spacetime is static.
The contribution of the scalar fields in the energy-momentum tensor, T
(1)
µν ,
T
(2)
µν will be consistently independent of t and ϕ if for the scalar fields we use the
following ansatz, which is similar to the one used for rotating, multifield, multi-
frequency boson stars and even for chains [176],
Φ(1) =φ(r,θ)eiωt−imϕ; Φ(2) =φ(r,θ)eiωt+imϕ. (6.9)
Here m is and integer called winding number. Moreover, the opposite sign
of this parameter for each field in Eq. (6.9) can be interpreted as having contra-
rotating scalar fields distributions. It is not difficult to obtain that with this elec-
tion of winding numbers, T
(1)
tϕ = −T
(2)
tϕ , consistent with the Einstein tensor com-
ponent G tϕ being zero for the metric in Eq. (6.8).
Again, analyzing the components of the Einstein tensor we can elucidate the
anzatz for the field Aµ. Two possibilities for the electromagnetic four-potential
are compatible with the spacetime at hand: the purely poloidal (Ar = Aθ = 0) and
the purely toroidal (At = Aϕ = 0) magnetic fields2 [170, 171], however only the
first possibility can be realized given the ansatz (6.9) chosen for the scalar fields
since only the Jϕ source of the Maxwell equations is nonzero3, which additionally
implies that At = constant, therefore we adopt
Aµdxµ = C(r,θ)dϕ. (6.10)
The resulting number density currents, given in Eqs. (6.36) and (6.36), imply
Q = 0 since Jµnµ = 0.
2In both cases, the circularity property of spacetime is not broken and the metric tensor takes the
form (6.8).
3In the single field charged rotating star, also the J t component is nonzero, in our case however
jt
1
= jt
2
, see Appendix 6.6.
Magnetostatic boson star 127
6.3 Numerical solutions
6.3.1 Boundary conditions and numerical method
In order to construct magnetostatic solutions of boson stars, the Einstein-Klein-
Gordon-Maxwell system is solved. This means solving for the five functions
{φ,C,F0,F1,F2}
and the unknown parameter ω, imposing the appropriate symmetries and bound-
ary conditions. The full elliptic system of coupled partial differential equations
(PDEs) in r and θ is given in the Appendix 6.6.
First, we impose even parity with respect to reflections at the equatorial plane
of the five unknown functions which in particular implies that derivatives with re-
spect to θ at θ =π/2 vanish and also that the required integration domain reduces
to 0≤ θ ≤π/2, 0≤ r <∞.
Asymptotic flatness implies that the following outer boundary conditions must
be imposed,
φ|r→∞ = 0, C|r→∞ = 0;
F0|r→∞ = 0, F1|r→∞ = 0, F2|r→∞ = 0.
(6.11)
Also the condition ω < µ is necessary in order to have φ|r→∞ = 0. Regularity
of the solution at the origin and on the symmetry axis require,
φ|r=0 = 0, C|r=0 = 0;
∂rF0|r=0 = 0, ∂rF1|r=0 = 0, ∂rF2|r=0 = 0,
F1|r=0 = F2|r=0.
(6.12)
φ|θ=0,π = 0, C|θ=0,π = 0;
∂θF0|θ=0,π = 0, ∂θF1|θ=0,π = 0, ∂θF2|θ=0,π = 0,
F1|θ=0,π = F2|θ=0,π.
(6.13)
The regularity conditions for φ for the case m = 0 are different from those of the
previous expressions, however this case reduce to the widely studied spherical,
nonrotating, neutral boson star, and will not be addressed in this manuscript
except for comparison.
The nonlinear PDEs are solved numerically using the spectral solver Kadath
[16, 17] which implements a Newton-Raphson iteration. This library, which has
been successfully applied to solve a wide variety of PDEs in theoretical physics
and in particular in relativity, was also used in the construction of rotating boson
stars [117].
128 6.3 Numerical solutions
Chebyshev polynomials have been used in the spectral method as basis func-
tions for the expansions of the five unknown functions. The spatial domain is di-
vided into 8 spherical shells with boundaries located at r = {2,4,8,16,32,64,128}.
The regularity conditions in Eqs. (6.12) and (6.13) are either imposed by the spec-
tral basis4 for a given function on the corresponding domain, or checked that they
hold up to numerical accuracy. On the other hand the outer boundary conditions
in Eq. (6.11) are imposed “exactly” (without the need of a cutoff radius) given the
compactification of the radial variable at the outermost spherical shell.
An initial guess for the functions is required in order to start the iteration.
For each value of m this needs to be done only once. The expressions
N := eF0 = 1− (1−N0)e−r2/r2
0 , F1 = F2 = 0, C = 0, (6.14)
also used in [85], and
φ=φ0(rsinθ)me−(x2/2+2z2)m/r2
0 (6.15)
with x = rsinθ, z = r cosθ, proposed in Ref. [117], have proven to be good guesses
given certain choice of φ0, r0 fixing the coupling constant q = 0 and the lapse at r =
0, N0 ≈ 0.95. The last condition prevents convergence to the trivial φ= 0 solution
and also leads to a Newtonian configuration (ω ∼ µ). Once the first solution is
obtained the rest of the solutions are obtained by varying N0 and increasing q by
small steps.
In addition to the Komar expression on Eq. (6.5), the ADM mass definition can
be used to obtain the total mass of the star. Both quantities should coincide given
the stationarity of the spacetime we are considering [175], therefore the difference
between the ADM and the Komar masses can be used as an indicator of the nu-
merical accuracy and provide an estimation of the numerical error of the solution.
An expression for the ADM mass suitable for our case is the expression [175],
M =−
1
8π
lim
S→∞
∮
S
[
∂
∂r
(F1 +F2)+
F2 −F1
r
]
r2 sinθdθdϕ, (6.16)
where the limit indicates integration over a sphere S of radius r →∞.
One can also verify that the value of the unknown frequency ω converge ex-
ponentially to a finite value with the number of collocation points. This error
indicator has been used together with the relative difference of the ADM and Ko-
mar masses to monitor accuracy along the sequence of numerical solutions and
to carry out convergence tests of the solutions with increasing number of spectral
coefficients.
4Details on how this is implemented in terms of the Chebyshev spectral basis in the innermost
shell and on the symmetry axis for similar problems can be found in [117] and [116].
Magnetostatic boson star 129
−40 −20 0 20 40
x
−40
−30
−20
−10
0
10
20
30
40
z
q = 1, ω = 0.96µ
−40 −20 0 20 40
x
−40
−30
−20
−10
0
10
20
30
40
z
q = 1, ω = 0.96µ
−40 −20 0 20 40
x
−40
−30
−20
−10
0
10
20
30
40
z
q = 120, ω = 0.96µ
−40 −20 0 20 40
x
−40
−30
−20
−10
0
10
20
30
40
z
q = 120, ω = 0.96µ
Figure 6.1: Scalar field φ isocontours (left) and magnetic field lines (right) in a
plane of constant ϕ with x = rsinθ and z = r cosθ for m = 1 and ω = 0.96µ using
different values of q.
6.3.2 Structure of the stars
One can use the code to solve for boson stars of several types. The spectral code
has been able to reproduce sequences of solutions already presented in the liter-
ature such as the single field static and the rotating mini-boson stars, as well as
the multifield ℓ-boson stars and the toroidal static boson stars. In this section
we present new solutions that correspond to magnetostatic boson stars. These
configurations generalize the toroidal static boson stars, incorporating a new pa-
rameter, q, in addition to the frequency ω and the winding number m.
Typical solutions for magnetized (q 6= 0) boson stars are presented in Fig. 6.1,
where isocountours of the scalar field function φ and the ϕ component of Aµ are
plotted for m = 1. In the first place we can notice from the φ contours, that the
star has a toroidal structure just like the q = 0 case, secondly we observe from the
isocontours of C, which can be interpreted as the magnetic field lines (see Sec. 6.4),
that the expected poloidal magnetic field distribution preserves as q increases,
however, as we will show next in this paragraph, near to the location of maximum
130 6.3 Numerical solutions
0 10 20 30 40 50
µr
0.0
0.2
0.4
0.6
0.8
1.0
m− qC
φ/max(φ)
0 10 20 30 40 50
µr
0.0
0.2
0.4
0.6
0.8
1.0
m− qC
φ/max(φ)
0 10 20 30 40 50
µr
0.0
0.2
0.4
0.6
0.8
1.0
m− qC
φ/max(φ)
Figure 6.2: Profiles of φ and C at the equatorial plane for solutions with m = 1
and ω = 0.96µ. Left panel: Neutral (q = 0). Center panel: q = 25. Right panel:
q = 140.
φ a region of constant C (zero magnetic field) is formed which grows in size. This
effects on the structure of the star and the morphology of the magnetic field can be
seen more clearly, plotting the profiles of the scalar field and the electromagnetic
potential on the equatorial plane. This is done in Fig. 6.2 where we have plotted
m− qC instead of C, to observe an interesting property: Above certain value of q
(which in the ω= 0.96µ case of Fig. 6.2 is at q ≈ 50, between the second and third
panel), the quantity qC approaches but never exceeds m in a region that grows
as q increases. In all of the solutions presented in this paper we have obtained
C(r)< m/q for all r.
The sources of gravitational field are also enlightening regarding the structure
of the star as well as its global properties, as we will see in the next section. Re-
stricting to the equatorial plane, θ =π/2, the complete contributions of the energy
momentum tensor (see Appendix. 6.6) are given by,
E|θ=π/2 =
[
ω2
e2F0
+
(m− qC)2
e2F2 r2
]
φ2 +
1
e2F1
(
∂φ
∂r
)2
+µ2φ2 +EB, (6.17)
Sr
r|θ=π/2 =
[
ω2
e2F0
−
(m− qC)2
e2F2 r2
]
φ2 +
1
e2F1
(
∂φ
∂r
)2
−µ2φ2 +EB, (6.18)
Sθ
θ|θ=π/2 =
[
ω2
e2F0
−
(m− qC)2
e2F2 r2
]
φ2 −
1
e2F1
(
∂φ
∂r
)2
−µ2φ2 −EB (6.19)
S
ϕ
ϕ|θ=π/2 =
[
ω2
e2F0
+
(m− qC)2
e2F2 r2
]
φ2 −
1
e2F1
(
∂φ
∂r
)2
−µ2φ2 +EB (6.20)
Where we have defined EB as the purely electromagnetic contribution to the
energy density (at the equatorial plane),
EB :=
1
2r2e2F1+2F2
(
∂C
∂r
)2
. (6.21)
Magnetostatic boson star 131
0 5 10 15 20 25
µr
−1× 10−6
0× 10−6
1× 10−6
2× 10−6
EB
Sr
r
Sθ
θ
Sϕ
ϕ
0 5 10 15 20 25
µr
−2× 10−6
0× 10−6
2× 10−6
4× 10−6
EB
Sr
r
Sθ
θ
Sϕ
ϕ
0 5 10 15 20 25
µr
−2× 10−5
−1× 10−5
0× 10−5
1× 10−5
2× 10−5
EB
Sr
r
Sθ
θ
Sϕ
ϕ
Figure 6.3: Source terms at the equatorial plane for boson stars with m = 1 and
ω = 0.96µ. Left panel: Neutral (q = 0). Center panel: q = 25. Right panel: q =
140.
Regarding the stress tensor components, which are usually identified as com-
ponents of the pressure, the system is completely anisotropic even in the q = 0
case, where for instance the difference Sθ
θ
−S
ϕ
ϕ ∝φ2e−2F2 /r2 is not zero, although
suppressed by r2. In the left panel of Fig. 6.3 we plot the sources of the q = 0 case
where in fact the difference between Sθ
θ
−S
ϕ
ϕ cannot be appreciated. The middle
and right panels of Fig. 6.3 show the magnetized q = 25 and q = 140 cases respec-
tively. Notice that the extrema of S
ϕ
ϕ decrease in magnitude with respect to the
extrema of Sθ
θ
and Sr
r. In particular near r = 0, the minimum of S
ϕ
ϕ increases
due to the EB contribution. The behavior of the pressure term S
ϕ
ϕ is relevant in
the analysis of the effect of q on global quantities, as for example the magnetic
dipole moment and the total mass. This will be discussed in the next section.
6.3.3 Sequence of magnetic boson stars
For m = 1,2 we obtained a family of configurations by means of slowly varying the
parameters of the solution starting from a Newtonian solution, as stated before.
First, we have verified that in the case q = 0, m = 1 we obtain the known sequence
of toroidal static boson stars [104]. Thereafter, starting from this set of solutions,
we have slowly increased the value of q, generating in this way sequences of mag-
netized boson stars.
In Fig. 6.4, the global quantity M is shown vs. the scalar field frequency ω for
m = 1 and five chosen values for q. Some interesting aspects arise from these
solutions: firstly the mass of the star decreases monotonically with q; this is
the opposite behavior to that obtained in models of neutron stars with magnetic
fields [167,168], where their structure begins from spherical morphology at zero
magnetic field (for the static case), and flattens, increasing the circumferential
radius of the star, as the magnitude of the magnetic fields increase, with a cor-
responding increase in the mass of the star. The observed structure dependence
on q of the magnetized boson stars, is also opposite to the corresponding depen-
dence of charged boson stars as discussed in the previous section. However, such
132 6.3 Numerical solutions
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
µ
M
q = 0
q = 1
q = 5
q = 25
q = 50
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
µ2N
−0.04
−0.02
0.00
0.02
0.04
0.06
0.08
0.10
µ
M
−
µ
2
N
q = 0
q = 1
q = 5
q = 25
q = 50
Figure 6.4: Left panel: Frequency vs. mass for the m = 1 magnetostatic boson star
solutions using different values of the coupling constant q. Right panel: Binding
energy M−µN for m = 1.
observed decrease on the size and total mass M as the coupling constant q grows,
is also observed in the magnetized Bose-Einstein condensate stars [177].
The second aspect to consider about Fig. 6.4, regards the existence of equilib-
rium configurations with coupling constant q above the value qcrit ≈ 1/
p
2. Indeed,
as reported in [165,178], that value was an upper limit for stable charged and sta-
ble rotating-charged boson stars. In our model without total charge, that limit is
overcome. This interesting result is related to the fact that the Lorentz force,
fν := FναJα =−∇µ(TEM)
µ
ν, (6.22)
points everywhere outwards for the charged mini-boson stars, while for the mag-
netostatic boson star it only points outward near the origin and points inward
outside the main distribution of scalar field. Therefore, the nonrelativistic ar-
gument regarding Coulomb repulsion vs. gravitational attraction does not apply
here. Instead, it is the stress anisotropy that ultimately determines the structure
and global properties of the star, as we will see below in relation to the decrease
in the total mass. Numerically we have not obtained any limiting value for the
parameter q, the equations are difficult to solve for large values of the coupling
constant due to the resolution required at the “edges of the plateau” that forms in
the function C (see the right panel of Fig. 6.2).
Anisotropic pressures are essential to obtain equilibrium configurations with
high compactness and large values for the mass. In [85] (see [133] for recent dis-
cussion on fluid anisotropic stars and [135] for shell-type configurations in the
Einstein-Vlasov system) it was shown that for ℓ-boson stars, small radial pres-
sures and big tangential pressures are related to an increase in the mass and
radius of the star in a way that resembles the forces on an arch. In our case, to
understand the decrease in size of the magnetic boson stars, we start by noticing
Magnetostatic boson star 133
0.00 0.01 0.02 0.03 0.04 0.05
max(C)
0.0050
0.0055
0.0060
0.0065
0.0070
0.0075
m
ax
(φ
)
m = 1
m = 2
0 20 40 60 80 100 120 140 160 180
q
6
8
10
12
14
16
18
µ
r
r of max(φ), m = 1
r of max(C), m = 1
r of max(φ), m = 2
r of max(C), m = 2
Figure 6.5: Sequence of solutions constant ω and increasing coupling q. Given
q we locate in the solution for the maximum of φ and C and its location. Left
panel: Maximum value of the scalar field with respect to the maximum value of
the four potential function C for sequence of solutions with m = 1 and m = 2 with
ω= 0.96µ. Right panel: Radius at which the maxima of φ and C is attained, as a
function of the coupling constant q.
that the tangential pressure is composed by two different contributions, Sθ
θ
and
S
ϕ
ϕ, which act together with the radial pressure (and with the Lorentz force in
some region) against gravity in order to support the configuration. The toroidal
shape, which is made possible by the S
ϕ
ϕ contribution, shrinks with increasing q
- right panel Fig. 6.5, given that the electromagnetic contribution to the energy-
momentum tensor in the region where the scalar field concentrates, is bigger for
Sr
r than for the tangential components and in particular with respect to S
ϕ
ϕ, as
discussed in Sec. 6.3.2. Our results indicate that this reduction in the size of
the star is accompanied by a reduction in the total mass that the boson star can
support.
Fig. 6.5 shows properties of the scalar field, φ and of the electromagnetic one,
C vs. q. More precisely, the figure shows the maximum of the functions and the
coordinate at which the maximum is attained. From the right panel we appreci-
ate the decrease in size of the torus as a function of q for fixed ω. In the left panel
we see another important property of the solutions: max(C) reaches a maximum
value and then begins to decrease with q. As a consequence, the magnetic dipole
moment M which can be obtained from the asymptotic behavior of the electro-
magnetic potential Aµ,
Aµdxµ ∼
µ0
4π
M sin2θ
r
dϕ, (6.23)
reaches a maximum and decreases thereafter. This behavior is shown in the right
panel of Fig. 6.6 for the sequence of m = 1 solutions using four selected values of
q and in the same panel can be seen for fixed ω, m = 1,2 and several values of
q. One can note from these plots that larger values for the maximum of M are
134 6.3 Numerical solutions
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0
1
2
3
4
5
√
µ
0
µ
2
M
q = 1
q = 5
q = 25
q = 50
0 20 40 60 80 100 120 140 160 180
q
0
2
4
6
8
10
√
µ
0
µ
2
M
m = 1
m = 2
Figure 6.6: Left panel: Frequency vs. magnetic dipole moment M for the m = 1
magnetostatic boson star solutions using different values of the coupling constant
q. Right panel: Magnetic dipole moment M as a function of q for configurations
with m = 1 and m = 2 and a fixed value of ω.
obtained, closer to ω = µ, as q increases. In Table 6.1 we provide data of the
maximum M configuration for a sample of values for the coupling constant. For
the explored q ≫ 1 configurations, the maximum of M increases more slowly, for
example, the q = 200 case, with max(M )= 4.84, is not far from the value obtained
for the q = 50 case, plotted in Fig. 6.6, so the maximum dipole moment seems to
tend asymptotically to a finite value, however to establish with precision the limit,
it would be necessary to solve the equations in the limiting case q →∞, which is
beyond the scope of this paper.
Finally, we wonder about the possibility of determining the magnetic dipole
moment from the asymptotic behavior of the metric functions. For example, in the
charged mini-boson star, the total electric charge of the configuration can be read
off the grr component by comparing with the Reissner-Nordström solution [163].
Some electrovacuum exact solutions (in General Relativity) for a mass endowed
with a magnetic dipole moment have been obtained in the literature, as for exam-
ple the Gutsunaev-Manko [179] and Bonnor [180] two-parameter family of solu-
tions. However, analyzing our solutions, we obtain that they do not match with
neither of those metrics, for instance the lapse function in all of the solutions that
we generate has the asymptotic behavior N2 = 1−2M/r+α/r2 +O (1/r3), with α
some constant, while according to [179], the lapse of the Gutsunaev-Manko metric
goes as N2 = 1−2M/r+O (1/r3). On the other hand, comparison with the Bonnor
solution, for which N2 = 1−2M/r+M2/r2, could seem to be a better alternative,
however we obtain from the analysis of our solutions that the coefficient α is a
function of both of M and M and in the M = 0 case, it is not proportional to M2.
Therefore the magnetic dipole moment of the star cannot be obtained from the
the metric components by comparison with any of the mentioned exact solutions,
allowing us to conclude that our solutions differ from those two spacetimes.
Magnetostatic boson star 135
Table 6.1: Maximum mass and magnetic dipole configurations.
q
p
µ0µ
2
M ω/µ µM µ2
N max(φ) µrmax(φ)
Max. M
0.1 0.0418 0.905 1.048 1.077 0.0156 4.66
1 0.417 0.905 1.047 1.076 0.0156 4.66
5 1.93 0.911 1.015 1.041 0.0147 4.82
25 4.62 0.970 0.627 0.633 0.00492 9.08
50 5.06 0.990 0.363 0.363 0.00154 16.8
100 5.17 0.997 0.195 0.195 0.000454 30.1
200 4.84 0.999 0.084 0.085 0.000132 36.0
Max. M
0 0 0.840 1.147 1.189 0.0290 2.76
0.1 0.0368 0.834 1.147 1.189 0.0303 2.65
1 0.366 0.834 1.146 1.188 0.0303 2.65
5 1.66 0.839 1.121 1.160 0.0296 2.69
25 2.46 0.848 0.960 0.989 0.0312 2.22
50 1.69 0.848 0.894 0.921 0.0337 1.85
6.4 Magnetic field
The electric and magnetic field as measured by an observer whose four-velocity is
nµ (Eulerian observer) are given by the formulas Eµ = Fµνnν and Bµ =− 1
2
ǫµναβnνFαβ.
where ǫ is the Levi-Civita tensor. For the metric (6.8) and the electromagnetic
four-potential (6.10), we obtain Eµ = 0 as expected and
Bµdxµ =
e−F2
sinθ
(
1
r2
∂C
∂θ
dr−
∂C
∂r
dθ
)
. (6.24)
Some examples of Bi for configurations with m = 1 are given in Fig. 6.7. The
distribution of the vector field resemble that of the magnetic field around a finite
size current loop. For reference we also plot the isocontour of half the maximum
value of the energy density. Rotation of this curve around the z axis generates a
torus. The figure also shows the region of zero magnetic field that forms in config-
urations with high values of q, where m− qC ≈ 0, see for instance the black line
region with
√
BiBi/(µ
p
µ0)< 10−6 inside the torus in the right panel of Fig. 6.7.
We have seen in the previous section that as q get closer to zero, the magnetic
moment decreases and the maximum values ofφ and C are reached at larger radii.
This explains why some of the configurations with relative low values of q, as for
example the q = 5 and q = 25 cases (Fig. 6.7, left and central panels), do not have
the maximum of Bi at the center of the star but in a toroidal region around the
center, while other configurations as for instance the q = 140 case (Fig. 6.7, right
panel), posses magnetic fields concentrated in a central region with maximum
136 6.4 Magnetic field
−5.0
−4.8
−4.6
−4.4
−4.2
−4.0
−3.8
−3.6
−5.0
−4.5
−4.0
−3.5
−3.0
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
Figure 6.7: Magnetic field in a plane of constant ϕ for configurations with m =
1 and ω = 0.96µ, for the coupling constant with values (from left to right) q =
5, 25 and 140. The thick black line correspond to the density isocountour E =
0.5max(E). The color bar indicates the norm of the quantity log10[Bi /(µ
p
µ0)]
The radius of the circle is µr = 32.
along θ = 0.
Until now it has not been required to specify the value for the mass param-
eter of the scalar field given that solutions with different µ are related to each
other by rescaling rules. In particular, we have used these rules to construct di-
mensionless quantities (e.g., µr, ω/µ, µM, Bµ/(µ
p
µ0), etc.), used in the numerical
implementation and in the results reported in previous sections. We now proceed
to recover units of different physical quantities in order to compare magnetostatic
boson stars with magnetized neutron star solutions. Restoring c and G, the di-
mensionless quantities related to the total mass of the star and the norm of the
magnetic field (B2 = BµBµ) are Gc−2µM and Gc−4B2/(µ2µ0). Furthermore, the
product
I :=
1
c4
√
G3
µ0
MB, (6.25)
is dimensionless and, more importantly, do not rescale with µ. Evaluating the
magnetic field at the center of the star we define Ic = I |r=0 and plot it along
m = 1 sequences in Fig. 6.8.
For a neutron star with mass in the range 1.1M⊙ . M . 2.1M⊙ and strong
magnetic fields at the star’s pole within the interval 109T . Bpole . 1011T [181],
the value of the product of mass and magnetic field is between 10−7 . Ipole .
3×10−5. Internal magnetic fields in magnetars have been estimated according
to simulations to be as high as5 1014T. An extended range for I at the center of
strongly magnetized neutron stars would be 10−7 .Ic . 3×10−2.
5Restricting to the Einstein-Maxwell-Euler self-consistent models of neutron stars with equations
of state independent of the magnetic field, the maximum magnetic field Bc , which is attained at the
center of the star, is approximately only one order of magnitude bigger than Bpole (see e.g., [167]).
Magnetostatic boson star 137
0.2 0.4 0.6 0.8 1.0 1.2
µM
10−2
10−1
100
101
I c
q = 1
q = 5
q = 25
q = 50
magnetar
Figure 6.8: Dimensionless quantity I defined in Eq. (6.25), evaluated at r = 0 for
configurations with m = 1 and different values of q. The yellow region corresponds
to the upper region of the interval I at the center of strongly magnetized neutron
stars (see the text for details).
As can be appreciated from Fig. 6.8, some of the individual configurations in-
tersect with the interval 10−7 . Ic . 3×10−2, which means that it is possible
to find a value of µ such that Bc and M are within the range of neutron stars.
In particular some of the low mass solutions obtained with q = 1 are in this re-
gion, while typical, compact solutions with q = 1,5,25 and 50 might have stronger
magnetic fields (larger masses) than magnetars if we assume similar values of the
mass (magnetic fields) of the boson stars to those of the neutron star models. In
order to perform a numerical application we restrict to the configurations with
q = 1 and Mµ = 0.7 and choose M = 1.5M⊙. This fixes µ(c~) = 1.8× 10−10eV
which in turn sets the magnitude of all other physical variables, for instance the
magnitude of the magnetic field at the center of coordinates takes then the value
of Bc ≈ 1×1014T, which is within the expected values of magnetic fields inside
magnetars. Furthermore, the size of this bosonic configuration, which can be
estimated from the size of the torus, is of order rmax(φ) ≈ 9 km, obtaining a com-
pactness comparable to those of neutron stars.
6.5 Conclusions
In the present work we have constructed magnetized solutions of boson stars,
which are static, axisymmetric, everywhere regular and asymptotically flat so-
lutions of the Einstein-Maxwell-Klein-Gordon system characterized by the mass
parameter of the scalar field µ, the azimuthal harmonic index (winding number)
m and the coupling constant q. The configurations consist of two contrarotating
oppositely charged tori, and we have seen that they give rise to an electrically neu-
tral current that generates a poloidal magnetic field, according to the observer at
138 6.6 Appendix
rest.
Comparing with the q = 0 case, which reduces to the toroidal static boson stars
found in [104], we obtained that the electromagnetic field affects the structure of
the star and can noticeably change their mass and size. Similarly to other boson
star models, in the magnetostatic solutions obtained in this work, a maximum
mass configuration was found for each q, and in all explored cases the sequence of
solutions contain a region of negative binding energy. Regarding the electromag-
netic part, the dipole magnetic moment M has been obtained and an important
difference is noted with respect to the rotating charged boson star, namely that
the maximum M configuration does not corresponds with the maximum mass
configuration for every q, and is shifted towards the zero mass solution (ω=µ).
We showed that regions of zero magnetic field form at the inner part of the
torus which grow in size as one considers large enough values of the electromag-
netic coupling constant q. On the other hand, it has also been found that the elec-
tromagnetic contribution to the sources increases in relation to the scalar field
corresponding sources. For these reasons and since we have not found any bound
for q, it would be interesting to study in a future work, the numerically challeng-
ing solutions with q ≫ 1 and also analyze the asymptotic limit q →∞.
The magnetic field has been compared to that of strongly magnetized neutron
stars, obtaining that for similar values of the total mass of the star, the inner mag-
netic field is comparable to that of magnetars for q . 1 compact configurations and
greater, for larger values of q, making our objects, besides being interesting by
their own value, faithful mimickers of neutron stars. Since toroidal static boson
stars with q = 0 are known to be unstable we expect that the obtained solutions
(at least for small values of q) remain unstable, however there is a stabilization
(and formation) mechanism for neutral multifield boson stars [104] which might
be applied to the charged scalar field case. Starting from the conditions in which
these magnetized boson stars are stable, a mechanism for their formation could
be devised. Boson stars are useful entities in strong gravity research, in partic-
ular in dynamical studies and as toy models of more complex scenarios. In this
sense, the compactness and magnetic field magnitudes of magnetostatic boson
stars motivates the study of the collapse and the consequent emission in both the
gravitational and electromagnetic channels. Such collapse dynamics and multi-
messenger studies will be presented in future works.
6.6 Appendix of Chap. 6. 3+1 decomposition of Tµν and the
elliptic system of PDEs
In terms of energy-momentum tensor decomposed into the 3+1 quantities,
E = Tµνnµnν; Pα =−nνTµνγ
ν
α; Sαβ = Tµνγ
µ
αγ
ν
β. (6.26)
Magnetostatic boson star 139
where nα = (1/N,0,0,0), γα
β
= δα
β
+nαnβ and N := eF0 , the Einstein equations can
be written [117, 182] as the following system of elliptic equations for the metric
coefficients at Eq.(6.8),
∆3F0 = 4πe2F1
(
E+S
)
−∂F0∂
(
F0 +F2
)
, (6.27a)
∆2
[
(
NeF2 −1
)
rsinθ
]
= 8πNe2F1+F2 rsinθ
(
Sr
r +Sθ
θ
)
, (6.27b)
∆2
(
F1 +F0
)
= 8πe2F1 S
ϕ
ϕ−∂F0∂F0, (6.27c)
where,
∆3 :=
∂2
∂r2
+
2
r
∂
∂r
+
1
r2
∂2
∂θ2
+
1
r2 tanθ
∂
∂θ
, (6.28)
∆2 :=
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂θ2
, (6.29)
∂ f1∂ f2 :=
∂ f1
∂r
∂ f2
∂r
+
1
r2
∂ f1
∂θ
∂ f2
∂θ
. (6.30)
The source terms using the ansatz in Eqs. (6.8), (6.9) and (6.10) lead to the
following expressions:
E+S =
4
N2
ω2φ2 −2µ2φ2 +
e−2(F1+F2)
µ0r2 sin2θ
∂C∂C, (6.31)
Sr
r +Sθ
θ = 2
[
ω2
N2
−
e−2F2 (m− qC)2
r2 sin2θ
]
φ2 −2µ2φ2, (6.32)
S
ϕ
ϕ =
[
ω2
N2
+
e−2F2 (m− qC)2
r2 sin2θ
]
φ2 − e−2F1∂φ∂φ−µ2φ2 (6.33)
+
e−2(F1+F2)
2µ0r2 sin2θ
∂C∂C.
The two Klein Gordon Eqs., (6.3), and the Maxwell Eq., (6.4), reduce to,
∆3φ = e2F1
(
µ2 −
ω2
N2
)
φ−∂φ∂(F0 +F2)+ e2F1−2F2
(m− qC)2φ
r2 sin2θ
(6.34)
(2∆2 −∆3)C = −∂C∂(F0 −F2)−2µ0qe2F1 (m− qC)φ2. (6.35)
We have obtained the following number density currents,
j
µ
1
=
(
ω
φ2
N2
,0,0,
e−2F2 (m− qC)φ2
r2 sin2θ
)
, (6.36)
and
j
µ
2
=
(
ω
φ2
N2
,0,0,−
e−2F2 (m− qC)φ2
r2 sin2θ
)
, (6.37)
140 6.6 Appendix
which have been inserted in Eq. (6.4) to obtain Eq. (6.35), and corresponds to the
nontrivial remaining Maxwell equation, Aϕ−R
ϕ
ϕ =−µ0Jϕ.
The Eqs. (6.27), (6.34) and (6.35) make up the elliptic system of PDEs for the
model (6.1) using the ansatz presented at section 6.2.
CHAPTER 7
TWO BOSON STARS IN EQUILIBRIUM1
Contents
7.1 Mathematical background . . . . . . . . . . . . . . . . . . . . . . 142
7.2 Some results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.1 Spherical boson star parenthesis . . . . . . . . . . . . . 145
7.2.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2.3 Flat spacetime static equilibria . . . . . . . . . . . . . . 146
I
n this chapter we construct solutions to the Einstein-Friedberg-Lee-Sirlin sys-
tem, representing two boson stars in equilibrium made of a complex scalar
field whose mass is given to it by a real scalar field with a finite vacuum
expectation value generated via a symmetry breaking potential.
Solutions in full General Relativity interpreted as two non-spinning static
boson stars in equilibrium where first obtained in [183]. And subsequently the
(mini) two boson star system have been obtained as a special case within more
complex families, some of them consisting of several more scalar field accumula-
tions [176], some of them out of axial symmetry [166] and some other using multi-
ple scalar fields [104]. The system of two boson stars in equilibrium is also known
as a dipolar boson star. The equilibrium obtained for such kind of system is due to
the odd symmetry of the complex scalar field with respect to the plane of symmetry
that separates both lumps which implies a phase difference between both lumps
and consequently a repulsive interaction between them, which together with the
gravitational attraction allows obtaining equilibrium solutions.
According to the fully-nonlinear numerical evolutions in 3+1 dimensions per-
formed in [104], the two boson star solutions with the standard massive non-self
interacting complex scalar fields, are unstable. In the same Ref. it was also found
a stabilization mechanism for the dipolar boson stars, which consists in super-
pose a sufficient amount of a spherical mode of an extra scalar field (complex,
1This chapter will be incorporated into a paper to be published with C. Herdeiro, D. Núñez and E.
Radú. The work contained in this chapter as well as the numerical implementation described here is
under development.
141
142 7.1 Mathematical background
canonical and of the same mass). Of course, the stable scenarios in this con-
text no longer represent a simple two-star system and therefore the search for
stabilization mechanisms for this system has continued2. Attempts to use the
self-interacting parameters of the scalar field to achieve dynamically robust sys-
tems have apparently not been successful, however there is one more possibility
at hand to be explored.
7.1 Mathematical background
The Friedberg-Lee-Sirlin model [184] is a composite system which contain a com-
plex scalar field Φ and a real scalar field Ψ. In this model the complex scalar field
acquires mass due to a coupling m with the real field. On the other hand, the real
scalar field possesses a mass parameter µ and a vacuum expectation value v in
such a way that when Ψ→ v the system reduce to the Klein-Gordon case with a
complex scalar field of mass mv. Relevant for our work are the solitonic solutions
in the Friedberg-Lee-Sirlin model (in Minkowski spacetime), obtained in [185] in
the limit µ→ 0.
This system can be minimally coupled to gravity and the full action reads
S =
∫
d4x
p
−g
[
1
4α2
R− gµν∇µΦ∇νΦ
∗
−
1
2
gµν∇µΨ∇νΨ−m2
Ψ
2|Φ|2 −µ2
(
Ψ
2 −v2
)2
]
,
(7.1)
please be aware that the references [129, 186, 187] have typos in the Lagrangian
of the matter fields or in the ψ Klein-Gordon equation (and sometimes in both)
however the action (7.1) is consistent with that originally presented in [184] ex-
cept for multiplicative factor changes. Here m and µ are the mass parameters
of the complex and real scalar fields, respectively. We have used the signature
(−,+,+,+), c = 1 units and α2 = 4πG.
Boson stars and rotating solutions with horizons (hairy black holes) in this
theory have already been mentioned in Chap. 4 of this manuscript. This interest-
ing solutions where obtained in Ref. [129]. In this reference are presented rotating
solutions of parity-even and parity-odd boson stars3, we note that the latter have
been much less studied in the literature even in the case of mini boson stars. The
purpose in this chapter is to address the non-rotating static case with a scalar
field in the odd symmetry case.
2Carlos Herdeiro, Eugen Radu. Personal communication
3This refers to the allowed boundary conditions adopted for Φ in the equatorial plane which main-
tain the reflection symmetry θ→π−θ: Φ= 0 for parity-odd and ∂θΦ= 0 for parity-even at θ =π/2.
Two boson stars in equilibrium 143
The Einstein equations obtained from (7.1) are:
Rµν−
1
2
R gµν = 2α2Tµν;
Tµν = 2∇(µΦ∇ν)Φ
∗+∇µΨ∇νΨ
− gµν
(
gαβ∇αΦ∇βΦ
∗+
1
2
gαβ∇αΨ∇βΨ+m2
Ψ
2|Φ|2 +µ2
(
Ψ
2 −v2
)2
)
(7.2)
The equation of motion for the complex scalar fields Φ and Ψ are
gµν∇ν∇µΦ= m2
Ψ
2
Φ , gµν∇ν∇µΨ= 2
[
m2|Φ|2 +2µ2
(
Ψ
2 −v2
)
]
Ψ. (7.3)
We are looking for static solutions in axial symmetry:
gµνdxµdxν =−e2F0(rθ) dt2 + e2F1(r,θ)(dr2 + r2 dθ2)+ e2F2(r,θ)r2 sin2θdϕ2, (7.4)
Φ=φ(r,θ)e−iωt , Ψ=ψ(r,θ). (7.5)
The 3+1 source terms are
E = Tµνnµnν; Pα =−nνTµνγ
ν
α; Sαβ = Tµνγ
µ
αγ
ν
β. (7.6)
with nα = (1/N,0,0,0), γα
β
= δα
β
+nαnβ. The Einstein equations for the metric
(7.4) are the following,
∆3F0 = α2 A2
(
E+S
)
−∂F0∂
(
F0 +F2
)
(7.7)
∆2
[
(
NB−1
)
rsinθ
]
= 2α2N A2Brsinθ
(
Sr
r +Sθ
θ
)
(7.8)
∆2
(
F1 +F0
)
= 2α2 A2S
ϕ
ϕ−∂F0∂F0, (7.9)
where N = eF0 , A = eF1 y B = eF2 and
∆3 :=
∂2
∂r2
+
2
r
∂
∂r
+
1
r2
∂2
∂θ2
+
1
r2 tanθ
∂
∂θ
(7.10)
∆2 :=
∂2
∂r2
+
1
r
∂
∂r
+
1
r2
∂2
∂θ2
(7.11)
∂ f ∂g :=
∂ f
∂r
∂g
∂r
+
1
r2
∂ f
∂θ
∂g
∂θ
. (7.12)
Only the source terms remain to be determined explicitly. Using our ansatz
for metric and fields we arrive at the following,
E+S = 4
ω2
N2
φ2 −2m2ψ2φ2 −2µ2
(
ψ2 −v2
)2
(7.13)
Sr
r +Sθ
θ = 2
ω2
N2
φ2 −2m2ψ2φ2 −2µ2
(
ψ2 −v2
)2
(7.14)
S
ϕ
ϕ =
ω2
N2
φ2 −
1
A2
∂φ∂φ−
1
2A2
∂ψ∂ψ−m2ψ2φ2 −µ2
(
ψ2 −v2
)2
.(7.15)
144 7.1 Mathematical background
Explicitly,
E =
ω2
N2
φ2 +
1
A2
∂φ∂φ+
1
2A2
∂ψ∂ψ+m2ψ2φ2 +µ2
(
ψ2 −v2
)2
, (7.16)
Sr
r =
ω2
N2
φ2 +Y −m2ψ2φ2 −µ2
(
ψ2 −v2
)2
, (7.17)
Sθ
θ =
ω2
N2
φ2 −Y −m2ψ2φ2 −µ2
(
ψ2 −v2
)2
, (7.18)
with
Y =
1
A2
[
(
∂φ
∂r
)2
−
1
r2
(
∂φ
∂θ
)2
+
1
2
(
∂ψ
∂r
)2
−
1
2r2
(
∂ψ
∂θ
)2
]
(7.19)
The equation of motion for Φ and Ψ are
∆3φ= A2
(
m2ψ2 −
ω2
N2
)
φ−∂φ∂(F0 +F2) (7.20)
and
∆3ψ= 2A2
[
m2φ2 +2µ2
(
ψ2 −v2
)
]
ψ−∂ψ∂(F0 +F2) (7.21)
Asymptotic flatness implies that the following outer boundary conditions must
be imposed,
φ|r→∞ = 0, ψ|r→∞ = v;
F0|r→∞ = 0, F1|r→∞ = 0, F2|r→∞ = 0.
(7.22)
Also the condition ω < µ is necessary in order to have φ|r→∞ = 0. Regularity
of the solution at the origin and on the symmetry axis require,
φ|r=0 = 0, ∂rψ|r=0 = 0;
∂rF0|r=0 = 0, ∂rF1|r=0 = 0, ∂rF2|r=0 = 0,
F1|r=0 = F2|r=0.
(7.23)
∂θφ|θ=0,π = 0, ∂θψ|θ=0,π = 0;
∂θF0|θ=0,π = 0, ∂θF1|θ=0,π = 0, ∂θF2|θ=0,π = 0,
F1|θ=0,π = F2|θ=0,π.
(7.24)
Then, we impose the spacetime to be invariant with respect to a reflection on
the θ = π/2 plane while on the other hand expect a change of sign in the complex
scalar field profile. In particular implies that derivatives with respect to θ at
θ =π/2 of the metric functions vanish
φ|θ=π/2 = 0, ∂θψ|θ=π/2 = 0;
∂θF0|θ=π/2 = 0, ∂θF1|θ=π/2 = 0, ∂θF2|θ=π/2 = 0,
(7.25)
Two boson stars in equilibrium 145
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ω/m
0
10
20
30
40
50
M
m
µ/m→∞
µ/m = 0.5
µ/m = 0
(µ/m = 0.25)
0 20 40 60 80
Qm2
0
10
20
30
40
50
M
m
µ/m = 0.5
µ/m = 0
Figure 7.1: Spherical Klein-Gordon and Friedberg-Lee-Sirlin boson stars. Our
µ/m = 0.25 solutions do not correspond to the µ/m = 0.25 solutions reported in
[129], but the µ/m = 0.5 curve does coincide with the later.
The required integration domain therefore reduces to 0≤ θ ≤π/2, 0≤ r <∞.
7.2 Some results
7.2.1 Spherical boson star parenthesis
Before proceeding to obtain the dipolar solutions, we solve the above equations for
parity-even Φ, substituting the boundary condition φ|r=0 = 0 and φ|θ=π/2 = 0 by
∂rφ|r=0 = 0, ∂θφ|θ=π/2 = 0. (7.26)
The two-dimensional solver converges to the spherical symmetric boson star in
the Friedberg-Lee-Sirlin model, reported in [129]. We reproduce Fig. 1 of [129].
In Fig. 7.1 we show quick numerical integrations using 11 spectral coefficients in
r and θ. Although we use the same conventions and rescalings as they do, the
solution curve they report as µ/m = 0.25 and α = 0.5, in our case corresponds to
the µ/m = 0.5 and α= 0.5 solutions.
7.2.2 Solutions
We consider the following initial guess to obtain a Einstein-Klein-Gordon dipolar
boson star (µ→∞ limit),
φ= rφ0e−x2/σ2
x−z2/σ2
z Y 0
1 (θ,ϕ), N = 1− (1−N0)er2
, ω= 0.965, (7.27)
where φ0 = 9.07× 10−3, σx =
p
200, σz =
p
50 and Y 0
1
(θ,ϕ) = 1
2
√
3
π
cosθ. For a
solution with N0 = 0.9404.
After that we construct the equilibrium two-boson star solution family in the
Einstein-Klein-Gordon theory, whose resulting mass as function of frequency and
146 7.2 Some results
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ω/m
0
20
40
60
80
100
M
m
µ/m→∞
µ/m = 0.5
µ/m = 0.25
µ/m = 0
0 50 100 150 200 250
Qm2
0
20
40
60
80
100
M
m
µ/m→∞
µ/m = 0.5
µ/m = 0.25
µ/m = 0
Figure 7.2: Two boson stars in equilibrium in Klein-Gordon and Friedberg-Lee-
Sirlin for different values of µ/m and α= 0.5.
Noether charge is shown in Figure 7.2 (black line). In this result we have used
α = 0.5 and that is why the mass looks so large. However in the µ/m →∞ limit
(and only in that limit) the constant α can be rescaled away. So by rescaling to
solutions with α2 = 4π i.e. units where G = 1, we have verified that the magnitude
reported in [104,166,188] for the dipole boson star mass is obtained.
α= 0.5 sequence of boson star solutions
Finite µ solutions are obtained using a corresponding equal-frequency Einstein-
Klein-Gordon solution as initial guess and slowly decreasing the value of µ. In
Figure 7.2 we display the total mass and Noether charge plots for the values µ/m =
0.5 and 0.25, as well as the limit of vanishing scalar potential µ= 0.
We have analyzed the ψ profiles and obtained that the field ψ becomes long-
ranged when µ= 0, as expected from the asymptotic behavior of ψ from Eq. (7.21).
ω= 0.9 sequence of solutions varying α
Figure 7.3 shows M as a function of the gravitational coupling for solutions with
ω/m = 0.9.
7.2.3 Flat spacetime static equilibria
Figure 7.3 already showed that there are solutions with zero or even negative
gravitational coupling constant α2. In the latter the negative sign could be passed
to the Tµν and thus speak of a ghost field soliton [118,124].
In reference [188] it was argued that the short-range character of the self-
interactions that normally allows to obtain Q-balls, does not work to obtain equi-
librium solutions composed by distant energy lumps, concluding that gravity, be-
Two boson stars in equilibrium 147
−0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 0.25
α2
101
102
M
m
µ/m = 0.5
µ/m = 0.25
µ/m = 0
Figure 7.3: ω/m = 0.9 sequence of solutions for different cases of µ/m. For the
µ/m = 0.5 and 0.25 cases we have not been able to obtain flat space solutions so
far.
ing long-range, is mandatory for configurations of a single scalar field. It is inter-
esting that only in the µ = 0 case we have been able to construct solutions with
α ≤ 0, but precisely this corresponds to the case in which the Ψ field becomes
massless and long-ranged, the latter being precisely the fact that allows to bal-
ance at a distance the scalar repulsive interaction due to the phase difference of
π between the two lumps. Reference [188] also provides a proof of why there are
no two energy lumps solutions in Minkowski and we might be interested in find-
ing out exactly where the argument fails. It is possible to follow the steps from
equation 5.11 to equation 5.20 of [188] using the total energy-momentum tensor
of our system and, using the odd-parity of φ and the even parity of ψ we obtain
an analogous to equation 5.22 of [188]:
∫∞
0
ρ
[
φ2
1 −
1
2
ψ2
0,ρ −µ2
(
ψ2
0 −v2
)2
]
dρ = 0, (7.28)
From which we can not conclude anything. We could also try to apply the proce-
dure on a part of the Tµν, for example the one associated to Φ:
Qµν = 2∇(µΦ∇ν)Φ
∗− gµν
(
gαβ∇αΦ∇βΦ
∗+m2
Ψ
2|Φ|2
)
, (7.29)
however due to the m2 term, this tensor is not conserved:
∇αQα
β =∇βΦ
∗(Φ−m2
Ψ
2
Φ)+∇βΦ(Φ
∗−m2
Ψ
2
Φ
∗)−2m|Φ|2Ψ∇βΨ. (7.30)
In any case we have obtained the sequence of solutions for α= 0 and µ= 0 two
static Q-balls in equilibrium. Figure 7.4 shows the Noether charge as a function
148 7.2 Some results
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ω/m
0
100
200
300
400
500
Q
m
2
Figure 7.4: α= 0, µ= 0 sequence of solutions. (For ω> 0.9 numerical error indi-
cators get large)
of ω. These results are reminiscent of the results obtained for Q-balls in the FLS
model in spherical symmetry [189].
Part III
Numerical evolutions

CHAPTER 8
STABILITY1
Contents
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.2 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.3 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.4 Time evolution and numerical results . . . . . . . . . . . . . . . 158
8.4.1 Spherical perturbation test . . . . . . . . . . . . . . . . . 160
8.4.2 Non-spherical perturbation: perturbing the energy den-
sity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.4.3 Non-spherical perturbation: perturbing the amplitude
of each mode . . . . . . . . . . . . . . . . . . . . . . . . . 162
8.5 Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . 167
8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
T
he ℓ-boson stars are static, spherical, multi-field self-gravitating solitons.
They are asymptotically flat, finite energy solutions of Einstein’s gravity
minimally coupled to an odd number of massive, complex scalar fields. A
previous study assessed the stability of ℓ-boson stars under spherical perturba-
tions, finding that there are both stable and unstable branches of solutions, as
for single-field boson stars (ℓ= 0). In this work we probe the stability of ℓ-boson
stars against non-spherical perturbations by performing numerical evolutions of
the Einstein-Klein-Gordon system, with a 3D code. For the timescales explored,
the ℓ-boson stars belonging to the spherical stable branch do not exhibit mea-
surable growing modes. We find, however, evidence of zero modes; that is, non-
spherical perturbations that neither grow nor decay. This suggests the branching
off towards a larger family of equilibrium solutions: we conjecture that ℓ-boson
stars are the enhanced isometry point of a larger family of static (and possibly
stationary), non-spherical multi-field self-gravitating solitons.
1This chapter is extracted with minor revisions from Ref. [103], which was written in collaboration
with N. Sanchis-Gual, J. Barranco, A. Bernal, J. C. Degollado, C. Herdeiro and D. Núñez. © American
Physical Society. Reproduced with permission. All rights reserved.
151
152 8.1 Introduction
8.1 Introduction
Boson stars [44,45] are remarkable gravitational solitons. These self-gravitating,
localised energy lumps of a complex, massive scalar field have appealing theo-
retical properties. A key one is their dynamical stability. For spherical boson
stars there is a stable branch of solutions against perturbations. Indeed, a va-
riety of studies including linear perturbation theory [106, 190, 191], catastrophe
theory [192] and numerical simulations [56, 193] agree that boson stars are per-
turbatively stable, as long as the amplitude of the scalar field is smaller than a
critical value. When the latter is attained, boson stars acquire their maximum
mass.
Being dynamically stable legitimates inquiring about the possible (astro)physical
role of boson stars. Albeit exotic, lacking undisputed observational evidence, bo-
son stars have found important applications in strong gravity and astrophysics.
For instance, boson stars provide a common model for a black hole mimicker [49,
159, 194]. Being dynamically tractable, one can then compare dynamical space-
time properties, such as waveforms of binary boson star systems, with those of
black holes [57, 58, 195]. This is particularly timely in view of the recently initi-
ated gravitational-wave astronomy era [196, 197], which provides data for both
models to be compared with.
A second important application is in relation to a central mystery of contem-
porary science: the nature of dark matter. An increasing attention has been
dedicated to models that consider dark matter as an ultra-light bosonic parti-
cle [26,89–91]. The bosonic nature allows this sort of dark matter to form coher-
ent macroscopic excitations. In this context, bosons stars can model, in particular,
the core of dark matter galactic halos [98,113,198,199].
In their original guise, the Einstein-Klein-Gordon (EKG) model contains a
massive, free scalar field, and the solitonic solutions are called mini-boson stars.
A variety of generalisations ensued. Boson stars for scalar field theories with self-
interactions have been reported, starting with the case of quartic self-interactions
considered by Colpi et al. [132]. Spacetime angular momentum was introduced for
mini-boson stars in [164], giving rise to stationary (but not static) self-gravitating
solitons. A cousin model with a complex, massive vector (rather than scalar) field
yields Proca stars [52]. These and other examples use single (complex) field mod-
els; however, multi-field boson stars have also been reported. One example is given
by multi-state boson stars [113, 199, 200]. More recently, multi-field boson stars
with an arbitrary odd number, N = 2ℓ+1, ℓ ∈N0, of equal mass, uncoupled (ex-
cept through gravity) complex scalar fields with harmonic time dependence were
introduced [99]; they are dubbed ℓ-boson stars and they will be the focus of this
paper.
ℓ-boson stars are described by spherically symmetric and static metrics. For
Stability 153
ℓ= 0 they are simply the usual mini-boson stars. For ℓ> 1, the 2ℓ+1 scalar fields
have an angular dependence given by the corresponding 2ℓ+1 spherical harmon-
ics Y ℓm. Then, if the radial dependence for all fields is the same, corresponding to
choosing the amplitude of the spherical harmonics equal at all radial distances,
static, spherical configurations are obtained, regardless of the energy-momentum
tensor of each individual field being angular dependent. This is an example of
symmetry non-inheritance: the (spherical) spacetime and the (non-spherical) in-
dividual matter fields do not share spherical symmetry. The usual boson stars
already have a version of symmetry non-inheritance: the (time oscillating) scalar
field and the (static) spacetime do not share time-translation symmetry. Consis-
tency requires only that the spacetime geometry and the total energy-momentum
tensor share the same symmetries, not the individual matter fields.
Generic ℓ-boson stars have been shown to exhibit similar properties to those
of the standard ℓ = 0 stars. In particular, ℓ- boson stars have a stable branch of
solutions against spherical perturbations [101]. The main goal of this paper is to
assess the stability of ℓ-boson stars (in this branch) against generic, non-spherical
perturbations. As we shall see, our analysis will show that, in this respect, generic
ℓ-boson star do not exactly mimic the ℓ = 0 case. Although no instabilities are
observed, the analysis provides a glimpse of a larger landscape of solutions, of
which ℓ-boson stars are just the enhanced symmetry point.
Departure from spherical symmetry is physically relevant. Firstly, spherical
objects – such as ℓ-boson stars – need to be stable against non-spherical pertur-
bations, in order to be dynamically viable. Secondly, astrophysical bodies are not,
typically, perfectly spherical, in particular due to angular momentum. So one
must assess if some perturbations actually deform ℓ-boson stars into acquiring
new degrees of freedom. In this respect, it was recently proposed that multi-field
boson stars, in the non-relativistic regime, could have non-spherical stable config-
urations. This provides an extra motivation to inquire about the behaviour of rel-
ativistic ℓ-boson stars under non-spherical perturbations Finally, assessing non-
spherical configurations and perturbations often yields a richer phenomenology.
As a fruitful example, it was recently found that spinning, single-field mini-boson
stars are unstable against non-axisymmetric perturbations, either decaying into
a non-rotating boson star or collapsing into a Kerr black hole [58,63]. By contrast,
spinning Proca stars do not present instabilities under non-axisymmetric pertur-
bations and furthermore, they can form dynamically [63]. This example shows
how the study on non-spherical perturbations unveiled a new relevant dynamical
property of boson stars.
We shall investigate the behaviour of ℓ-boson stars under non-spherical per-
turbations using fully non-linear numerical simulations of the corresponding Einstein-
Klein-Gordon system. As initial data, we use configurations found in [99] which
154 8.1 Introduction
are then perturbed in two different ways. The first type of perturbation tests the
stability against non-axially symmetric perturbations, targeting potential bar-
mode instabilities. The second type of perturbation tests the stability against
a relative change in the amplitude of the internal fields. In none of the two cases
measurable growing modes were found, either by perturbing the total mass den-
sity or by perturbing each of the constituent fields, as long the ℓ-boson star belongs
to the stable branch against spherical perturbations. By following the evolution of
distortion parameters (defined below) we found, however, evidence for long-lived
perturbations, which we interpret as zero modes. These modes, in turn, are in-
terpreted as evidence for a larger family of equilibrium solutions.
Consider the Schwarzschild black hole of vacuum General Relativity. It has
been shown to be mode stable in the renowned works of Regge and Wheeler [201]
and Zerilli [202]. No gravitational perturbations grow. However, a perturbation
that carries angular momentum yield not decay. The Schwarzschild solution mi-
grates to a small angular momentum Kerr solution and oscillates around this
new ground state. Similarly, a perturbation which electric charge will not decay
and the spacetime will oscillate around a small charge Reissner-Nordström solu-
tion. These special perturbations are zero modes. Such modes are often found
when a spacetime is unstable against some sort of perturbations, at the thresh-
old between stable and unstable modes. An example occurs for the superradiant
instability of the Kerr spacetime due to a massive bosonic field. The zero modes
indicate the bifurcation of the Kerr family towards a new family of black holes
with bosonic hair [203]. But zero modes can also occur even if there is no in-
stability, as in the Schwarzschild example, indicating, nonetheless, an enlarged
family of solutions (Kerr or Reissner-Nordström), of which the initial spacetime
(Schwarzschild) is a special case. Thus, one of the outcomes of our analysis is the
conjecture that ℓ-boson stars are the enhanced isometry point of a larger family
of static (and possibly stationary), non-spherical multi-field self-gravitating soli-
tons.
In the rest of this work we will focus on configurations with ℓ = 1. Such ℓ-
boson stars are described by N = 3 fields, with m =−1,0,1 respectively. In order
to follow the dynamics of the perturbed system a numerical code that solves the
Einstein-N-Klein-Gordon system is required. We have used the Einstein Toolkit
framework [14, 15, 204] with the Carpet package [205, 206] for mesh-refinement
capabilities to achieve our goal.
As a technical step we perform a Cauchy (3+1) decomposition on each scalar
field that constitutes the star and solve the full Einstein-N-Klein-Gordon system.
This is done implementing an arrangement in the Einstein Toolkit, a thorn, to
solve N scalar fields using finite differences [63].
This paper is organized as follows: Section 8.2 addresses the construction of
Stability 155
initial data to set up perturbed ℓ-boson stars. Section 8.3 describes the diagnostic
tools used to monitor the evolution and some aspects used to decide on whether
instabilities are present. The numerical results are described in Section 8.4 and
in Section 8.5 our conclusions and final remarks are presented. In this work we
use units where G = 1= c.
8.2 Initial Data
Following previous works on ℓ-boson stars [99,101], we consider a set of N = 2ℓ+1
complex scalar fields, with mass µ and no self-interaction within the Einstein
theory of gravity, for which the energy-momentum tensor is given by:
Tαβ =
N
∑
i=1
T
(i)
αβ
, (8.1)
where the index i labels each field and the stress-energy-momentum for each field
is given by
T
(i)
αβ
=
(
∇αΦi ∇βΦ
∗
i +∇βΦi ∇αΦ
∗
i
)
+gαβ
(
∇σΦi∇σ
Φ
∗
i +
1
2
µ2|Φi|2
)
. (8.2)
Complex conjugation is denoted by ‘*’. Following [99, 100] we propose a set of
scalar fields of the form
Φ
(i)(t, r,ϑ,ϕ)=ψℓ(r, t)Y ℓm(ϑ,ϕ) , (8.3)
where the angular momentum number ℓ is fixed, and m, which plays the role of
index i in equation (8.3), takes the values m = −ℓ,−ℓ+1, . . . ,ℓ (hence the total
number of fields needed for a fixed value of ℓ will be 2ℓ+1), Y ℓm are the spherical
harmonics defined over the unitary 2D-sphere. Then we assume that the ampli-
tudes ψℓ(r, t) are the same for all m. It was shown in [99] that if the N fields have
all the same amplitude ψℓ, the stress-energy tensor (8.1) has spherical symmetry
regardless if the fields have angular dependence. See also [100] and for a detailed
discussion on the procedure, see [122].
Assuming the harmonic time dependence
ψℓ(r, t)=φℓ(r)e−iωt , (8.4)
where φℓ(r) and the frequency ω are both real-valued, the stress-energy tensor
becomes time independent. Under these assumptions it is possible to find self
gravitating static, spherically symmetric equilibrium configurations by solving
the EKG system of equations. Those configurations are parametrized by the an-
gular momentum number ℓ, hence the name, ℓ-boson stars.
156 8.2 Initial Data
In order to obtain initial data suitable for numerical evolution, we construct
equilibrium ℓ-boson stars, to be subsequently perturbed. Considering a spheri-
cally symmetric spacetime with a line element given by:
ds2 =−α(r)2 dt2 + A(r)dr2 + r2(dϑ2 +sin2ϑdϕ2) , (8.5)
where α and A, are functions of r, and the assumptions mentioned above for ψℓ,
the EKG system yields
∂2
rφℓ = −∂rφℓ
(
2
r
+
∂rα
α
−
∂r A
2A
)
+ Aφℓ
(
µ2 +
ℓ(ℓ+1)
r2
−
ω2
α2
)
, (8.6)
∂r A = A



(1− A)
r
+4πrA
[
(∂rφℓ)2
A
+ φ2
ℓ
(
µ2 +
ℓ(ℓ+1)
r2
+
ω2
α2
)





, (8.7)
∂rα = α
[
(A−1)
r
+
∂r A
2A
− 4πrAφ2
ℓ
(
µ2 +
ℓ(ℓ+1)
r2
)
]
. (8.8)
By studying the Klein-Gordon equation in the vicinity of r = 0 one finds that
the scalar field behaves as φ∼φ0rℓ in that region. For a fixed value of the angular
momentum number ℓ, a given value of the parameter φ0, and the boundary con-
dition at infinity requesting that φℓ decays exponentially, the system of equations
(8.6-8.8) becomes a nonlinear eigenvalue problem for the frequency ω. We solve
this set of equations in a finite size grid by means of a shooting method using the
frequency ω as the shooting parameter. For numerical purposes we take the mass
parameter µ= 1.
Fig. 8.1 shows a plot of the Arnowitt-Deser-Misner (ADM) mass M versus the
frequency ω for the ℓ-boson stars. In Ref. [99] it was shown that ℓ-boson stars
with ℓ> 0 have similar properties to those of single-field mini-boson stars, i.e. the
ℓ = 0 case. For instance, given a value of ℓ, the mass M of the equilibrium con-
figurations as a function of ω has a maximum, which gets larger as ℓ increases,
yielding more compact stars. Furthermore, as in the case of 0-boson stars, the
maximum value of the mass separates the space of solutions into two branches.
These branches correspond to stable and unstable configurations against spheri-
cal perturbations, as shown in [101].
As mentioned above, the hypothesis that all the fields must have the same
amplitude is essential to keep the spherical symmetry of the configuration. If one
wants to consider different amplitudes of each constituent field, the assumption
of spherical symmetry has to be relaxed. However, hitherto there has been no
evidence that the resulting states may be equilibrium solutions of the Einstein-
N-Klein-Gordon system. In this work we will show that deviations from spherical
symmetry may indeed lead to new equilibrium solutions.
To proceed further with our non-spherical analysis we transform the solutions
of the previous system of equations to Cartesian coordinates, xµ = (t, r,ϑ,ϕ) →
Stability 157
Figure 8.1: ADM mass vs. frequency for static ℓ-boson stars. The properties of
models M1, M2 and M3 are listed in Table 8.1.
xµ = (t, x, y, z). Then we perform a full non-linear numerical evolution of the per-
turbed stationary solutions.
8.3 Diagnostics
In order to test the stability of the static solutions we perform two different types
of perturbations:
(i) The first type consists in perturbing the energy density of the star given by
ρ = nαnβTαβ, where nα is the four velocity of Eulerian observers in the 3+1 space
time decomposition. The perturbed energy density is obtained by adding a non
spherically symmetric small amplitude term to the homogeneous density in the
following way [207]:
ρ = ρ0

1+κ
(
x2 − y2
R2
99
)

 (8.9)
where ρ0 is the energy density of the equilibrium configuration, obtained from
the solution of Eqs. (8.6)-(8.8), and R99 is the radius enclosing 99% of the config-
uration’s mass. In our simulations we choose κ= 0.1. This type of perturbations
could trigger a potential bar-mode instability because it only affects the Ixx and
I yy components of the quadrupole moment defined as
Ixx =
∫
ρ(y2 + z2)dV , I yy =
∫
ρ(x2 + z2)dV . (8.10)
Since the value of κ is small κ ≪ 1, this perturbation can be considered linear,
initially; more importantly, it breaks the spherical symmetry of the original solu-
tion.
(ii) The second type of perturbations consist in varying separately the ampli-
tude of each field. With these perturbations it is possible to study the stability of
158 8.4 Time evolution and numerical results
the stars against variations on each mode m and break the spherical symmetry.
We choose the following form
φℓ,m = (1+ǫ)φℓ, (8.11)
where φℓ is the unperturbed solution of the system of Eqs. (8.6)-(8.8). This per-
turbation introduces an additional constraint violation, besides the well known
numerical error, but its magnitude is controlled by choosing a small ǫ, which, in
general, depends on the ℓ,m-mode. Note that if ǫ is the same for all m, the per-
turbation is spherical.
In order to assess the stability properties of the stars during the numerical
simulation, we monitor the mass of the star, its angular momentum, and its den-
sity. We also follow the change in the quadrupole moment of the star, as shown
below. Following the technique described in [207, 208] to examine the stability
of rotating neutron stars, we monitor the behaviour of the distortion parameter
defined as
ηz :=
Ixx − I yy
Ixx + I yy
, (8.12)
which is a good measure of the magnitude of the bar-mode instability for pertur-
bation (i). This parameter has been used to study the stability of rapidly, differen-
tially rotating stars [207]. It has been observed that when the star is dynamically
unstable, ηz grows exponentially up to a maximum value; then the maximum
value of ηz remains constant on dynamical timescales. For stable stars, on the
other hand, the maximum initial value of ηz remains constant throughout the evo-
lution. Thus the monitoring of ηz provides a good tool to determine the properties
of the star against bar-mode perturbations. In this work we also use
ηy :=
Ixx − Izz
Ixx + Izz
, (8.13)
as a measure of the deformation of the star.
As further diagonostics, the maximum of the density and the lapse function
are used to determine whether the configuration disperses or is undergoing a col-
lapse. We have used the thorn AHFinder [209] to follow the formation of an appar-
ent horizon (AH) during the evolution. We have also computed the Hamiltonian
constraint [13] to check the fourth order convergence of the implementation - see
the Appendix for details on this procedure.
8.4 Time evolution and numerical results
In this section we present the results from dynamical spacetime simulations from
the perturbed ℓ-boson stars. We have compared the evolution of the equilibrium
ℓ-boson stars with the perturbed stars.
Stability 159
Model ℓ ω/µ µR99 µMADM
M1 1 0.882 13.45 1.133
M2 1 0.836 12.75 1.176 (maximum)
M3 1 0.783 7.53 1.122
Table 8.1: Frequency, radius and ADM mass for the configurations analysed.
We numerically integrate the EKG system using fourth-order spatial discretiza-
tion within the Einstein Toolkit framework. The Einstein Toolkit solves the
Einstein equations within the ADM 3+1 framework and evolves the spacetime
using the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Ein-
stein equations [210] through the McLachlan thorn [211]. All the evolutions were
made using the 1+log time slicing condition for the lapse α, and the Gamma-driver
condition for the shift βi [10].
We use the Method of Lines thorn to solve the equations in time by using a
fourth order Runge-Kutta scheme. The equations for the scalar fields are solved
using a finite difference scheme of fourth order. We also employ the mesh refine-
ment capabilities provided by the Carpet arrangements. The fixed mesh refine-
ment grid hierarchy used consists of nested cubes with 3 levels of refinement. The
finest is set in such a way that it covers the entire star.
We set the spatial resolution on the finest level to {dx,dy,dz} = 0.8 (and the
coarsest to {dx,dy,dz} = 3.2) in order to fully capture the properties of the star.
We follow the formation of an AH after the collapse of unstable stars.
More details on the resolution, as well as numerical convergence are given in
the Appendix.
The three stationary configurations we chose to illustrate the general behaviour
of the stars are represented with a square over the curve in Fig. 8.1 denoted as
(M1, M2, M3). Some of the properties of these stars are summarized in Table 8.1.
Both spherical and non-spherical perturbations to the stationary solutions are
induced by increasing or decreasing the amplitude of the different constituent
fields, see Eq. (8.11). In our case of study, ℓ = 1 and thus, for each configuration
M1, M2, M3 there are three fields φℓ,m: {φ1,−1, φ1,0, φ1,1}. We use the position
of sub-index in the models (M1m=−1,m=0,m=1) to label the mode (field) that is be-
ing perturbed. We use + or − to ascribe an increase (ǫ > 0) or decrease (ǫ < 0) of
the amplitude, we use 0 to represent that no perturbation was introduced in that
mode (ǫ= 0). In this way, for instance, M1+0− means that M1 has been perturbed
in the following way: the first scalar field, φ1,−1, has been perturbed with ǫ > 0;
the second field φ1,0 has not been perturbed (ǫ = 0), and the third field φ1,1 has
been perturbed with ǫ < 0. In summary, we have perturbed M1, M2, M3 in the
following ways: perturbing all fields with the same amplitude as a test (spher-
160 8.4 Time evolution and numerical results
Model M1 (ω/µ= 0.882) ǫ Collapse
Run m =+1 m = 0 m =−1
M1000 0 0 0 No
M1+++ +0.01 +0.01 +0.01 No
M1−−− −0.01 −0.01 −0.01 No
Model M2 (ω/µ= 0.836) ǫ Collapse
M2000 0 0 0 No
M2+++ +0.01 +0.01 +0.01 Yes
M2−−− −0.01 −0.01 −0.01 No
Model M3 (ω/µ= 0.783) ǫ Collapse
M3000 0 0 0 Yes
M3+++ +0.01 +0.01 +0.01 Yes
M3−−− −0.01 −0.01 −0.01 No
Table 8.2: List of simulations performed for the case where all fields are perturbed
with the same amplitude (spherical perturbations). These cases are similar to the
simulations performed in [101].
ical perturbation), introducing a non-axisymmetric bar-mode perturbation, and
finally, we perturbed each constituent field using different amplitudes.
8.4.1 Spherical perturbation test
First, we perform numerical evolutions of the models listed in Table 8.1 with
spherical perturbations. We induce perturbations in each field of a ℓ-boson star
with all the perturbations having the same amplitude. In this way we guarantee
that the spherical symmetry is preserved. This type of perturbations is done in
order to compare and validate our results with those found using a spherically
symmetric 1D code reported in [101]. While perturbing the initial equilibrium
configurations adding perturbations (with a positive or negative value for ǫ) that
preserve the spherical symmetry, we find that the configuration that was reported
to be stable in Ref. [101] (model M1) remains stable in the timescale we reach in
the 3D simulations, run M1000. The values of the amplitude of the perturba-
tions for these perturbed configurations are reported in Table 8.2 as M1+++ and
M1−−−, in which we perturb each field adding or subtracting |ǫ| = 0.01 to each
mode.
Our results are also consistent with models of ℓ-boson stars that are unstable
in spherical symmetry. According to the results in [101], the configuration M3000
is unstable in the 1D simulations. When we perturb the amplitudes of the fields
adding (run M3+++) or subtracting (run M3−−−) the same amount, the configura-
tion collapses or migrates to the stable branch respectively, as described in Table
Stability 161
8.2. We monitor the behaviour of the metric coefficients during the evolution, and,
in particular, we use the lapse and the formation of an AH as an indicator of the
collapse of the star and the formation of a black hole.
The model M2 deserves special mention since it corresponds to
the critical solution: the star with maximum mass. We found that pertur-
bations increasing the amplitude of the field (ǫ > 0, run M2+++) make the star
collapse whereas perturbations that decrease the amplitude (ǫ < 0, run M2−−−)
drive the configuration to a new stable state as described in Table 8.2. These
results are consistent with the results reported in [101] for perturbations that
increase of decrease the mass of the star.
8.4.2 Non-spherical perturbation: perturbing the energy density
In order to determine whether ℓ-boson stars develop a bar mode instability, we
took as initial data a stationary model and modified the energy density in accor-
dance with eq. (8.9). We have performed this analysis for configurations with
ℓ = 0 and ℓ = 1, both stable against spherical perturbations. In the case ℓ = 0
we have taken the equilibrium configuration corresponding to ω/µ= 0.937 and for
ℓ= 1 the configuration with ω/µ= 0.882, M1 in Table. 8.1. By choosing κ= 0.01,
the momenta of inertia Ixx and I yy change by less than 0.5% with respect to the
equilibrium solution, hence we consider that the induced initial perturbation is
small. Then we evolve the perturbed system via the Einstein-N-Klein-Gordon
equations and monitor the behaviour of ηz.
In Fig. 8.2 we show ηz as a function of time for perturbed and unperturbed con-
figurations for ℓ= 0 (top panel) and ℓ= 1 (bottom panel). For ℓ= 0, the distortion
ηz oscillates around zero for the perturbed case, indicating the star maintains,
essentially, the spherical symmetry.
On the other hand, for ℓ= 1 in the case where the perturbation was included,
the initial perturbation induces a small deviation from spherical symmetry there-
fore ηz acquires a nontrivial value during the evolution. This non-zero value of ηz
indicates that the shape of the star deviates from spherical and becomes oblate.
During the evolution time considered (t ∼ 3500) we did not find any signal of a
bar-mode instability for the models considered: no exponential growth in ηz was
measured. Most importantly ηz does not grow as it happens for unstable stars
[63]. This change in the shape of the ℓ = 1 configuration, illustrated by ηz 6= 0,
is compared with the case where M1 is not perturbed (black solid line). For the
unperturbed case, ηz simply oscillates around zero. The conclusion, therefore, is
that the perturbed configuration lingers, neither collapsing nor dissipating, thus
showing a non-spherical distribution that is either stable or long-lived, without
signs of instability. It is worth emphasising the key difference with the ℓ= 0 case,
for which the evolution oscillates around a spherical distribution, in agreement
162 8.4 Time evolution and numerical results
Figure 8.2: Evolution of ηz as a function of time for unperturbed (black solid line)
and perturbed density as defined by Eq. (8.9) (red solid line) with κ = 0.01. Top
panel: the unperturbed configuration is a single-field boson star (ℓ = 0). Bottom
panel: the unperturbed configuration is a multi-field boson star (ℓ= 1). In neither
perturbed case has a bar instability been observed. Instead, a long lived departure
from spherical symmetry occurs for ℓ = 1, but not for ℓ = 0, as long as ηz 6= 0.
In contrast, the unperturbed configurations preserve spherical symmetry as ηz
oscillates around zero in both cases.
with the fact that such a distribution is the only equilibrium configuration.
We found that after some time, the stars acquire a small linear momentum
due to the numerical error and thus the deviation parameters can not be obtained
accurately. Once this becomes noticeable we stop the evolution.
8.4.3 Non-spherical perturbation: perturbing the amplitude of each mode
In this section, we describe the evolutions we have performed implementing non-
spherical perturbation by varying the amplitude of each field of the ℓ-boson star.
Stability 163
Model M1 (ω/µ= 0.882) ǫ Collapse
Run m =+1 m = 0 m =−1
M1+00 +0.01 0 0 No
M1−00 −0.01 0 0 No
M10+0 0 +0.01 0 No
M10−0 0 −0.01 0 No
M100+ 0 0 +0.01 No
M100− 0 0 −0.01 No
M1++0 +0.01 +0.01 0 No
M1+−+ +0.01 −0.01 +0.01 No
M1+0+ +0.01 0 +0.01 No
M1−0+ −0.01 0 +0.01 No
M10(−)0 0 −0.1 0 No
Table 8.3: List of simulations performed for model M1 under the second type of
perturbations. The parenthesis indicate a larger amplitude on the perturbations.
Notice that gravitational collapse was not observed in any of the simulations.
Figure 8.3: Distortion parameter ηy, for model M1 for runs shown in Table (8.3)
. Departure from spherical symmetry is shown for those configurations that have
been perturbed differently in all the three fields φ1,−1, φ1,0 and φ1,1. On the
contrary, the configuration M1000 has the same perturbation for all the fields
(spherical perturbation), and it shows ηy = 0 at all times.
164 8.4 Time evolution and numerical results
Non-spherical perturbation of M1
In order to illustrate the procedure to perturb the star, let us consider first model
M1. Different perturbations have been applied to M1 and all them are sum-
marised in Table 8.3. Subscripts indicate which fields have been perturbed and if
the amplitude is increased by ǫ> 0 (subscript +), decreased by ǫ< 0 (subscript −)
or it has been left without perturbation ǫ= 0 (subscript 0), as described before.
All our evolutions show that M1 remains stable without collapsing (none AH
was found), independently of the perturbation. Thus, from the results summarised
in Table 8.3 we can conclude that the configurations in the stable branch (against
spherical perturbations), that is, to the right of configuration M2 in Fig. 8.1, are
also stable under non-spherical perturbations, against collapse.
Let us now turn to another result that can be extracted by studying the distor-
tion parameters ηz and ηy. For spherical configurations and for those configura-
tions that are spherically perturbed, these are zero. On the other hand, for non-
spherical perturbations a small deviation from spherical symmetry is induced.
In other words, non-trivial values of ηy are obtained throughout the evolution of
M1. This behaviour of ηy as a function of time is shown in Fig. 8.3. Indeed, non-
zero values of ηy for the evolution of M1+00, M100,+, M1−00, M100−, M10+0, and
M10−0, are obtained. No instability is observed, but the deformation does not die
off either. These long-lived deformed configurations, arising as dynamical solu-
tions of the Einstein-N-Klein-Gordon with three fields with different m are not
spherically symmetric. The corresponding perturbations appear to be zero modes,
suggesting a larger family of solutions.
As expected, we observe that the equilibrium configuration M1000 has ηy = 0
at all times of the evolution. Besides, we have found that the behaviour of ηy and
ηz during the evolution is the same for perturbations in the modes m = 1 and
m =−1 with the same values of ǫ. This suggests that the resulting configurations
are axially symmetric.
Finally, we report in Fig. 8.4 the time evolution of the total mass of M1 under
different spherical and non-spherical perturbations. As expected, the mass of
the perturbed configurations decreases or increases when ǫ < 0 or ǫ > 0. Notice,
however, the change in mass is the same whether we perturb the m =−1 or m =
1 modes, for the same the sign of ǫ. This fact supports the assertion that the
resulting configurations are axially symmetric.
The total mass of the models decreases with time, showing a small drift, even
for the unperturbed solution. We have checked that this drift is due to the numer-
ical error, since it is reduced when the grid resolution is increased (see Appendix
A).
Fig. 8.5 displays a series of snapshots of projections in the planes xy and xz of
the energy density for the run M10(−)0 (With a large amplitude in the perturbation
Stability 165
Figure 8.4: Evolution of the mass of the model M1 subjected to spherical and
non-spherical perturbations listed in Tables 8.2 and 8.3 respectively. The mass,
as expected, is increased for those perturbations with ǫ > 0 and decreases when
ǫ< 0.
in the mode m = 0) . The initial perturbation is introduced in the mode m = 0,
decreasing the mass of the star and inducing a small deformation. Notice that we
have taken the largest perturbation presented in this section (model M10(−)0), so
that the deformation can be appreciated in these projections of the energy density.
At later times the system evolves and settles down into a configuration without
collapsing or exploding. Although the configuration looks almost spherical, the
value of ηy at late times is slightly different from zero (ηy ∼ 0.05). We have evolved
this configuration for t ∼ 10000 and remains in the same state not showing any
signs of instability, or returning to a ℓ-boson star.
At this point it is important to mention that perturbations in the mode m =
0 do not modify the value of the total angular momentum, while perturbations
in the modes m = 1 or m = −1 do. Specifically, M1−00 and M100+ (M1+00 and
M100−) have a positive (negative), non-trivial and constant value of total angular
momentum. As we could expect, runs like M1+0+ have zero angular momentum.
This particular result was also obtained for the perturbations of the models that
will be presented below.
Non-spherical perturbation of M2 and M3
The results of the previous section indicate that those configurations (M1) that are
stable under spherical perturbations do not show non-spherical growing modes.
Furthermore, perturbations to the fields φ1,−1 and φ1,1 applied to those configura-
tions provide evidence for zero modes, producing new equilibrium configurations
that are dynamically stable, and have small departures from the spherical con-
figurations.
166 8.4 Time evolution and numerical results
Figure 8.5: Three snapshots of the projection of the rest mass density in two
planes. In the second snapshot the star expands and thus the maximum value
of the density decreases. In the third snapshot the star returns to its original
state. This repetitive behaviour is present during all the evolution time. The
mesh represent a box with sides 1
2 R99 of the unperturbed star.
Now we are interested in studying non-spherical perturbations on configura-
tions that might undergo gravitational collapse to a black hole. Those configu-
rations are M2 and M3. In particular M2, as we have mentioned, corresponds
to the critical configuration with maximum ADM mass. Configuration M2, as is
shown in Section 8.4.1 and reported in the literature [101], divides stable from un-
stable configurations. The latter are those configurations that can collapse into
a black hole. We now go one step further and study them under non-spherical
perturbations.
The list of perturbation applied to each field of M2 is summarised in Table
8.4 and the result of the evolution is reported in the fifth column of the same
Table. The results confirm that M2 is the configuration that separates stable from
unstable configurations. Indeed, as it can be observe in the results of Table 8.4,
all those configurations which have perturbations that increased the total mass of
the configuration undergo a collapse, while those configurations which have been
perturbed and reduced the total mass of the configuration did not collapse to a
black hole. In this respect, run M2−0+ is of special interest. The perturbation did
not change the total mass of the configuration, and the result of their evolution is
that it did not collapse.
Finally, we have considered non-spherical perturbations of model M3. The
results of the evolution of the different models studied are summarised in Table
Stability 167
Model M2 (ω/µ= 0.836) ǫ Collapse
Run m =+1 m = 0 m =−1
M2+00 +0.01 0 0 Yes
M2−00 −0.01 0 0 No
M20+0 0 +0.01 0 Yes
M20−0 0 −0.01 0 No
M200+ 0 0 +0.01 Yes
M200− 0 0 −0.01 No
M2++0 +0.01 +0.01 0 Yes
M2+−+ +0.01 −0.01 +0.01 Yes
M2+0+ +0.01 0 +0.01 Yes
M2−0+ −0.01 0 +0.01 No
Table 8.4: List of simulations performed from the model M2 under non-spherical
perturbations. Those configurations that increased the total mass by the addition
of the perturbation did collapse to a black hole. Configurations that did not change
the total mass or did not decrease the total mass of the configuration did not
collapse.
8.5. This configuration is on the so called unstable branch. The results mimic, to
some extent, those observed for M2. Perturbations that reduced the total mass
of the configuration led to a migration to the stable branch. On the other hand,
those that increased the mass of the configuration let to a collapse into a black
hole. But a key difference is seen for M3−0+. This perturbation did not change
the total mass of the configuration, and contrary to M2−0+, it did collapse to a
black hole. This result, combined with the spherical perturbations mentioned
in 8.4.1, further confirms the special status of the maximum mass configuration
M2: it marks the threshold of unstable configurations of ℓ-boson stars, for both
spherical and non-spherical perturbations.
8.5 Discussion and Outlook
In this paper we performed dynamical simulations in the fully non-linear EKG
model to investigate the stability of ℓ-boson stars. Unlike previous works we have
considered non-spherical perturbations. An expected result is that those config-
urations known to be unstable under spherical perturbations, are also unstable
under more general perturbations. The most interesting question, however, was
if the configurations known to be stable under spherical perturbations would re-
main stable under more general ones. Here, our conclusions are two-fold. Firstly,
no growing modes have been measured in our simulations. In this sense ℓ-boson
stars are stable against non-spherical perturbations. However, when deformed
168 8.6 Appendix
Model M3 (ω/µ= 0.783) ǫ Collapse
Run m =+1 m = 0 m =−1
M3+00 +0.01 0 0 Yes
M3−00 −0.01 0 0 No
M30+0 0 +0.01 0 Yes
M30−0 0 −0.01 0 No
M300+ 0 0 +0.01 Yes
M300− 0 0 −0.01 No
M3++0 +0.01 +0.01 0 Yes
M3+−+ +0.01 −0.01 +0.01 Yes
M3+0+ +0.01 0 +0.01 Yes
M3−0+ −0.01 0 +0.01 Yes
Table 8.5: List of simulations performed form the model M3 under non-spherical
perturbations. Those configurations that increased the total mass by the addition
of the perturbation did collapse to a black hole. Configurations that decreased
the total mass of the initial configuration, migrated to the stable branch. The run
M3−0+ that did not change the total mass did collapse to a black hole.
away from sphericity, ℓ-boson stars do not return to a spherical state. They appear
to oscillate around a new (slightly) non spherical state. We take this as evidence
that new, multi-field, equilibrium configurations of the Einstein-N-Klein Gordon
system exist, which are non-spherical. This conjecture is our second conclusion.
If our conjecture is proven correct, the spherically symmetric ℓ-boson stars are
only an enhanced isometry point of a larger family of solutions of the Einstein-N-
Klein Gordon. As discussed in the introduction, this is analogous to the Schwarzschild
BH being the isometry enhancement point of the Kerr family. It is well known
that the Kerr solution brings about qualitatively novel features with respect to
the Schwarzschild solution. So it will be quite interesting to understand the nov-
elties brought by the enlarged family of solutions that this work is suggesting.
The conjecture on the existence of these new non-spherical, multi-field config-
urations can be tested by solving the Einstein-N-Klein Gordon system for static or
stationary (i.e. spinning) configurations, without assuming spherical symmetry.
Research in this direction is already ongoing.
8.6 Appendix of Chap. 8. Code validation
For run M1000 we report the time evolution of the mass and violations of the
Hamiltonian constraint, with different resolutions, {dx,d y,dz}= 3.2, {dx,d y,dz}=p
2 1.6 and {dx,d y,dz} = 1.6, where dx, d y and dz are the sizes of the coarsest
level of refinement
Stability 169
Figure 8.6: Convergence for run M1000: Evolution of the L2-norm of the Hamilto-
nian constraint for three different resolutions rescaled to show fourth order con-
vergence. Green line shows L2-norm of the Hamiltonian constrain for a perturbed
run, initially, the violation of the constraint due to the perturbation is evident, as
time passes the magnitude of the error is comparable with the error of the unper-
turbed runs.
The L2-norm of the Hamiltonian constrain, given by |H|2 =
√
∑N
i=1
H2
i
N
, where
N is the number of points in the grid, is shown in Fig. 8.6. Here we conclude that
the constraint equations converge as H reduces when the resolution is increased,
the black (solid) and the blue (dashed) line have been multiplied by the factors 4
and 16, showing fourth order convergence. The low resolution (solid line, coarsest
grid dx = 3.2) corresponds to the one used in all the simulations presented in this
work. The L2-norm of H increases with time, however tends to a constant value
which approaches zero as {dx,d y,dz}→ 0.
We plot the mass for run M1000. As the resolution is increased the mass con-
verge to a constant value and the overall drift is reduced. In Fig. 8.6 and Fig. 8.7
we have plotted until time equal to 1300, however the ({dx,d y,dz} = 3.2) simu-
lation extends up to t ∼ 3500, where the final total mass differs from the initial
value by 0.6%.
170 8.6 Appendix
Figure 8.7: Convergence for run M1000: Evolution of the mass for three different
resolutions.
CHAPTER 9
COLLISIONS1
Contents
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.2 Models for ℓ-boson star binaries . . . . . . . . . . . . . . . . . . 174
9.2.1 Coherent and incoherent states . . . . . . . . . . . . . . 175
9.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . 176
9.3.1 Initial data . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.3.2 Evolution Scheme . . . . . . . . . . . . . . . . . . . . . . 178
9.4 Head-on Collision Dynamics . . . . . . . . . . . . . . . . . . . . . 180
9.4.1 Analysis Quantities . . . . . . . . . . . . . . . . . . . . . 181
9.4.2 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . 182
9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
9.5.1 Aligned stars . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.5.2 Non-aligned stars . . . . . . . . . . . . . . . . . . . . . . 194
9.6 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
F
ully non-linear numerical evolutions of the Einstein-(multi)–Klein-Gordon
equations are performed to study head-on collisions of ℓ-boson stars. De-
spite being spherically symmetric, ℓ-boson stars have a (hidden) frame
of reference, used in defining their individual multipolar fields. To assess the
impact of their relative orientation, we perform simulations with different an-
gles between the axes of the two colliding stars. Additionally, two scenarios are
considered for the colliding stars: that they are composites of either the same or
different scalar fields. Despite some model-specific behaviours, the simulations
generically indicate that: 1) the collision of two sufficiently (and equally) massive
stars leads to black hole formation; 2) below a certain mass threshold the end
1This chapter is extracted with minor revisions from Ref. [212], which was written in collaboration
with N. Sanchis-Gual, J. Barranco, A. Bernal, J. C. Degollado, C. Herdeiro, M. Megevand and D.
Núñez. © American Physical Society. Reproduced with permission. All rights reserved.
171
172 9.1 Introduction
result of the evolution is a bound state of the composite scalar fields, that neither
disperses nor collapses into a black hole within the simulation time; 3) this end
product (generically) deviates from spherical symmetry and the equipartition of
the number of bosonic particles between the different scalar fields composing the
initial boson stars is lost, albeit not dramatically. This last observation indicates,
albeit without being conclusive, that the end result of these collisions belongs to
the previously reported larger family of equilibrium multi-field boson stars, gener-
ically non-spherical, and of which ℓ-boson stars are a symmetry enhanced point.
We also extract and discuss the waveforms from the collisions studied.
9.1 Introduction
The advent of the gravitational wave (GW) era promises to deliver invaluable
information on some of the most prominent challenges in theoretical physics.
Amongst these is the nature of the dark Universe. At the time of writing, the
LIGO-Virgo and now Kagra (LVK) collaborations released three public catalogues
from the first three science runs O1+O2 [213], O3a [214] and O3b [215], report-
ing 85 events. These events are providing invaluable information about black hole
populations [216], constraints on dark energy models [217–220] and even tanta-
lizing hints about the nature of dark matter [95].
LVK searches are performed using “matched filtering”, a data analysis tech-
nique to detect characteristic signals in noisy data, which requires a library of
theoretical waveform models. In this respect, the issue of degeneracy has been an
understated caveat in GW detections. The black hole interpretation seems vindi-
cated within the Kerr black hole paradigm of general relativity. There is, however,
a lack of alternative models for which waveforms have been accurately produced,
in order to assess whether matched filtering really selects general relativity black
holes within a more extensive library of theoretical templates.
Within this empty landscape of alternatives, bosonic stars offer a unique op-
portunity. First constructed in the late 1960s [44,45] for massive, complex scalar
fields (and more recently for massive complex vector fields [52]) minimally coupled
to Einstein’s gravity, these are self-gravitating solitonic solutions that are both
compact [54,221] and dynamically robust in regions of their parameter space [46,
56, 60, 63], forming from fairly generic initial data and reaching the equilibrium
state by "gravitational cooling" [47, 61, 63], possibly complemented by GW emis-
sion (in non-spherically symmetric evolutions). Thus, they offer an (non-black
hole) alternative relativistic two body problem which, unlike neutron stars, need
not have electromagnetic counterparts. As a matter of critical importance, the
well-posedness/hyperbolicity of the evolutions is under control, and currently avail-
able computational infrastructures can be used with fairly minor adaptations,
Collisions 173
such as the Einstein Toolkit [14, 204, 222]. This state of affairs contrasts with
modified gravity, wherein even promising models may face fundamental issues,
such as the breakdown of hyperbolicity, see e.g [223]. Thus, one can use bosonic
star binaries to produce waveform banks that share features, but also present
differences, with respect to those generated by the vanilla black hole binaries of
general relativity. In fact, one such recent analysis raised the intriguing possibil-
ity that a real GW event [224] could be interpreted as a collision of vector bosonic
stars [95], showing, at least as a proof of concept, how such interpretation would
lead to the discovery of a fundamental, ultralight dark matter particle.
The simplest bosonic stars arise in single (complex) field models. A realiza-
tion of the last few years, however, is that there is a wider landscape of bosonic
star families when allowing multi-field models. Among these different possibili-
ties, one of the most interesting configurations are the ℓ-boson stars [99], as it has
been proved that they are the only stable configuration [104], bestowing them with
a central role in the possible scalar field configurations. ℓ-boson stars are spher-
ically symmetric solutions arising in a model with 2ℓ+1 complex scalar fields,
wherein the individual fields carry a multipolar structure but the composite ob-
ject is spherical and static. Note that ℓ-boson stars reduce to standard boson
stars in the particular case where ℓ = 0. Subsequent studies showed these so-
lutions are dynamically robust in regions of the parameter space [101, 102] and
also unveiled they can be regarded as part of a wider family of multi-field, multi-
frequency bosonic stars [85, 103, 104]. ℓ-boson stars in contrast to ℓ = 0 regular
boson stars have greater compactness and they are hollow in the central region.
This empty space becomes larger as ℓ increases as well as their mass and reaches
the maximum compactness as ℓ → ∞ [85]. The maximum compactness for the
ℓ-boson stars almost doubles the maximum compactness pf ℓ= 0 boson stars. Fol-
lowing this line of thoughts it becomes an interesting problem to evolve ℓ-boson
stars in binaries, both to further test their dynamical robustness and to obtain
new waveform templates, that enlarge the effort of constructing a vaster library
of non-Kerr waveforms. These are precisely the goals of this paper.
In this work we shall study head-on collisions of ℓ-boson stars, as the sim-
plest “binaries" of these objects. Even in this simplest scenario, the multi-field
nature of ℓ-boson stars allows more possibilities than when a single field is con-
sidered. Firstly, despite being described by a spherically symmetric metric and
total energy-momentum tensor, the composite nature of ℓ-boson stars endows the
individual fields with a (hidden) frame of reference with respect to which the mul-
tipolar structure of the individual fields is defined. Thus, even for head-on colli-
sions starting from rest, there is the additional degree of freedom of misaligning
the hidden spin axes of the individual stars. Secondly, since one is entertaining
the possibility of many scalar fields, it fits such rationale to allow the ℓ-boson stars
174 9.2 Models for ℓ-boson star binaries
to be composed by the same, or by different, scalar fields. We shall dub the former
(latter) scenarios as tackling coherent (incoherent) states. As we shall see, both
choices, i.e. alignment and coherence, lead to important changes in the dynamics,
although some generic trends can also be inferred from the sample of simulations
performed.
We have focused on binary ℓ-boson stars with ℓ= 1, using several initial config-
urations and evolving them while analysing the spatial distribution of the scalar
fields and computing the GW emission. When the individual stars are massive
and compact enough, it is found that the remnant is a black hole. However, if
the sum of the masses of both stars does not greatly exceed the maximum mass
of the corresponding family of solutions, (see Fig. 9.1 below), the collision forms
a gravitationally bound scalar field configuration. Whereas in the merger of two
standard ℓ = 0 boson stars some of the final configurations clearly tend towards
another ℓ= 0 boson star [57–59, 62, 195], our results indicate that the merger of
two ℓ-boson stars, albeit remaining in a bound state, do not necessarily lead to an
ℓ-boson star.
The waveforms generated from the head-on collision of the ℓ-boson stars, like
in the usual case of boson stars with ℓ= 0, present very peculiar features, which
make them significantly different from the waveform of a black hole collision, even
nudus oculus. As we show in the present work, the waveform produced by the
head-on collision of ℓ-boson stars has a richer structure depending on the param-
eters of the initial configuration (such as the relative alignment), which makes a
stronger case for the generation of catalogues to be included in the LVK libraries.
This paper is organized as follows. In the next section, we present the main
ideas needed to construct ℓ-boson stars, in particular explaining the two different
scenarios that shall be considered: when taking two spatially separated lumps
composed by the same fields, called coherent states, or when taking two spatially
separated lumps composed of different fields, called incoherent states. Next, we
describe the numerical implementation of these configurations and in section 9.4
we define the quantities that will be analyzed, together with the GWs during the
evolution of the system. We present our results in section 9.5 and conclude with
some final remarks. Throughout the paper we use natural units, c =G = ~= 1.
9.2 Models for ℓ-boson star binaries
A single ℓ-boson star is described by an odd number N of complex scalar fields,
each with a harmonic time dependence, of the form
Φℓm(t, r,ϑ,ϕ)= e−iωtφℓ(r)Yℓ,m(ϑ,ϕ) , (9.1)
where Yℓ,m(ϑ,ϕ) are the standard spherical harmonics. Notice that the angular
momentum number ℓ for a given solution is fixed and a single star has N = 2ℓ+1
Collisions 175
fields, corresponding to each possible value of m within the range −ℓ, . . .0, . . . ,+ℓ.
A key ingredient to get a spherically symmetric solution of the Einstein-Klein-
Gordon (EKG) system is that the field amplitude φℓ(r) is precisely the same for
all m.
We shall be considering models without self-interactions amongst the different
scalar fields. These fields, therefore, only see each other via gravity. Still, when
considering two stars we may choose that the composing fields of the stars are
equal or are different. In this work we shall consider both possibilities. This
is reminiscent of the description of coherent and incoherent states in quantum
mechanics; for the former case, a macroscopic number of quanta all pile into the
same momentum state, being used to describe lasers and superfluids.
Our approach is as follows. The binary system we consider is governed by the
EKG theory and the field equations for the metric gµν are
Rµν−
1
2
gµνR = 8π
(
T(1)
µν +T(2)
µν
)
:= 8πTµν , (9.2)
where Rµν is the Ricci tensor and R = gµνRµν. The matter content is given by
either one or two sets of 2ℓ+1 complex scalar fields Φℓm, each with a stress-energy
tensor of the form
T(i)
µν =
1
2
ℓ
∑
m=−ℓ
[
∇µΦ̄
(i)
ℓm
∇νΦ
(i)
ℓm
+∇µΦ
(i)
ℓm
∇νΦ̄
(i)
ℓm
− gµν
(
∇αΦ̄
(i)
ℓm
∇α
Φ
(i)
ℓm
+µ2
Φ̄
(i)
ℓm
Φ
(i)
ℓm
)
]
,
(9.3)
where i = 1,2; Φ̄ℓm denotes the complex conjugate of Φℓm and µ is the mass of the
scalar field particle, which we assume is the same for all fields. This assumption
amounts to consider that all different scalar fields belong to a larger multiplet.
As expected, each complex scalar field satisfies the Klein-Gordon equation:
gµν∇µ∇νΦ
(i)
ℓm
−µ2
Φ
(i)
ℓm
= 0 . (9.4)
9.2.1 Coherent and incoherent states
As mentioned above, in this work we consider two possible systems. We shall refer
to the first system as coherent state and to the second as incoherent state. The
functional description of both states is as follows.
i) Coherent states. For this case, both ℓ-boson stars are made up of the same
set of scalar fields. Such a scenario is modeled with a single set of fields, initially
accumulated at two (essentially) disjoint spatial regions, that is:
Φ
(1)
ℓm
6= 0 , and Φ
(2)
ℓm
= 0 , ∀ m . (9.5)
For this system there are 2ℓ+1 independent fields to describe the binary. In this
scenario a single set of fields fills up spacetime, which can nonetheless clump at
two different locations, forming two ℓ-boson stars centered at different positions.
176 9.3 Numerical implementation
ii) Incoherent states. Here, each star is composed by a set of 2ℓ+1 fields, being
different for each star. This requires to turn on both sets of fields discussed
Φ
(1)
ℓm
6= 0 , and Φ
(2)
ℓm
6= 0 , ∀ m . (9.6)
Consequently, there are 2(2ℓ+1) independent fields to describe the binary. Notice,
however, that in both systems the interaction between any of the fields is only
through gravity.
9.3 Numerical implementation
In order to describe the dynamics of the binary it becomes necessary to evolve
2ℓ+1 complex fields for the coherent system and 2(2ℓ+1) complex fields for the
incoherent system. For concreteness, we focus in this work on the simplest non-
trivial ℓ= 1 case yielding 3 fields for coherent states and 6 for incoherent states.
9.3.1 Initial data
Initial data for the binary system are obtained using a superposition of two iso-
lated ℓ-boson stars. The construction of single isolated ℓ-boson stars is described
in detail in Ref. [99]; here we outline a brief description of the procedure.
The starting point is to consider a static and spherically symmetric spacetime
of the form
ds2 =−α2dt2 +γ jkdx jdxk =−α2dt2 +a2dr2 + r2 dΩ2 , (9.7)
where α and a are functions of r, dΩ2 is the line element on the unit 2-sphere
and the scalar fields that compose the ℓ-boson stars have a harmonic time depen-
dence given by (9.1). According to this assumptions it has been shown in [99] that
even though the scalar field oscillates in time the stress-energy tensor is time in-
dependent and the EKG equations yield static solutions that are described by the
following set of ordinary differential equations:
φ′′
ℓ =−φ′
ℓ
(
2
r
+
α′
α
−
a′
2a
)
+aφℓ
(
µ2 +
ℓ(ℓ+1)
r2
−
ω2
α2
)
, (9.8)
a′
a
=
(1−a)
r
+4πra


(φ′
ℓ
)2
a
+φ2
ℓ
(
µ2 +
ℓ(ℓ+1)
r2
+
ω2
α2
)

 , (9.9)
α′
α
=
(a−1)
r
+
a′
2a
−4πraφ2
ℓ
(
µ2 +
ℓ(ℓ+1)
r2
)
, (9.10)
where a prime denotes derivative with respect to r. By studying the Klein-Gordon
equation close the origin r = 0 one finds that the scalar field behaves as φℓ ∼φ0rℓ.
Collisions 177
For a given value of φ0, and demanding that the scalar field has an exponential
decay and the metric is Minkowski at infinity, the EKG system becomes a non-
linear eigenvalue problem for the frequency ω.
The equilibrium configurations are found by integrating numerically Eqs. (9.8)-
(9.10), considering appropriate boundary conditions, by means of a shooting method
using the frequency ω as the shooting parameter. The solutions are identified by
the value of ω, although for some ranges of ω there may be more than one solution,
defining different branches –see Fig. 9.1 (left).
ℓ-boson stars share many features with the single field ℓ= 0 boson stars. Both
exist only for a limited range of frequencies and achieve a maximum Arnowitt-
Deser-Misner (ADM) mass. Fig. 9.1 (left) displays the mass of ℓ-boson stars versus
ω. The maximum mass solution separates stable from unstable configurations as
described in [102]. In this work we shall only consider configurations in the stable
branch. More concretely, the stars we shall use as initial data for the evolution
below are marked with a box on the existence curve in Fig. 9.1. We define the
boundary of the star as the radius of the spherical surface that encloses 99% of
the mass; this radius is referred to as R99 and it is displayed for the solutions
in Fig. 9.1 (right). We define the compactness of the stars as C := M/R99 and
it is displayed as an inset in the same panel. In order to use the infrastructure
provided by the Einstein Toolkit we transform the solutions to the usual Cartesian
coordinates, xµ = (t, r,ϑ,ϕ)→ xµ = (t, x, y, z) as
x = r cosϕsinϑ , y= rsinϕsinϑ , z = r cosϑ . (9.11)
In our present investigation we also include configurations that involve an inter-
mediate rotation in the angles to describe a relative misalignment between the
stars, as already discussed above and described in detail below. The initial data
for the binaries used in this work are obtained by a linear superposition of the
isolated solutions of two stars given by the spatial metric and a set of scalar fields
{γ(−)
jk
,Φ(−)
ℓm
} and {γ(+)
jk
,Φ(+)
ℓm
} for star 1 and star 2, respectively. We shall consider
the stars are centered at (xc,0,0) and (−xc,0,0). We also consider the stars to be
initially at rest (at t = 0).
In this work we restrict our attention to ground state solutions, for which the
amplitudes Φℓm have no nodes. These are the solutions exhibited in Fig. 9.1.
We take as initial data for the spatial metric the superposition [57–59,62,195,
225]:
γ jk = γ(+)
jk
(x− xc, y, z)+γ(−)
jk
(x+ xc, y, z)− γ̂ jk(x, y, z). (9.12)
where γ̂µν is the flat spatial metric. For the scalar fields we need to distinguish
between the two systems presented above. For coherent states we construct the
field as
Φ
(1)
m (t = 0, x, y, z)=Φ
(+)
m (x− xc, y, z)+Φ
(−)
m (x+ xc, y, z) , (9.13)
178 9.3 Numerical implementation
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0.0
0.2
0.4
0.6
0.8
1.0
1.2
µ
M
0
N
A
B
C
D
A
BC
D
ℓ = 1
ℓ = 0
0.70 0.75 0.80 0.85 0.90 0.95 1.00
ω/µ
0
10
20
30
40
50
60
µ
R
9
9
N
A
B
C
D
0.90 0.95 1.00
0.00
0.03
0.05
0.08
C
N
A
B
C
D
ABC
D
ℓ = 1
ℓ = 0
Figure 9.1: ADM mass (left) and radius (right) vs. frequency for static ℓ-boson
stars with ℓ= 0,1. The maximal value of the boson star mass for ℓ= 0 and ℓ= 1
is µM0 = 0.63 and µM0 = 1.18 respectively. The black squares correspond to the
solutions used in the head-on collision analysis. In the inset of the right plot the
compactness is shown as a function of the frequency.
and Φ
(2)
m (t, x, y, z)= 0. Whereas for incoherent states, the field is constructed as
Φ
(1)
m (t = 0, x, y, z)=Φ
(+)
m (x− xc, y, z) ,
Φ
(2)
m (t = 0, x, y, z)=Φ
(−)
m (x+ xc, y, z) . (9.14)
Due to the nonlinearity of Einstein equations these initial data introduce con-
straint violations. How this effect has been tracked and controlled throughout the
simulations is discussed in Appendix 9.7.
By means of the transformation (9.11), the stress-energy tensor of each star
is defined with respect to a Cartesian frame (x, y, z). For the initial data, instead
of taking the spherical harmonics of each star defined with respect to the same
Cartesian frame, we may consider a relative misalignment of the two correspond-
ing Cartesian frames, by performing a rigid rotation. This allows a more general
scenario, in which the stars have arbitrary initial orientations. As we shall see,
this has an interesting impact on the dynamics during the merger and in the re-
sulting configuration.
In order to model such non aligned stars we define an intermediate set of
coordinates x′ = r cosϕsinϑ, y′ = rsinϕsinϑ, z′ = r cosϑ, and perform a transfor-
mation of the form x= Ri(δ)x′, where i = x, y, z are the Einstein Toolkit Cartesian
coordinates and Ri(δ) is the rotation matrix for an angle δ around the i-axis. The
effect of the rotations Rz(π), Rz(π/2) and Ry(π) can be visualized in Fig. 9.2.
9.3.2 Evolution Scheme
Our numerical simulations are performed using the open source Einstein Toolkit
infrastructure [14]. The Einstein equations are integrated in time using the
Collisions 179
I Rz(π) Ry(π)
|Y
1
,−
1
(ϑ
,ϕ
)|
|Y
1
,0
(ϑ
,ϕ
)|
|Y
1
,1
(ϑ
,ϕ
)|
Figure 9.2: Amplitude of spherical harmonics for the modes m =−1, m = 0, m = 1,
that compose a boson star with ℓ= 1. The first column is taken as a reference for
the usual alignment; the second column shows the effect of a rotation Rz(π); and
the third column represents a rotation Ry(π). As a reference, we have drawn a a
red and a blue line on the surfaces. This is particularly useful when comparing
the first two columns, which would otherwise look identical.
180 9.4 Head-on Collision Dynamics
Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation [210]. We use the
Method of Lines of the MoL thorn to solve the equations, via a fourth order Runge-
Kutta scheme provided by the McLachlan thorn [211]. The scalar field evolution
code has been recently employed to study the stability of ℓ-boson stars [103] and
it is described in more detail in that reference.
All the evolutions were made using the 1+log time slicing condition for the
lapse α, and the Gamma-driver condition for the shift βi [10]. We have used
the thorn AHFinder [209] to follow the formation of an apparent horizon (AH)
during the evolution. We have also monitored the Hamiltonian and momentum
constraints [13] to check the convergence during the evolution.
The numerical grid consists of nested cubes with six refinement levels and it
is handled using the fixed mesh refinement grid hierarchy included in the Carpet
arrangements.
In all simulations the stars have their centers placed at µxc = 25. The com-
putational domain consists of a box with µxmax = 500 = µymax = µzmax, with a
grid structure of six refinement levels. The waveform extraction is made at µr =
100,120. The spatial domain of the refinement levels is {500,50,50,25,25,10}. We
set the resolution on the finest level to µ{∆x,∆y,∆z}= 0.3125. This corresponds to
at least 86 points across the initial stars’ diameter (defined as 2R99). The choice
of a fixed mesh refinement is mainly motivated by simplicity. The last refinement
level that do not contain the stars is included in order to have enough resolution
in case a black hole forms. Refinement levels of the grid can produce reflections.
However, we do not see any significant impact of the crossing of the mesh bound-
aries on the dynamics and on the determination of the different quantities (mass,
particle number, Hamiltonian constraint). The Cartesian grid can induce a m = 4
perturbation (see, for instance, [226,227]).
9.4 Head-on Collision Dynamics
Once we have constructed the initial data, we allow the systems to evolve freely
while monitoring the Hamiltonian and momentum constraints to check the ac-
curacy of the results. As described above, both the evolution and the number
of scalar fields are different depending on the system. For the coherent states
we evolve 3 complex fields whereas for the incoherent states we evolve 6 fields.
We have found that the main differences between the two systems occur near the
plunge and this is reflected in the emitted GW signal and in the posterior outcome
of the plunge. We quantify the differences using some analysis quantities, such as
the mass of the final configuration, the number of particles, and the GW signal.
Collisions 181
9.4.1 Analysis Quantities
Starting from the static superposition initial data, the stars approach each other
and eventually collide. In some of the scenarios we have established that the final
object after the merger is a black hole. To diagnose its appearance in the evo-
lution we use the AHFinder thorn and then compute the mass of the black hole
through the apparent horizon area A, using the relation MBH =
p
A/16π, which
is valid for a Schwarzschild black hole. The use of the Schwarzschild relation re-
lies on no-hair theorems for static spherically symmetric black holes, ruling out
final equilibrium configurations with scalar hair, even with a harmonic time de-
pendence. Such no-hair theorems are, however, circumvented for spinning black
holes [203].
We focus primarily on the configurations for which the end state has no horizon
and the rest mass density is non zero. For the final time reached in our evolution,
the final remnant is a localized, perturbed distribution of the different complex
scalar fields. However, it is not possible to determine whether that object corre-
sponds to an ℓ-boson star. We describe some of its properties in the next sections.
For the total gravitational mass of localized solutions we use the Komar inte-
gral.
M =−
∫
Σ
p
γd3xα (2T t
t −Tµ
µ) , (9.15)
where Σ is a spacelike slice extending up to spatial infinity, γ is the determinant
of the 3-metric induced on that slice and α is the lapse function. To describe the
end state of the collision we compute the Noether charge associated with the total
bosonic number of particles N, which is defined as
N =
∫
Σ
p
γd3xα g0µ j(i)
µ , (9.16)
where j
(i)
µ =
∑
m
i
2
(Φ̄(i)
m ∇µΦ
(i)
m −Φ
(i)
m ∇µΦ̄
(i)
m ). This conserved current is associated
with the global invariance of the theory under the action of a U(1) group for each
field. We will use this quantity to classify the remnant of the merger. Note that
the Noether charge can also be computed for each field. The integrals for the
Komar mass (9.15) and the number of particles (9.16) are performed in the entire
numerical grid on each time step.
We monitor the energy density of matter as ρ = nµnνTµν during the evolution
as a measure of the energy left after the collision, where nµ is the unitary normal
vector to Σ.
Furthermore, in order to determine the deviations from spherical symmetry
of the post merger configuration, we compute the moments of inertia Ixx, I yy and
Izz defined by
Ixi xi =
∫
Σ
p
γd3xαρ(r2 − xi2
) . (9.17)
182 9.5 Results
9.4.2 Gravitational Waves
Gravitational radiation is extracted from the numerical simulations by comput-
ing the Newman-Penrose scalar Ψ4 = Cαβγδkαmβkδmγ, where Cαβγδ is the Weyl
tensor and k and m are two vectors of the null Kinnersley tetrad [228, 229]. Far
from the source Ψ4 represents an outward propagating wave and has been used
as a measure of the gravitational radiation emitted during the merger of compact
objects. In order to analyze the structure of the radiated waves it is convenient to
decompose the signal in -2 spin weighted spherical harmonics as
Ψ4(t, r,ϑ,ϕ)=
∑
l,k
Ψ
l,k
4
(t, r)−2Yl,k(ϑ,ϕ) . (9.18)
According to the peeling theorem the leading order decay of Ψ4 is 1/r [66]. We use
this fact to check the accuracy in the computation of the gravitational waveforms.
Our description will focus directly on the strongly dominant component l = 2, k =
2.
9.5 Results
We have investigated five different cases of the ℓ-boson star datasets correspond-
ing to different values of compactness. All the initial configurations correspond
to two stars of the same type, localized on the stable branch, and with total mass
(sum of the two stars’ mass) that is larger than the maximal mass of the model.
The stars have radii (as defined before) ranging from µR99 = 56.5 (model N), to
µR99 = 13.5 (model D), corresponding to a compactness ranging from C = 0.0074
to C = 0.0838, respectively. In this section, we extensively discuss the dynamics
for the various cases.
The physical attributes of the initial data and some properties of the end state
of the collisions are summarized in Table 9.1 (for coherent states) and Table 9.2
(for incoherent states). Note that the properties of the initial data coincide in
both tables, but not the end state. The models CHl1N and INl1N that appear in
both tables will appear only on the analyzes related to the study of the final state.
Concerning the initial data the tables show the typical size R99, the frequency,
the Komar mass of each star M0, the number of particles in one of the fields of
each star (i.e. 1/3 of the total number of particles of each star, as described in
Appendix 9.7) and the compactness.
We define the time of collision tc, as the time at which the spheres given by
R99 of each star intersect. Furthermore, in order to give a simple estimation of
the object’s size after the merger, we still use the aforementioned definition of R99
as the radius of a sphere containing 99% of the total mass, even when the object is
not spherically symmetric. The center of such sphere is set at the center of mass,
x = y= z = 0.
Collisions 183
Coh. ℓ= 1
Model µR99 ω/µ µM0 µ2N0 C Rem. µR ∼ µtc
CHl1N 56.5 0.990 0.418 0.419 0.0074 BS - 0
CHl1A 31.5 0.970 0.697 0.703 0.0221 BS 24 0
CHl1B 27.5 0.962 0.775 0.784 0.0282 BS 22 70
CHl1C 24.7 0.954 0.837 0.849 0.0391 BH µrAH = 2.1 150
CHl1D 13.5 0.883 1.13 1.17 0.0838 BH µrAH = 3.4 240
Table 9.1: Coherent cases for ℓ = 1. R99 is the radius that contains 99% of the
mass of the star. ω is the frequency, M0 is the mass of each star, N0 is the number
of particles of each star and C is the compactness. The end state of the simulation
can be a localized boson configuration (BS) or a black hole (BH). R is the radius
that encloses 99% of the mass for scalar field remnant at µt = 2500 while for black
holes it labels the radius of the apparent horizon.
Incoh. ℓ= 1
Model µR99 ω/µ µM0 µ2N0 C Rem. µR ∼ µtc
INl1N 56.5 0.990 0.418 0.419 0.0074 BS - 0
INl1A 31.5 0.970 0.697 0.703 0.0221 BS 30 0
INl1B 27.5 0.962 0.775 0.784 0.0282 BS 23 0
INl1C 24.7 0.954 0.837 0.849 0.0391 BH µrAH = 2.2 0
INl1D 13.5 0.883 1.13 1.17 0.0838 BH µrAH = 3.6 230
Table 9.2: Same as Table 9.1 for incoherent states.
184 9.5 Results
9.5.1 Aligned stars
As discussed above, ℓ-boson stars are spherically symmetric at the level of the
total energy-momentum tensor but have an internal frame of reference with re-
spect to the different modes. The phrase “aligned stars” refers to both stars having
the same orientation. The properties of the remnants are presented in the last
columns of Table 9.1 for coherent states and the last columns of Table 9.2 for in-
coherent states. In particular, the mass of the merger remnant is computed using
the Komar integral (9.15). Also, some snapshots of the scalar field energy density
ρ, during the coalescence are displayed in Fig. 9.3 and 9.4. Fig. 9.3 exhibits the
evolution of a coherent state (model CHl1B) whereas Fig. 9.4 exhibits the evolu-
tion of an incoherent state (model INl1B).
 
 
 
x 
y 
y 
z 
Figure 9.3: Aligned evolutions of a coherent state. Snapshots of the scalar field
energy density, ρ/µ2, for model CHl1B on the z = 0 (top) and x = 0 (bottom) planes.
The maximum density of the final configuration for these models is one order of
magnitude larger than the progenitors. Time is given in units of the scalar field
mass, µ. Cycle refers to the iteration number.
Despite the fact that in all cases the initial mass of the system is above the
maximal mass of the model, we observe two qualitatively distinct behaviours. For
the most massive and compact models (C and D) a black hole is formed. But for
the less compact models (A and B) neither a black hole forms during the simula-
tion time, nor the field disperses away after the collision. In such cases, a bound
scalar field configuration remains after the merger, albeit the system is still evolv-
ing at the end of the simulation. Whereas one cannot state with certainty the final
outcome of the system, the results suggest that the system does tend to an equi-
librium lump of scalar fields. This is an asymptotic process, in which the gravita-
tional cooling mechanism plays a key role to allow relaxation, by slowly decreas-
ing the mass of the system. This slow mass ejection can be observed in Fig. 9.5,
where we plot the total mass for models CHl1A, CHl1B, and INl1A, INl1B as a
Collisions 185
 
 
 
y 
z 
x 
y 
Figure 9.4: Aligned evolutions of an incoherent state. Snapshots of the scalar
field energy density, ρ/µ2, for model INl1B on the z = 0 (top) and x = 0 (bottom)
planes. The maximum density of the final configuration for these models is one
order of magnitude larger than the progenitors.
0 500 1000 1500 2000
µ(t− tc)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
µ
M
CHl1B
2µM0 CHl1B
CHl1A
2µM0 CHl1A
0 500 1000 1500 2000
µ(t− tc)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
µ
M
INl1B
2µM0 INl1B
INl1A
2µM0 INl1A
Figure 9.5: Aligned evolutions. Collision remnant mass, for coherent states, mod-
els CH1A and CH1B (left) and for incoherent states, models IN1A and IN1B
(right).
function of time after the merger. The horizontal dotted lines indicate the value
of twice the initial mass of each star. From Figs. 9.3 and 9.4 one can observe that
the early phase of the encounter is qualitatively similar for both states; but once
the objects make contact, the dynamics becomes complex and model dependent.
Nonetheless, we observe the same separation between black hole formation and
scalar remnant formation, regardless of the states being coherent or incoherent.
The natural question is: what is the system evolving towards, when no black hole
forms? In particular, since ℓ-boson stars involve precisely equal amplitudes of
the scalar fields involved, these collisions test how fine-tuned these configurations
are, and if such a destructive process readily creates an asymmetry between the
different fields.
186 9.5 Results
1.0
1.5
µ
2
∑
N
(1
)
m
2µ2N0 (CHl1A)
2µ2N0 (CHl1B)
−5× 10−8
0× 10−8
1
−
N
(1
)
+
1
/N
(1
)
−
1
CHl1A
CHl1B
0 500 1000 1500 2000
µ(t− tc)
0.02
0.04
1
−
N
(1
)
+
1
/N
(1
)
0
Figure 9.6: Aligned evolutions. (Top) Total number of particles of the remnant of
the merger of coherent states CH1A, CH1B. The dotted lines represent twice the
initial number of particles present in the binary. (Middle) Ratio between the num-
ber of particles associated to the fields Φ
(1)
m with m =+1 and m =−1. It remains
constant at unity. (Bottom) Ratio between the number of particles associated to
the fields Φ
(1)
m with m =+1 and m = 0. At t = 0 this quantity is not exactly unity,
this is effect of the superposition and appears only in this (coherent) case. Dur-
ing the coalescence a small percentage of their initial value is lost. There is no
leaking or exchange of particles of individual fields during the evolution.
To tackle this question one may look at the Noether charge, or particle num-
ber, in each field. First, we observe that (as discussed in Appendix 9.7) for a single
ℓ-boson star the total number of particles, N, is equally distributed amongst each
field. Denoting the Noether charge of each field as Nm, then Nm = N
2ℓ+1
. Thus,
we can investigate if such equipartition of the Noether charge remains after the
collision. To this end we have computed the time evolution of the number of par-
ticles associated to each scalar field. This is displayed for the coherent models
CHl1A, CHl1B in Fig. 9.6 and for the incoherent models INl1A, INl1B in Fig.
9.7. One observes that, independently of the state, after the merger there is a
monotonic loss of Noether charge, consistent with the gravitational cooling pro-
cess. Moreover, this scalar field leaking is rather democratic; the equipartition
between the m±1 modes is kept to high accuracy, and the relative difference be-
tween the m = 0 and m = ±1 modes is kept to a few percent. Thus, none of the
individual Noether charges (corresponding to the individual fields) suffers dra-
matic preferential losses, even though there is a slight suppression of the m =±1
modes with respect to the m = 0 mode.
To gain further insight into the post-merger dynamics, in Fig. 9.8 we have
looked at the spatial distribution of the Noether charge during the evolution for
model CHl1B. One observes that the m = 0 field keeps its morphology throughout
the evolution. The dynamics arises from the m = ±1 modes, which yield a non-
Collisions 187
0.6
0.8
µ
2
∑
N
(1
)
m
µ2N0 (INl1A)
µ2N0 (INl1B)
−5× 10−6
0× 10−6
5× 10−6
1
−
N
(1
)
+
1
/N
(1
)
−
1
0 500 1000 1500 2000
µ(t− tc)
0.00
0.02
0.04
1
−
N
(1
)
+
1
/N
(1
)
0
INl1A
INl1B
0.6
0.8
µ
2
∑
N
(2
)
m
µ2N0 (INl1A)
µ2N0 (INl1B)
−5× 10−6
0× 10−6
5× 10−6
1
−
N
(2
)
+
1
/N
(2
)
−
1
0 500 1000 1500 2000
µ(t− tc)
0.00
0.02
0.04
1
−
N
(2
)
+
1
/N
(2
)
0
INl1A
INl1B
Figure 9.7: Same as Fig. 9.6, but now for incoherent states INI1A and INI1B.
trivial dynamics as the two lumps become superimposed, in particular altering
their morphology. This can be corroborated in the left panel of Fig. 9.9, where a
difference can be observed between the moments of inertia Ixx and I yy ≈ Izz after
the collision. In the incoherent case, the asymmetry is also evident, as can be seen
in the right panel of Fig. 9.9. In both cases a difference between Ixx and I yy, Izz
remains after µt = 2500. The above analysis is consistent with the remnants be-
 
 
 
x 
y 
x 
z 
Figure 9.8: Aligned evolutions. Individual currents jm/µ for model CHl1B. Top
panel: m = 0 in the y= 0 plane. Bottom panel: m =+1 in the z = 0 plane.
ing undergoing a relaxation process towards an equilibrium configuration which
is a multi-field, but not necessarily spherically symmetric, boson star [103, 104].
In particular, for the coherent state, it could be close to an ℓ-boson star, by virtue
of the evolution of the moment of inertia, Fig. 9.9 (left panel). See also [59, 113]
for the discussion of multistate boson stars in the context of the standard ℓ = 0
boson stars.
We also look at the evolution of the oscillation frequency of the scalar field(s).
188 9.5 Results
500 750 1000 1250 1500 1750 2000 2250 2500
µ(t− tc)
0
200
400
600
800
1000
1200
Coherent
µ3Ixx
µ3Iyy
µ3Izz
500 750 1000 1250 1500 1750 2000 2250 2500
µ(t− tc)
0
200
400
600
800
1000
1200
Incoherent
µ3Ixx
µ3Iyy
µ3Izz
Figure 9.9: Aligned evolutions. Moments of inertia of the post merger configura-
tion as defined in Eq. (9.17) for the models CHl1B (left panel) and INl1B (right
panel). Note that the curves of I yy and Izz coincide perfectly in both cases. In the
incoherent case, the configuration does not seem to tend to a spherically symmet-
ric distribution of scalar field. Despite the fact that the moments of inertia used
in Eq. 4.3 are gauge dependent they represent a good indicator of the presence of
nonlinear stability, as has been described in [207].
During the merger process, this frequency changes, which can be investigated by
performing a spectral analysis. To this end we have evaluated the discrete Fourier
transform (DFT) in time of each scalar field component, m =−1, 0 and 1. For more
accuracy, we have actually evaluated the DFT at five different points and then
calculated the average. Figure 9.10 shows the DFT for the case CHl1B. We notice
0
2
4
6
8
10
12
14
16
18
20
0 0.2 0.4 0.6 0.8 1
CHl1B
0.1
1
10
0.85 0.86 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 0.95
D
F
T
[a
u
]
ω/µ
m = −1
m = 0
m = 1
Figure 9.10: DFT in arbitrary units (au) vs. frequency. The inner figure shows a
zoom-in of he most relevant region in logarithmic scale.
that more than one frequency peak arises, correspond to several bound modes
with ω< µ. Moreover, at least up to the resolution we could achieve, the spectra
of the three m components coincide, not only in the peaks locations, but also in
Collisions 189
their amplitude. From this viewpoint, therefore, the different modes remain in
synchrony, as for the spectrum of the equilibrium states, the ℓ-boson stars. The
key difference is, however, that the ℓ-boson stars spectra would only show a single
peak.
To better resolve the excited modes, one would have to increase the frequency
resolution. This resolution, however, is proportional to the inverse of the evolution
time, which implies a high computational cost to improve the frequency resolution.
In fact, the case shown here was run up to µt = 14231, a much longer evolution
than for every other model in this work. For this reason we have not considered
the spectrum of the other models.
Let us now turn our attention to the GWs emission. In Fig. 9.11 we show the
real part of the dominant quadrupolar (l = 2, k = 2) mode of the Newman-Penrose
scalar Ψ4 as a function of time for both coherent states listed in Table 9.1 and
for incoherent sates listed in Table 9.2. The signal is extracted at µrext = 100
and we have scaled the amplitude with a factor r to better capture the asymp-
totic behaviour. The horizontal axis has been shifted as t → t− rext. For models
CHl1A, INl1A with relatively low compactness (C = 0.0221) the GWs signal dis-
play some differences between the two. This differences are still visible for models
with larger compactness CHl1B and INl1B (C = 0.0282). But for models CHl1C
and INl1C the final remnant collapses and the differences in the signal are almost
negligible. The waveform indeed resembles the signal of the collision of two black
holes. Finally for the most compact binary CHl1D and INl1D, the final product of
the merger forms a black hole and the waveforms are identical for both states.
To quantify the differences between waveforms as well as the similarities with
the head-on collision of black holes, mentioned in previous statements, we per-
form the following pair of analyses. First, we calculate the Fourier transform
of Ψ4 to obtain the frequencies of the gravitational signal for the different mod-
els. The results are shown in Fig. 9.12, from which it can be confirmed that the
main differences in the signal occur in relation to the compactness of the stars;
the peak frequency increases by a factor of 4 between model A and C, whether
the superposition is coherent or incoherent. Second, we compute the difference
between the collapsing models, CHl1C, CHl1D, INl1C and INl1D. To this end,
we compare in each case with the gravitational signal from a head-on collision of
equal mass black holes hole with the same total mass as the boson star system.
In Fig. 9.13 these gravitational waveforms are shown. The more compact model,
CHl1D overlaps better with the black hole signal, in particular after the collapse,
at the ringdown phase. For the model CHl1C the GW signal differentiates from
the black hole signal qualitatively and quantitatively at the first stage, however it
matches roughly the frequency and phase of the black hole collision at ringdown.
The model CHl1C is close to the BH/BS remnant limit (see Table 9.2), it involves
190 9.5 Results
more complex dynamics before collapse, which is imprinted in the GW at early
times.
100 200 300 400 500 600 700 800
µ(t− rext)
−0.0002
−0.0001
0.0000
0.0001
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 31.5 CHl1A, INl1A
coherent
incoherent
100 200 300 400 500 600 700 800
µ(t− rext)
−0.0006
−0.0004
−0.0002
0.0000
0.0002
0.0004
0.0006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 27.5 CHl1B, INl1B
coherent
incoherent
100 200 300 400 500 600 700 800
µ(t− rext)
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 24.7 CHl1C, INl1C
coherent
incoherent
100 200 300 400 500 600 700 800
µ(t− rext)
−0.003
−0.002
−0.001
0.000
0.001
0.002
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 13.5 CHl1D, INl1D
coherent
incoherent
Figure 9.11: Waveforms for ℓ = 1 binaries described in Table 9.1 and Table 9.2.
The extraction radius is µrext = 100.
The more compact the binary, the larger the amplitude of the gravitational
waveform; for the most compact objects the amplitude is almost one order of mag-
nitude larger than for their less compact counterparts. This result applies for
both coherent and incoherent states. This phenomenon is related with the size
and geometry of the binary. For ℓ-boson stars the maximum energy density is not
located at the geometrical center of the star; the shape of these objects is more like
Collisions 191
0.00 0.02 0.04 0.06 0.08 0.10
f/µ
10−8
10−7
10−6
10−5
D
F
T
Ψ
2,
2
4
[a
u]
CHl1A
CHl1B
CHl1C
CHl1D
0.00 0.02 0.04 0.06 0.08 0.10
f/µ
10−8
10−7
10−6
10−5
D
F
T
Ψ
2,
2
4
[a
u]
INl1A
INl1B
INl1C
INl1D
Figure 9.12: Fourier transform of the mode l = 2, k = 2 of Ψ4 for coherent (left)
and incoherent (right) models.
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
CHl1C
BH
200 300 400 500 600 700 800
µ(t− rext)
0
1
2
3
R
el
.
D
iff
er
en
ce
−0.004
−0.002
0.000
0.002
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
CHl1D
BH
100 200 300 400 500 600 700 800
µ(t− rext)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
R
el
.
D
iff
er
en
ce
Figure 9.13: Difference between the black hole end state evolutions (CHl1C,
CHl1D, INl1C and INl1D) and the corresponding same mass black hole head-on
collision. Rel. Difference = |Ψ2,2
4
−Ψ
2,2
4,BH
|/max(Ψ
2,2
4,BH
).
192 9.5 Results
a spherical shell. As the value of ℓ increases, the maximum of the density tends to
the external boundary leaving a region with almost zero density at the geometri-
cal center. For very compact binaries the radius R99 is considerably smaller than
for the less compact ones and the encounter is more violent producing a stronger
gravitational signal.
To further investigate the post-merger behavior, let us compare it for the ℓ= 1
model B and for the merger of two standard ℓ = 0 boson stars, with the same
radius R99 as the ℓ = 1 model. The ℓ = 0 corresponding collisions were already
discussed in [57], where it was shown that the remnant approaches another ℓ= 0
boson star. In Fig. 9.15 (left) we can see that the mass decreases in the collision
process, reaching below the maximal mass allowed for ℓ= 0 boson stars. The mass
loss can be roughly divided into two phases. There is an initial slower decrease,
partly due to GW emission, followed by a larger decrease rate, due to scalar field
emission, i.e. gravitational cooling, that remains throughout the evolution. The
remnant oscillates around a spherical distribution; in fact an equilibrium boson
star with ℓ= 0, with the oscillation amplitude decreasing with time. We stress the
main relaxation process to attain the equilibrium state is gravitational cooling.
This can be confirmed in Fig. 9.14, where the energy radiated by GW is shown;
when compared to the total mass loss, in Fig. 9.5, it is seen that the emitted
gravitational radiation is at least one order of magnitude smaller than the the
total mass loss, in all the cases.
0 500 1000 1500 2000
µ(t− tc)
10−8
10−7
10−6
10−5
10−4
10−3
10−2
E
ra
d
/M
0
CHl1A
CHl1B
CHl1C
CHl1D
0 500 1000 1500 2000
µ(t− tc)
10−8
10−7
10−6
10−5
10−4
10−3
10−2
E
ra
d
/M
0
INl1A
INl1B
INl1C
INl1D
Figure 9.14: Total energy radiated by the gravitational wave.
The right panel of Fig. 9.15 exhibits the analogous behavior for the collision
of the stars with ℓ= 1. Similarities with the previous case are observable: there
is a mass loss (together with a decrease in the number of particles), with a GW
Collisions 193
0 2000 4000 6000 8000 10000 12000
µ(t− tc)
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
ℓ = 0 (CHl0B)
µ2
∑
Nm
µM
µMcrit (ℓ = 0)
0 5000 10000 15000 20000
µ(t− tc)
1.0
1.1
1.2
1.3
1.4
1.5
1.6
ℓ = 1 (CHl1B)
µ2
∑
Nm
µM
µMcrit (ℓ = 1)
Figure 9.15: Mass and total number of particles for long simulations of the merg-
ers of the ℓ= 0 CHl0B (Left) and ℓ= 1 CHl1B (Right).
emission component occurring initially and an expelling of scalar field throughout
the entire evolution. In this case, however, it is not clear that the final state oscil-
lates around a spherically symmetric configuration, say an ℓ-boson star. It can be
mentioned that the system, as in the previous case, keeps oscillating, but the os-
cillation amplitudes decrease more slowly in this case. Also binary configurations
consisting of boson stars with masses smaller than the CHl1A model have been
explored, of particular interest are those cases where the total mass of the system
is smaller than the maximum mass of the ℓ= 1 family of solutions, Mmax
0
= 1.176,
the results for these cases are very similar to those shown in Fig. 9.15, that is,
highly dynamic behavior at late times and a continuous and very slow loss of
mass compared to the corresponding ℓ = 0 models with the same value of R99.
For instance, low compactness models CHl1N and INl1N, which can be found in
Tables 9.1 and 9.2, are an example of this behaviour since the initial total mass
is smaller by half compared to Mmax
0
. In Appendix 9.7 we present the wave forms
of boson stars mergers with ℓ = 0 and ℓ = 1 in order to stress the role of the ℓ
parameter in the GW emission.
To establish how spherical the after merger is, one may investigate the evolu-
tion of the components of the inertia tensor, described above. For the ℓ= 0 boson
stars merger the difference between the diagonal components averages to zero,
confirming the tendency to sphericity. For the ℓ-boson stars with ℓ = 1 case, on
the other hand, this is not so. This supports the conclusion that the end state in
the merger of the ℓ= 0 boson stars tends towards a new ℓ= 0 boson star, whereas
for the ℓ-boson stars with ℓ = 1 merger, it does not seem to tend towards a new
ℓ-boson star with ℓ= 1, albeit still remaining a bound state of the scalar field. It
is possible that the asymptotic end state is a localized configuration with less sym-
metries; our simulations, however, can only raise this possibility, not establish it.
194 9.5 Results
1.0
1.5
µ
2
∑
N
(1
)
m
2µ2N0 (CHl1B)
Ry(π)
Ry(π/2)
Ry(π/4)
−5× 10−6
0× 10−6
5× 10−6
1
−
N
(1
)
+
1
/N
(1
)
−
1
0 500 1000 1500 2000
µ(t− tc)
−0.02
0.00
0.02
1
−
N
(1
)
+
1
/N
(1
)
0
1.0
1.5
µ
2
∑
N
(1
)
m
2µ2N0 (CHl1B)
Rz(π)
Rz(π/2)
Rz(π/4)
−5× 10−6
0× 10−6
5× 10−6
1
−
N
(1
)
+
1
/N
(1
)
−
1
0 500 1000 1500 2000
µ(t− tc)
0.000
0.025
0.050
1
−
N
(1
)
+
1
/N
(1
)
0
Figure 9.16: Non aligned stars. Total and individual (associated with each field)
number of particles. (Left) Rotations around y-axis. (Right) Rotations around
z-axis. As in Fig. 9.6, 1−N
(1)
+1
/N
(1)
0
shows expected deviations from unity at t = 0
due to coherent superposition.
9.5.2 Non-aligned stars
As discussed above, a binary system of ℓ-boson stars may be given a relative ori-
entation, despite the sphericity of the individual stars. We shall now discuss the
impact of this feature on the head-on collisions. Let us take the orientation of the
left (centered at µx = −25) ℓ-boson star fixed and rotate the right star (centered
at µx =+25) using the Ri matrices presented at the end of section 9.3.1. It turns
out that this relative orientation has very noticeable consequences in the outcome
of the merger. Specifically we will make rotations around the y and z axes (recall
that the collision is along the x axis) for the coherent model B, identifying the cases
by the transformation performed, i.e., the Ry(δ) and Rz(δ) respectively, where we
have chosen the angles of rotation as δ=π, π
2
and π
4
.
As a general observation before describing specific cases, we do not obtain ap-
preciable differences in the evolution of the mass or of the total number of particles
with respect to the aligned case, see the upper panels of Fig. 9.16, despite some
particularities that we will address later. Another similarity with the aligned co-
herent case is that the number of particles in the m =+1 mode is equal to that of
the m =−1 mode throughout the evolution, as indicated by the middle panels of
Fig. 9.16.
On the other hand, the relative behaviour of the number of particles in the
modes m = +1 and m = 0 can be different in the non aligned case, although the
relative difference is still small, never exceeding 10%. A difference found with
respect to the aligned configuration, is that that there are cases where the three
momenta of inertia become different.
Let us now discuss the specific cases with Ry(π), Ry(π/2) and Ry(π/4); the last
Collisions 195
two cases, however, are qualitatively similar. An ℓ = 1 boson star is a composite
of an m = 0 and m =±1 modes. The m = 0 mode has a dipole-like energy distribu-
tion, whereas the m =±1 modes are toroidal. In Fig. 9.17a we show snapshots of
the energy density for the z = 0 (top panels) and x = 0 (bottom panels) planes. One
observes that the maximum of the density can reach the origin. This behavior is
characteristic of this type of merger (rotation Ry(π)). The bottom panels confirm
that the symmetry along the (collision) x-axis is preserved during the merger.
Fig. 9.17b shows snapshots for the individual currents for the same collision.
The top panels correspond to the mode m = 0; one observes that the quadrupolar
shape is maintained all along the evolution, presenting a clearly repulsive effect
(attributed to the phase difference of π in this mode after the rotation) thus pre-
venting this mode from concentrating at the origin. The bottom panels show the
current for the mode m = +1 where there is not a well defined shape after the
merger.
 
 
 
x 
y 
y 
z 
(a) Scalar field energy density, ρ/µ2, for model CHl1B. Top panel: z = 0 plane. Bottom panel:
x = 0 plane.
 
 
 
 
x 
z 
x 
y 
(b) Individual currents jm/µ for model CHl1B. Top panel: m = 0 in the y= 0 plane. Bottom
panel: m =+1 in the z = 0 plane.
Figure 9.17: Non aligned stars Ry(π).
Regarding the Ry(π/2) collision, there is an even more intricate interaction
196 9.5 Results
amongst the modes of each star. In Fig. 9.18a we present snapshots of the energy
distribution in the z = 0-plane (top panels) and in the x = 0-plane (bottom panels).
The noticeable feature is that the symmetry of the configuration with respect to
the collision axis is lost. Another characteristic is that the maximum density can
reach the origin after the merger, despite the hollow shape of each star. Fig. 9.18b
shows the individual currents for the m = 0 (top panels) and the m =+1 (bottom
panels) modes. These examples show that the spherical symmetry of the end
 
 
x 
y 
y 
z 
(a) Scalar field energy density, ρ/µ2, for model CHl1B. Top panel: z = 0 plane. Bottom panel:
x = 0 plane.
 
 
 
x 
y 
x 
z 
(b) Individual currents jm/µ for model CHl1B. Top panel: m = 0 in the y= 0 plane. Bottom
panel: m =+1 in the z = 0 plane.
Figure 9.18: Non aligned stars Ry(π/2).
product of the merger is lost. One can further support this statement by looking
at the moments of inertia of the final configuration. Fig. 9.19 shows the moments
of inertia as a function of time for the rotated case Ry(π) and Ry(π/2). For the
case with CHl1B-Ry(π) one of the three moments of inertia is different. For the
case with CHl1B-Ry(π/2) the three moments of inertia remain all different. We
have also studied the evolution of a system where the non alignment is due to a
rotation around the z axis. This case shows several effects similar to the previous
case, for example, the attraction/repulsion of the modes and a loss of sphericity.
Collisions 197
500 750 1000 1250 1500 1750 2000 2250 2500
µ(t− tc)
0
200
400
600
800
1000
1200
Ry(π)
µ3Ixx
µ3Iyy
µ3Izz
500 750 1000 1250 1500 1750 2000 2250 2500
µ(t− tc)
0
200
400
600
800
1000
1200
Ry(π/2)
µ3Ixx
µ3Iyy
µ3Izz
Figure 9.19: Moments of inertia for the non aligned models CHl1B, with Ry(π)
(left) and Ry(π/2) (right). .
In Fig. 9.20a, we present the evolution of the energy density for the Rz(π)
rotation, and in Fig. 9.20b we present the corresponding evolution of the modes.
The energy density diagnosis shows, as in the rest of the cases analyzed, a loss of
the original symmetry of each star. As in the aligned case, this kind of rotations
keeps unchanged the initial m = 0 mode, see Fig. 9.2 (recall that the spherical
harmonic ℓ = 1, m = 0 does not depend on the angle ϕ). During the evolution it
can be seen that individually the m = 0 mode maintains its morphology, while
the m = ±1 have a drastic change in morphology, yielding the observed change
in the energy density. The Rz(π/2) rotation case is shown in Fig. 9.21a. We see
a twisting effect in the evolution of the energy density distribution in the z =
0−plane. Concerning the individual modes, the m = 0 mode tends to keep the
original morphology, while the m± 1 modes loose it, a behavior similar to the
previous case.
To conclude, we have observed that the rotations around the y−axis, Ry, pro-
duce more significant changes in the final morphology than the rotations along
the z−axis, which also modify the final morphology, but keep the shape of the
remnant m = 0 mode.
Regarding the GW profiles, we looked for any significant difference that could
indicate a waveform dependence (particularly in its amplitude) on the angle and
direction of rotation. As shown in Fig. 9.22, however, in none of the alignments
studied for the model B we have observed significant differences; the amplitude of
the seven cases is of the same order, although the first peak appears earlier in the
non-aligned cases, with a maximum difference in time with respect to the aligned
case of µ∆t ∼ 50 for the Ry(π) case. This is true for all Ψl,k
4
but in particular for
the dominant l = 2 k = 0,2 modes shown in Fig. 9.22.
Rotations Ry of the initial data cause slightly larger differences in the wave-
forms than the Rz cases. This is consistent with the fact, already pointed out, that
198 9.6 Final remarks
 
 
x 
y 
y 
z 
(a) Scalar field energy density, ρ/µ2, for model CHl1B. Top panel: z = 0 plane. Bottom panel:
x = 0 plane.
 
 
 
x 
y 
x 
z 
(b) Individual currents jm/µ for model CHl1B. Top panel: m = 0 in the y= 0 plane. Bottom
panel: m =+1 in the z = 0 plane.
Figure 9.20: Non aligned stars Rz(π).
unexpected dynamics on the m = 0 scalar field may occur, essentially because Ry,
unlike Rz, modifies the distribution of all the individual fields of the rotated star
and not only the m =±1 modes.
The GW signals just discussed can be compared with the ones previously pre-
sented for coherent (aligned) superposition and the incoherent one. For example,
comparing the upper right panel of Fig. 9.11 and 9.22, one observes that at the
level of the GWs. We remark that in hypothetical astrophysical scenarios an ex-
act alignment should be accidental, and the generic case should be of misaligned
stars. Concerning the end state, however, it is unclear if the coherence could have
a greater importance than the relative alignment of the stars.
9.6 Final remarks
In this paper we have studied head-on collisions of ℓ-boson stars, starting from
rest. These are composite self-gravitating solitons, made up of 2ℓ+ 1 complex,
Collisions 199
 
 
 
 
x 
y 
y 
z 
(a) Scalar field energy density, ρ/µ2, for model CHl1B. Top panel: z = 0 plane. Bottom panel:
x = 0 plane.
 
 
 
x 
y 
x 
z 
(b) Individual currents jm/µ for model CHl1B. Top panel: m = 0 in the y= 0 plane. Bottom
panel: m =+1 in the z = 0 plane.
Figure 9.21: Non aligned stars Rz(π/2).
massive scalar fields [99]. In such scalar lumps, the different scalar fields have
precisely the same amplitude, which raises the concern of possible self-tuning.
Yet, it has been shown that these solutions are dynamically robust in regions of
the parameter space [101, 102], at least against small perturbations. Here, we
test these solutions against more violent processes: head-on collisions of two such
equal stars.
Our simulations consider a variety of cases. As the two main scenarios we con-
sider that the two colliding ℓ-boson stars are made up of the same, or of different,
scalar fields. These two cases are dubbed coherent and incoherent, respectively.
Additionally, we consider different possible orientations of the colliding stars. It
may sound strange that such spherical stars have an "orientation"; yet they do.
The point is that the composing fields of each star have a multipolar structure - c f .
Fig. 9.2 -, defined with respect to a pre-established Cartesian reference frame. We
can thus choose that these reference frames coincide, or not, for the two colliding
stars. These two cases are dubbed aligned and non aligned, respectively.
Independently of the specific characteristics of each model that we have stud-
200 9.6 Final remarks
250 300 350 400 450 500 550 600 650 700
µ(t− rext)
−0.0010
−0.0005
0.0000
0.0005
0.0010
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
Ry
Ry(δ = 0)
Ry(δ = π/4)
Ry(δ = π/2)
Ry(δ = π)
250 300 350 400 450 500 550 600 650 700
µ(t− rext)
−0.0010
−0.0005
0.0000
0.0005
0.0010
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
Rz
Rz(δ = 0)
Rz(δ = π/4)
Rz(δ = π/2)
Rz(δ = π)
250 300 350 400 450 500 550 600 650 700
µ(t− rext)
−0.0010
−0.0005
0.0000
0.0005
0.0010
(r
/µ
)R
e(
Ψ
2,
0
4
)| µ
r=
10
0
Ry(δ = 0)
Ry(δ = π/4)
Ry(δ = π/2)
Ry(δ = π)
250 300 350 400 450 500 550 600 650 700
µ(t− rext)
−0.0010
−0.0005
0.0000
0.0005
0.0010
(r
/µ
)R
e(
Ψ
2,
0
4
)| µ
r=
10
0
Rz(δ = 0)
Rz(δ = π/4)
Rz(δ = π/2)
Rz(δ = π)
Figure 9.22: Waveforms for non aligned ℓ = 1 binary. CHl1B model. Top panels
show the real part of the l = 2, k = 2 mode. Bottom panels show the real part of
the l = 2, k = 0 mode. The extraction radius is µrext = 100.
ied, a first generic conclusion one can put forward is the following. The collision of
sufficiently massive (and therefore compact stars) stars forms a black hole. This is
what happens, e.g., for models C and D in Fig. 9.1, corresponding to the collisions
CH11C and CH11D in Table 9.1 and IN11C and IN11D in Table 9.2. But below
a certain mass threshold, the collisions do not lead to horizon formation; a bound
state of the different scalar fields remains, that neither disperses nor collapses.
This is seen, for instance, for models A and B in Fig. 9.1, corresponding to the
collisions CH11A and CH11B in Table 9.1 and IN11A and IN11B in Table 9.2.
Collisions 201
Concerning the end state of collisions that do not form a black hole, the sim-
ulations reveal different specificities, depending on the chosen characteristics of
the stars and of the collision. Still one sees two generic features. First, the democ-
racy between the different composing fields of each ℓ-boson star is lost, albeit not
dramatically. This feature can be diagnosed from the Noether charge in each field
- see e.g. Fig. 9.6 (for aligned coherent states), Fig. 9.7 (aligned incoherent states)
and Fig. 9.16 (for non aligned states). Observe, nonetheless, that the balance of
particles in the m =±1 modes is kept to high accuracy, and the slight imbalance
with the m = 0 mode does not exceed a few percent. Second, the collision after-
math deviates from spherical symmetry. This can be quantified by looking at
the moments of inertia, see Fig. 9.9 (for aligned collisions) and Fig. 9.19 (for non
aligned collisions). Observe that Ixx = I yy in all cases, except the non aligned col-
lisions under a rotation Ry(π/2); also note that in some cases the three momenta
of inertia, Ixx, I yy, Izz seem to converge after some time, which may be interpreted
as a glimpse of a tendency towards sphericity.
Overall, the above description, albeit not entirely conclusive, allows us to an-
swer the question we have set out to investigate: how dynamically robust are ℓ-
boson stars? The answer seems to be two-fold. Exact, equilibrium ℓ-boson stars,
with precise equipartition of the number of particles amongst the 2ℓ+1 fields are
indeed fine tuned, and will not withstand generic perturbations. This is no sur-
prise, and it was already anticipated by considering non-spherical perturbations
of equilibrium ℓ-boson stars [103]. Yet, ℓ-boson stars as a particular symmetry
enhanced point of a larger family of multi-field bosonic stars, as reported in [104],
can be considered to be dynamically robust and long lived. In fact, the violent
collisions we have considered, could only produce an imbalance in the Noether
charges, of the order of a few percent. Of course, the end states of our simula-
tions remain oscillating and have not yet reached a stationary state. We cannot
rule out that gravitational cooling and GW emission will work towards a spherical
distribution, getting rid of the non-spherical modes. However, our simulations do
not exhibit strong, generic evidence for this possibility. The very slow convergence
towards a final state is not a consequence of using models with initial total mass
larger than the maximum mass of the final expected state, an ℓ= 1 boson star, as
can be concluded from the CHl1N and INl1N cases presented.
Finally, let us comment that our collisions have generated waveforms which
can be quite different from those of BHs. As emphasized in the Introduction, it
will be interesting to continue this effort, in order to generate large libraries of
alternative waveform templates, to compare with observational data. This effort,
of course, will require going towards orbiting binaries.
202 9.7 Appendix
9.7 Appendices of Chap. 9
Appendix: Code validation
In all binary boson star configurations considered in this paper we have chosen
to separate the stars 50 coordinate units (µxc = 25). This responds to the fact
that this is a distance for which, at least in the solutions studied, the absolute
maximum of the violation of the Hamiltonian constraint H at t = 0 is approxi-
mately of the same order as the floor error of the numerical implementation, es-
timated from the single isolated star simulations. We verified that the L∞ norm
i.e., the maximum of the absolute value of the Hamiltonian constraint, decreases
as the separation of the stars is increased. For example, the value obtained if
the binary CHl1B is separated by a distance of 70, 50 and 30 coordinate units,
is L∞(H)(t = 0) = 9×10−5,10−4,2×10−3 respectively. Thus, for the chosen sep-
aration of 50, the violation induced for the constraint is of the same order of the
interpolation error. L∞(H) and the L2 norm of H, defined as L2(H)=
√
∑
H2
i
/Ng,
where Ng is the number of points of the grid, are shown in Fig. 9.23 as functions
of t in the case where the separation of the stars is 50 units. The left panel shows
that part of the initial data numerical error dissipates until µt = 500. The right
panel indicates that the maximum of
∣
∣H
∣
∣ remains contained in the same order of
magnitude throughout the simulation, in this plot, we show the results using two
different computational domains; in the first one, using a blue line we show the
case used in the simulations throughout the article, a box of size µxmax = 500, in
the second case a box of size µxmax = 120. The maximum of |H| is located (and
remains) near the center of the grid.
To evaluate the evolution of the small constraint violations induced by the ini-
tial data superposition of the two boson star solutions and test for convergence,
we have analyzed the Hamiltonian and momentum constraint together with some
of the analysis quantities and GW outputs using three different resolutions. The
coarsest level in the low resolution case is set to µ{∆x,∆y,∆z} = 20, while for the
medium and high resolutions we have µ{∆x,∆y,∆z} =
p
2 10, µ{∆x,∆y,∆z} = 10,
respectively. In the medium and low resolutions we have placed six refinement
levels with the same spatial distribution as in the high resolution case (which
is the one used in this work) described in section 9.3.2. In Fig. 9.24 we have
plotted the gravitational waveform of CHl1A, which converges. The left and the
bottom panels of Fig. 9.23 show that the Hamiltonian constraint also converges
with resolution. The momentum constrains have been corroborated to be consis-
tent with zero at t = 0, and later in the evolution the three components converge
with increasing resolution, as expected. At the middle panel of Fig. 9.24 the dif-
ferences between the gravitational signal at different resolutions are given, this
helps to establish convergence of the GW and also gives an estimate of 1×10−3
Collisions 203
0 250 500 750 1000 1250 1500 1750 2000
µt
10−7
10−6
L
2
(H
)
CHl1B, high res.
CHl1B, med res.
CHl1B, low res.
(ℓ = 0) INl0B
0 250 500 750 1000 1250 1500 1750 2000
µt
10−4
3× 10−5
4× 10−5
6× 10−5
2× 10−4
L
∞
(H
)
CHl1B, domain: µxmax = 500
CHl1B, domain: µxmax = 120
(ℓ = 0) INl0B, domain: µxmax = 500
0 250 500 750 1000 1250 1500 1750 2000
µt
10−5
10−4
10−3
L
∞
(H
)
CHl1B, high res.
CHl1B, med res.
CHl1B, low res.
(ℓ = 0) INl0B
Figure 9.23: L2 and L∞ norms of the Hamiltonian constraint for the model
CHl1B. In this plot, as in the results presented throughout this article the stars
have been separated 50 coordinate units.
for the relative difference when comparing the high and the medium resolutions.
Finally, we have generated GW output at different radii in order to check consis-
tency and accuracy of the results; the bottom panels of Fig. 9.24 show that Ψ2,2
4
and Ψ
2,0
4
overlap when properly rescaled by the factor 1/r. The extraction surfaces
are therefore within the “wave-zone”.
Appendix: Number of particles
In this appendix we show that the number of boson particles in a single isolated
ℓ-boson stars, for a fixed value of ℓ, is equally distributed among the different
modes m, that compose the star. There is an equipartition of the total Noether
charge.
The total conserved (Noether) charge associated to all scalar fields of an ℓ-
boson star is given by the zero component of the total current j0, as
N =
∫
j0α
p
γdx3 :=
∫
(
ℓ
∑
m=−ℓ
jm
)
α
p
γdx3 , (9.19)
204 9.7 Appendix
where j0 is given by the sum
j0 =
ℓ
∑
m=−ℓ
[
i
2
g0b(Φ̄m∇bΦm −Φm∇bΦ̄m)
]
=
ℓ
∑
m=−ℓ
jm . (9.20)
For a single ℓ-boson star the scalar field has a time dependence of the form Φm =
e−iωtφℓ(r)Y ℓm(ϑ,ϕ) and the metric is given by (9.7). The individual currents jm,
can be written, after some simplifications, as
jm = −
1
α2(r)
ωφ2
ℓ(r)
∣
∣
∣Y ℓm(ϑ,ϕ)
∣
∣
∣
2
. (9.21)
Integration of Eq. (9.21) over the 3-element of volume gives the number of parti-
cles associated to each field
Nm =
∫
α jm
p
γdx3 =−ω
[
∫∞
0
dr
a(r)r2
α(r)
φ2
ℓ(r)
][
∫π
0
∫2π
0
dϑdϕsinθ
∣
∣
∣Y ℓm(ϑ,ϕ)
∣
∣
∣
2
]
;
(9.22)
the second integral is equal to 1 due to the normalization of the spherical har-
monics. Therefore
Nm =−ω
∫∞
0
dr
a(r)r2
α(r)
φ2
ℓ(r) , (9.23)
and from Eq. (9.19) one obtains
N =
ℓ
∑
m=−ℓ
∫
α jm
p
γdx3 = (2ℓ+1) Nm . (9.24)
Consequently, the total number of particles of a single ℓ-boson star is divided
equally into the associated number of particles stored in each field
Nm =
1
2ℓ+1
N . (9.25)
Appendix: Comparison with ℓ= 0 boson stars
The purpose of this appendix is to continue the discussion regarding the com-
parison between boson stars mergers with ℓ = 0 and ℓ = 1, presented at the end
of section 9.5.1, identifying the effect of the ℓ parameter on the GW signal. The
dynamics and GW signatures of the merger of boson star binaries have been dis-
cussed extensively in the past, e.g. [46, 58]. It has been shown that the gravita-
tional waveform may be very different from that of black holes specially in the
early phases. In our numerical experiments, we have obtained similar results to
the ones presented in [58, 59] for the standard ℓ = 0 boson stars merger, in par-
ticular that the waveform of the merger of incoherent states is much smaller and
occurs after its coherent counterpart.
Collisions 205
Coh. ℓ= 0
Model µR99 ω/µ µM0 µ2N0 C Rem. µR ∼ µtc
CHl0A 31.5 0.985 0.296 0.298 0.009 BS 30 0
CHl0B 27.5 0.980 0.333 0.335 0.012 BS 30 0
CHl0C 24.7 0.976 0.364 0.367 0.015 BS 25 110
CHl0D 13.5 0.931 0.548 0.559 0.041 BH µrAH = 1.25 320
Table 9.3: Coherent cases for ℓ = 0. R99 is the radius that contains 99% of the
mass of the star. ω is the frequency, M0 is the mass of each star, N0 is the number
of particles of each star and C is the compactness. The final remnant can be
a localized boson configuration (BS) or a black hole (BH). R is the radius that
encloses 99% of the mass for scalar field remnant at µt = 2500 while for BHs it
labels the radius of the apparent horizon.
As discussed above, for the boson stars with ℓ= 1 mergers, the coherent and
incoherent configurations become more similar. Due to this fact, we focus our
description of the ℓ= 0 and ℓ= 1 comparison, only for the corresponding coherent
states. In order to perform such comparison of the gravitational waveforms we
select stars with the same R99.
In Table 9.3 we shown some properties of the stars. The radii of the stars are
the same as models listed in Table 9.1. In Fig. 9.25 it is shown the waveforms ψ22
4
,
for models listed in Tables 9.1 and 9.3. For models CHl0A and CHl1A with µR99 =
31.5, the mode ψ22
4
for boson stars with ℓ = 0 has the same order of magnitude
despite the fact the compactness is larger for the ℓ = 1 boson stars. For models
with µR99 = 27.5 (CHl0B and CHl1B) the maximum amplitude of the signal is
considerably larger for the ℓ = 1 case. For models with µR99 = 24.7 (CHl0C and
CHl1C) the final product of the collision for ℓ= 0 is a boson star but for ℓ= 1 the
remnant is a black hole. Both signals are clearly distinguishable.
Finally, for models with µR99 = 13.5 (CHl0D and CHl1D) the gravitational
imprints for ℓ= 0 boson stars and ℓ= 1 boson stars are quite different from each
other despite the fact a black hole forms after the merger in both cases. This
happens because in the ℓ = 1 boson star binary, the mass of the BH formed is
bigger.
206 9.7 Appendix
100 200 300 400 500 600 700 800
µ(t− rext)
−0.00010
−0.00005
0.00000
0.00005
0.00010
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µ∆x = 10
µ∆x =
√
2 10
µ∆x = 20
100 200 300 400 500 600 700 800
µ(t− rext)
−0.00010
−0.00005
0.00000
0.00005
0.00010
(r
/µ
)R
e(
Ψ
2,
0
4
)| µ
r=
10
0
µ∆x = 10
µ∆x =
√
2 10
µ∆x = 20
100 200 300 400 500 600 700 800
µ(t− rext)
−2
−1
0
1
2
3
4
R
e(
Ψ
2,
2
4
)| µ
r=
10
0
/µ
2
×
10
7
high-med
med-low
100 200 300 400 500 600 700 800
µ(t− rext)
−3
−2
−1
0
1
R
e(
Ψ
2,
2
4
)| µ
r=
10
0
/µ
2
×
10
7
high-med
med-low
100 200 300 400 500 600 700 800
µ(t− rext)
−0.00010
−0.00005
0.00000
0.00005
0.00010
(r
/µ
)R
e(
Ψ
2,
2
4
)| r
=
r e
x
t
µrext = 100
µrext = 120
100 200 300 400 500 600 700 800
µ(t− rext)
−0.00010
−0.00005
0.00000
0.00005
0.00010
(r
/µ
)R
e(
Ψ
2,
0
4
)| r
=
r e
x
t
µrext = 100
µrext = 120
Figure 9.24: Model CHl1A. Top panels show the real part of rΨ
2,2
4
and rΨ
2,0
4
at µrext = 100 using different resolutions. Their differences are displayed in the
middle panel. Bottom panels show overlap of waves extracted at different radii
when they are appropriately rescaled.
Collisions 207
100 200 300 400 500 600 700 800
µ(t− rext)
−0.0002
−0.0001
0.0000
0.0001
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 31.5 (coherent) CHl0A, CHl1A
ℓ = 0
ℓ = 1
100 200 300 400 500 600 700 800
µ(t− rext)
−0.0004
−0.0002
0.0000
0.0002
0.0004
0.0006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 27.5 (coherent) CHl0B, CHl1B
ℓ = 0
ℓ = 1
100 200 300 400 500 600 700 800
µ(t− rext)
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 24.7 (coherent) CHl0C, CHl1C
ℓ = 0
ℓ = 1
100 200 300 400 500 600 700 800
µ(t− rext)
−0.008
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006
(r
/µ
)R
e(
Ψ
2,
2
4
)| µ
r=
10
0
µR99 = 13.5 (coherent) CHl0D, CHl1D
ℓ = 0
ℓ = 1
Figure 9.25: GW signal for configurations described in Table 9.3. The extraction
radius is µr = 100.

CONCLUSIONS
C
lassical scalar fields are multipurpose mathematical objects that, when
coupled to gravity serve as excellent and simple models for the cosmolog-
ical dark matter and dark energy which in turn motivate the exploration
of the scalar field physics in astrophysical scales.
There are some examples of compact objects where the source of gravitational
field is the scalar field, such as the boson stars, some wormholes, many of the hairy
black hole solutions and other canonical-ghost configurations such as the E -boson
stars presented here. Emphasis should be placed on the fact that the mentioned
solutions, actively studied by the community, do not require any coupling that
takes the system out of the description of general relativity although some of
these solutions exist, in particular in a group of alternative theories called the
scalar-tensor theories. However, as has been argued throughout this manuscript.
Einstein’s theory of general relativity has passed all tests to date.
In this context, we have developed a kind of mold for stationary solution con-
struction procedures and their subsequent dynamical analysis in full 3+1 General
Relativity. This thesis addressed seven solutions among which some are spatially
homogeneous and others stationary in different symmetries. In Chap.2, as moti-
vation, we presented a dark matter two-scalar field solution (Sec. 2.2) where the
ultralight self-interacting model was combined with Higgs-like and axion fields
in the Friedmann cosmological evolution, finding that constrains from big bang
nucleosynthesis indicate that the ultralight field must make up at least about 40
percent of the total dark matter. Then in Sec. 2.3 we presented an exact solution
for a complex self-interacting model that behaves as quintessence at late times.
Part I was composed by the spherically symmetric compact object solutions. In
Chap. 3 ℓ-boson star solutions with ℓ≫ 1 where constructed by means of a spec-
209
210
tral method and rescaling properties. We found that the pressure anisotropy in
the configurations as well as their mass and compactness increase with ℓ reaching
values close to the Buchdahl limit. Chap. 4 introduced a real self-interacting ghost
scalar field and combined the gravitational soliton with a (standard) boson star
solution, obtaining composite system of a scalar field star with a phantom nucleus
which significantly increases the compactness even to values very close to black
hole limit while satisfying the null-energy condition everywhere. Then, using
multiple fields, but in this case all ghosts, we presented solutions with wormhole
topology in Chap. 5. We should mention with respect to this solution we expected
that the rotation parameter ℓ would help the stability of the configuration, how-
ever an independent study, outside the scope of this thesis, showed this was not
the case.
Part II dealt with axisymmetric solutions as the magnetic boson star, con-
structed in Chap. 6 and the two-boson star solutions in Chap. 7. Both solutions
have interesting applications in the dynamical scenario, which we are currently
studying, using them as initial data. Both axially symmetric Einstein-Maxwell-
Klein-Gordon and Einstein-Friedberg-Lee-Sirlin systems of Chaps. 6 and 7 re-
spectively, where solved using the library for spectral methods for theoretical
physics Kadath.
Then, in Part III, Chaps. 8 and 9 the initial data for ℓ-boson stars was inter-
polated to the Cartesian grid of Einstein Toolkit in order to analyze the stability
of this kind of boson stars and the head-on collisions. Both papers certainly found
similarities with the standard ℓ= 0 boson stars, however a notable difference ap-
peared, as evidence was found for the existence of multifield solutions outside of
spherical symmetry, which was confirmed shortly thereafter elsewhere.
Finally, we identify two opportunities in which the results presented in this
manuscript give rise to future work. First, we have found (numerical) evidence,
that some unstable E -boson stars migrate to a time-dependent periodic solution
with a ghost scalar field distribution remaining inside the canonical scalar field,
which would be interesting since we are not aware of other phantom field solu-
tions that form bound states in regular spacetimes. And second, the axial data
obtained by spectral methods would be very convenient to interpolate directly to
the Einstein Toolkit code in order to preserve numerical precision, obtaining more
reliable and stable simulations than those that use initial data obtained from fi-
nite difference schemes.
Appendices

APPENDIX A
BLACK HOLE SOLUTIONS WITH
MAGNETIC DIPOLE MOMENT
The full action of the Einstein-Maxwell system reads
S =
c4
16πG
∫
R
p
−gd4x−
∑
∫
mdτ+
∫
A i J
ip−gd4x−
1
4µ0
∫
FabFabp−gd4x
(A.1)
In this appendix we do not take c = 1.
Electrovacuum (J i = 0) imply the following:
Rµν−
1
2
R gµν =
8πG
c4
Tµν; ∇µFµν = 0 (A.2)
with
Tµν =
1
µ0
(
Fλ
µFνλ−
1
4
gµνFαβFαβ
)
. (A.3)
Kerr-Newman
The Kerr-Newman spacetime is a three-parameter family of solutions to the above
equations where the space-time is stationary and axisymmetric.
In Boyer Lindquist coordinates the solution is given by the line element
ds2 =−
∆
Σ
(cdt−asin2θdϕ)2 +
sin2θ
Σ
[
(r2 +a2)dϕ−acdt
]2
+Σ
(
dr2
∆
+dθ2
)
,
(A.4)
Aµdxµ =−
rQ
4πǫ0cΣ
(
cdt−asin2θdϕ
)
(A.5)
213
214
where
a =
J
Mc
, (A.6)
Σ= r2 +a2 cos2θ, (A.7)
∆= r2 − rsr+a2 + r2
Q (A.8)
and
rs =
2GM
c2
, r2
Q =
Q2G
4πǫ0c4
(A.9)
The first thing to notice with this solution is that it already has a magnetic
dipole moment. In the limit r →∞ it can be see how the dominant term of the
solution is an electric monopole and a magnetic dipole:
Aµ ≈
Q
4πǫ0cr
(
−1,0,0,asin2θ
)
(A.10)
the 0 component refers to the electric potential produced by a charge Q since in flat
space-time, in SI units and in coordinates (x0,x) it is satisfied that Aµ = (φ/c,A)
therefore
φ= cg00 A0 =
Q
4πǫ0r
, (A.11)
while the contravariant compontent Aϕ is
Aϕ = gϕϕAϕ =
Qa
4πǫ0cr3
=
µ0cQa
4πr3
. (A.12)
In order to compare with the three-vector A = Aϕ
eϕ, we place directly the
magnitude of the coordinate basis |eϕ| = rsinθ, we see then that
A=
µ0
4πr3
m×x, (A.13)
with m=Qcaez. Which is the magnetic moment of a circuit in xy of current I = Qc
πa
and area πa2. And it is (like the electron!), twice the magnetic moment of a charge
Q spinning in a ring of radius a at speed c: mclassical = (Q/T)A = (Qc/2πa)(πa2)=
Qca/2.
In addition to this identification of the situation in the asymptotic limit, New-
man wanted to identify the Kerr-Newman source with a charged rotating ring.
Carter then argued that it would not be possible to refer to the sources of this
space-time as a rotating charge distribution. However there have been several at-
tempts to give a source for these electromagnetic fields (see [230–232] and others).
All these attempts as far as I can see are in the limit G → 0, motivated by the fact
that Aµ does not depend on G. This path is the only situation where it seems to
be that it makes sense to talk about a source of rotating charges and therefore
possible to give an interpretation. There are some references where conductors
Black hole solutions with magnetic dipole moment 215
are used in vacuum or with infinite magnetic susceptibility that I do not under-
stand, but only for comparison. And others where it is solution, but its source is
the disk that generates the singular ring, which generates discontinuities and it
seems that infinite energy, etc..
A serious and moreover easy-to-read attempt to give a source to the Kerr-
Newman space-time is the one made by [233]. He obtains the source by taking the
G → 0 limit directly, but he considers the fact that the topology obtained is non-
trivial (though flat) and solves Maxwell equations there, proving that the source
is the ring (now spacelike) is unique. Taking the limit G → 0 of (A.4) we obtain
that rQ = 0= rs, and hence ∆= r2 +a2:
ds2 =−
r2 +a2
r2 +a2 cos2θ
(cdt−asin2θdϕ)2
+
sin2θ
r2 +a2 cos2θ
[
(r2 +a2)dϕ−acdt
]2
+ (r2 +a2 cos2θ)
(
dr2
r2 +a2
+dθ2
)
(A.14)
which reduces to
ds2 =−dt2 + (r2 +a2)sin2θdϕ2 + (r2 +a2 cos2θ)
(
dr2
r2 +a2
+dθ2
)
(A.15)
and which is Minkowski except for a change of coordinates resulting singular in
the ring.
On the other hand Aµ remains identical, and is a solution to Maxwell equa-
tions in space-time with such topology with sources located exclusively in the ring.
Ernst
There are magnetized black hole solutions, for example those described in [234,
235] where the homogeneous magnetic field at infinity is a test field. I also un-
derstand that there are exact solutions as for example those described in [236]
referring to a Black Hole solution found by Ernst using Harrison’s method, but as
with the Manko-Sibgatulin solutions, are obscure, besides that they have multi-
ple problems; there Aliev and Galtsov indicate which are these problems and how
to solve some of them.
Bonnor
If one looks for references to exact solutions (e.g. in the Refs. of Stephani-Kramer,
Griffiths-Podolsky, Chandasekhar), they refer (very briefly) to the solutions found
by Bonnor and those found by Manko and Sibgatullin. The Bonnor’s solution
( [237]) is the one that agrees the most with the idea (in vacuum) of giving a
216
possible description to the exterior of a massive object with a magnetic dipole
(a kind of magnet - magnetized sphere), the extension apparently corresponds
to a pair of extreme Reissner-Nordstrom black holes with magnetic charge (the
magnetic monopole solution) and separated a certain distance by a “strut”. The
solution, with its restored units reads,
ds2 =−
(
P
Y
)2
c2 dt2 +
P2Y 2
Q3Z
(dr2 +Z dθ2)+
Y 2Z sin2θ
P2
dϕ2, (A.16)
Aµdxµ =
c
√
4πǫ0G
rsbrsin2θ
P
dϕ, (A.17)
where
P = r2 − rsr−b2 cos2θ, (A.18)
Q =
(
r− rs/2
)2 −
(
r2
s /4+b2
)
cos2θ, (A.19)
Y = r2 −b2 cos2θ, (A.20)
Z = r2 − rsr−b2 (A.21)
with rs = 2GM/c2. According to Griffiths and Podolsky, this is a naked singularity.
However, the solution is asymptotically flat and an identification of its properties
can be made since it in r → and conserving some of the O (1/r) terms, we obtain
ds2 ≈−(1−2rs/r)c2 dt2 +
1
1−2rs/r
dr2 + r2(dθ2 +sin2θdϕ2), (A.22)
Aµdxµ ≈
c
√
4πǫ0G
rsbsin2θ
r
dϕ (A.23)
Clearly the spacetime mass is 2M.
Comparing Aµ with the equation for the Kerr-Newman four-potential, it is
clear that this solution has no electric charge, but does have a dipole. We define
Amu through an equation similar to the one that defines Q in the Kerr-Newman
solution,
b2 =
µ2G
4πǫ0r2
s c6
(A.24)
In terms of µ we have
Aϕ =
µ
4πǫ0c2
rsin2θ
P
(A.25)
and in the asymptotic limit,
Aϕ ≈
µ
4πǫ0c2
sin2θ
r
. (A.26)
Comparing one to one with the equation (A.10) which is the asymptotic ana-
logue of Kerr-Newmann, it is immediately seen that the magnetic moment asso-
ciated with this solution is
m=µez (A.27)
Black hole solutions with magnetic dipole moment 217
In units where G = c = 1/(4πǫ0) = 1 (Gaussian) one has Aµ = 2mb. Bonnor’s
paper on the other hand, is in units where G = c = 1/µ0 = 1, where the four-
potential Aµdxmu ≈ 1
Aµ
dxµ ≈ 1
Mbrsin2 θ
P dϕ reported by them is recovered.
The horizon is located at Z = 0 however it is a non-regular horizon because
gϕϕ = 0 there too. It evades the no-hair theorem. This solution does not reduce to
Schwarzschild but it does reduce to one of Zipoy’s solutions.
Manko-Sibgatullin
The first suerposition of a Kerr black hole with a magnetic dipole was made by
Kramer but for the hyperextremal case. Then, Manko, a long time ago, published
about 5 articles. His simplest solution was further simplified in the very recent
article [238]. These solutions depending on the parameters m, a, q and b, do
reduce to Schwarzschild in the limit a = q = b = 0. I now switch to units G = c = 1,
and the simplified, nonrotating, uncharged, case (a = 0= q) has metric,
ds2 =− f (dt−ωdϕ)2 + f −1
[
e2γ(dρ2 +dz2)+ρ2dϕ2
]
. (A.28)
with the metric coefficients determined from
f =
A2 −B2 +C2
(A+B)2
, e2γ =
A2 −B2 +C2
16κ4
+κ
4
−R+R−r+r−
, ω=−
Im[G(Ā+ B̄)+CĪ]
A2 −B2 +C2
(A.29)
with
A =κ2
−(m2 +b)(R−r−+R+r+)+κ2
+(m2 −b)(R−r++R+r−)−4b2(R−R++ r−r+),
B =mκ−κ+[m2(r−+ r+−R−−R+)+κmκ+(r−+ r++R−+R+)],
C =bκ−κ+[(κ++κ−)(R−−R+)+ (κ+−κ−)(r+− r−)],
G =κ2
+κ−(κ2
−+m2 +b)(R−r+−R+r−)
+κ2
−κ+(κ2
−+m2 −b)(R−r−−R+r+)
+mbκ+κ−[(κ++κ−)(r+− r−)− (κ+−κ−)(R+−R−)]−2zB,
I =mκ2
+[−ib(R−r++R+r−)]+mκ2
−[−ib(R−r−+R+r+)]
+2imb[m2(R+R−+ r+r−)+2κ2
+κ
2
−]
+ ibκ+κ−[2m(R+R−− r+r−)+4m2(R++R−− r+− r−)]− ziC+ ibB/m.
where
R± =
√
ρ2 +
[
z±
1
2
(κ++κ−)
]2
, r± =
√
ρ2 +
[
z±
1
2
(κ+−κ−)
]2
,
and
κ± =
√
m2 ±2b. (A.30)
218
A,B,C,G are real and I is imaginary. Hence, the solution would be simplified
(for the fourth time) to
f =
A2 −B2 +C2
(A+B)2
, e2γ =
A2 −B2 +C2
16κ4
+κ
4
−R+R−r+r−
, ω=−
CI
A2 −B2 +C2
. (A.31)
It is very likely that the equations can be further simplified in order to compare
with Bonnor’s solution.
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