Exoticcompactobjects interactingwithfundamentalfields · EFE EinsteinFieldEquations EMFE...
Transcript of Exoticcompactobjects interactingwithfundamentalfields · EFE EinsteinFieldEquations EMFE...
Exotic compact objectsinteracting with fundamental fields
Nuno André Moreira Santos
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisors: Prof. Dr. Carlos Alberto Ruivo Herdeiro
Prof. Dr. Vítor Manuel dos Santos Cardoso
Examination Committee
Chairperson: Prof. Dr. José Pizarro de Sande e Lemos
Supervisor: Prof. Dr. Carlos Alberto Ruivo Herdeiro
Member of the Committee: Dr. Miguel Rodrigues Zilhão Nogueira
October 2018
Resumo
A astronomia de ondas gravitacionais apresenta-se como uma nova forma de testar os fundamentos da física
− e, em particular, a gravidade. Os detetores de ondas gravitacionais por interferometria laser permitirão
compreender melhor ou até esclarecer questões de longa data que continuam por responder, como seja
a existência de buracos negros. Pese embora o número cumulativo de argumentos teóricos e evidências
observacionais que tem vindo a fortalecer a hipótese da sua existência, não há ainda qualquer prova
conclusiva. Os dados atualmente disponíveis não descartam a possibilidade de outros objetos exóticos, que
não buracos negros, se formarem em resultado do colapso gravitacional de uma estrela suficientemente
massiva. De facto, acredita-se que a assinatura do objeto exótico remanescente da coalescência de um
sistema binário de objetos compactos pode estar encriptada na amplitude da onda gravitacional emitida
durante a fase de oscilações amortecidas, o que tornaria possível a distinção entre buracos negros e outros
objetos exóticos. Esta dissertação explora aspetos clássicos da fenomenologia de perturbações escalares
e eletromagnéticas de duas famílias de objetos exóticos cuja geometria, apesar de semelhante à de um
buraco negro de Kerr, é definida por uma superfície refletora, e não por um horizonte de eventos. Em geral,
tais objetos registam instabilidades quando caracterizados por condições de fronteira totalmente refletoras.
No entanto, mostra-se que podem ser estáveis se se considerar condições de fronteira parcialmente ou
sobre-refletoras. Os resultados sugerem que, pelo menos no que respeita a esta instabilidade, estes objetos
exóticos podem ser viáveis do ponto de vista astrofísico.
Palavras-chave: instabilidade de ergo-região, objetos compactos, buracos negros, relatividade geral
Abstract
Gravitational-wave astronomy offers a novel testing ground for fundamental physics, namely by unfolding
new prospects of success in probing the nature of gravity. Current and near-future gravitational-wave
interferometers are expected to provide deeper insights into long-standing open questions in gravitation
such as the existence of black holes. Although a cumulative number of both theoretical and observational
arguments has been strengthening the black-hole hypothesis, some sort of proof is still lacking. Up-to-date
gravitational-wave data does not preclude other exotic compact objects rather than black holes from
being the ultimate endpoint of compact binary mergers. The late-time gravitational-wave ringdown signal
from compact binary coalescences has been argued to encode the signature of the compact object left
behind the merger, which hints at the possibility of distinguishing black holes from other exotic compact
objects. The present thesis addresses classical phenomenological aspects of scalar and electromagnetic
field perturbations of two families of Kerr-like exotic compact objects featuring a surface with reflective
properties instead of an event horizon. While these horizonless alternatives are prone to ergoregion
instabilities when their surface is perfectly-reflecting, it is shown that stability can be achieved when
considering partially- or over-reflecting boundary conditions. The results suggest that, at least in what
regards this instability, Kerr-like exotic compact objects may be astrophysically viable.
Keywords: ergoregion instability, exotic compact objects, black holes, general relativity
Table of contents
List of figures ix
List of tables xiii
Acronyms xv
1 Introduction 1
1.1 Black holes in general relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Black holes as an endpoint of stellar evolution . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Observing astrophysical black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Exotic compact objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Thesis scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 The Kerr metric 9
2.1 The Kerr metric in the Boyer-Lindquist form . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Continuous symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Curvature singularity and maximal analytical extension . . . . . . . . . . . . . . . . . . . 12
2.4 Zero angular momentum observer (ZAMO) and frame dragging . . . . . . . . . . . . . . . 15
2.5 Ergoregion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Penrose process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.7 Superradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Quasinormal modes 21
3.1 Black-hole perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Quasinormal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Methods for computing quasinormal modes . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Direct-integration shooting method . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Scalar perturbations of exotic compact objects 29
4.1 Klein-Gordon equation on Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
viii Table of contents
4.2 Quasinormal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Schwarzschild-like exotic compact objects . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.2 Kerr-like exotic compact objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2.3 Superspinars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Superradiant scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 Electromagnetic perturbations of exotic compact objects 47
5.1 The Newman-Penrose formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Maxwell’s equations on Kerr spacetime . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Electric and magnetic fields in the ZAMO frame . . . . . . . . . . . . . . . . . . . . . . . 53
5.4 Perfectly-reflecting boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Detweiler transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Quasinormal modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6.1 Schwarzschild-like exotic compact objects . . . . . . . . . . . . . . . . . . . . . . . 57
5.6.2 Kerr-like exotic compact objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.6.3 Superspinars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6 Conclusion and Future Work 65
References 67
Appendix A Teukolsky-Starobinsky identities 71
A.1 Definitions and operator identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.2 Teukolsky-Starobinsky identities for spin-1 fields . . . . . . . . . . . . . . . . . . . . . . . 72
List of figures
2.1 Maximal analytical extension of Kerr solution for a2 > M2. . . . . . . . . . . . . . . . . . 13
2.2 Carter-Penrose of the maximal analytical extension of Kerr spacetime along the axis of
symmetry (θ = 0) for a2 < M2 and a2 = M2. . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Proper volume of the ergoregion of Kerr spacetime as a function of |a/M |. . . . . . . . . . 16
4.1 Real and imaginary parts of the fundamental |l| = 1, 2 scalar quasinormal mode frequencies
of a Schwarzschild-like exotic compact object with a perfectly-reflecting (|R|2 = 1) surface at
r = r0 ≡ rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon of the corresponding
Schwarzschild black hole, as a function of δ/M , for both Dirichlet and Neumann boundary
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Real and imaginary parts of the fundamental l = m = 1 scalar quasinormal mode frequencies
of a Kerr-like exotic compact object with a perfectly-reflecting (|R|2 = 1) surface at
r = r0 ≡ rH + δ, δ ≪ M , where rH is the would-be event horizon of the corresponding
Kerr black hole, as a function of δ/M , for both Dirichlet and Neumann boundary conditions. 34
4.3 Critical value of the rotation parameter above which the fundamental l = m = 1 scalar
quasinormal mode frequency of a perfectly-reflecting (|R|2 = 1) Kerr-like exotic compact
object is unstable, for both Dirichlet and Neumann boundary conditions. . . . . . . . . . 36
4.4 Detailed view of the imaginary part of the fundamental l = m = 1 scalar quasinormal
mode frequencies of a Kerr-like exotic compact object with a perfectly-reflecting (|R|2 = 1)
surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon of the
corresponding Kerr black hole, as a function of the rotation parameter a/M in the range
[0.8,1], for both Dirichlet and Neumann boundary conditions. . . . . . . . . . . . . . . . . 36
4.5 Timescale of the scalar ergoregion instability of rapidly-rotating Kerr-like exotic compact
objects with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where
rH is the would-be event horizon of the corresponding Kerr black hole, as a function of
δ/M , for l = m = 1 and both Dirichlet and Neumann boundary conditions. . . . . . . . . 37
x List of figures
4.6 Imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequencies
of a Kerr-like exotic compact objects with a partially-reflecting (|R|2 < 1) surface at
r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr
black hole and δ/M = 10−5, as a function of a/M , for quasi-Dirichlet and quasi-Neumann
boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.7 Fit of the maximum value of the imaginary part of the fundamental l = m = 1 scalar
quasinormal mode frequency of a Kerr-like exotic compact object with reflectivity R in
the range [−0.9980,−1] (quasi-Dirichlet boundary conditions) to the polynomial (4.22), for
different values of δ/M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.8 Real and imaginary parts of the fundamental l = m = 1 scalar quasinormal frequencies of
a superspinar with a perfectly-reflecting (|R|2 = 1) surface at r = r0 > 0, as a function of
a/M , for both Dirichlet and Neumann boundary conditions. . . . . . . . . . . . . . . . . . 40
4.9 Critical value of the rotation parameter below which the fundamental l = m = 1 scalar
quasinormal mode frequency of a perfectly-reflecting (|R|2 = 1) superspinar is unstable, for
both Dirichlet and Neumann boundary conditions. . . . . . . . . . . . . . . . . . . . . . . 41
4.10 Timescale of the scalar ergoregion instability of superspinars with a perfectly-reflecting
(|R|2 = 1) surface at r = r0 > 0, as a function of r0, for l = m = 1. . . . . . . . . . . . . . 41
4.11 Imaginary part of the fundamental l = m = 1 scalar quasinormal mode frequencies of a
superspinar featuring a surface with reflectivity R at r = r0 => 0, as a function of a/M . . 41
4.12 Amplification factors for superradiant l = m = 1 scalar field perturbations scattered off
Kerr-like exotic compact objects with a/M = 0.9 and featuring a surface with reflectivity
R at r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr
black hole and δ/M = 10−5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.13 Numerical and analytical values for the amplification factors of superradiant l = m = 1
scalar field perturbations scattered off Kerr-like exotic compact objects with a/M = 0.9
and featuring a surface with reflectivity R at r = r0 ≡ rH + δ, where rH is the would-be
event horizon of the corresponding Kerr black hole and δ/M = 10−5. . . . . . . . . . . . . 45
5.1 Real and imaginary parts of the fundamental |l| = 1, 2 electromagnetic quasinormal
mode frequencies of a Schwarzschild-type exotic compact object with a perfectly-reflecting
(|R|2 = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon
of the corresponding Schwarzschild black hole, as a function of δ/M , for both Dirichlet and
Neumann boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 Real and imaginary parts of the fundamental l = m = 1 electromagnetic quasinormal
mode frequencies of a Kerr-like exotic compact object with a perfectly-reflecting (|R|2 = 1)
surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon of the
corresponding Kerr black hole, as a function of a/M , for both Dirichlet and Neumann
boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
List of figures xi
5.3 Detailed view of the imaginary part of the fundamental l = m = 1 electromagnetic
quasinormal mode frequencies of a Kerr-like exotic compact object with a perfectly-reflecting
(|R|2 = 1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon
of the corresponding Kerr black hole, as a function of a/M in the range [0.8,1[, for both
Dirichlet and Neumann boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.4 Timescale of the electromagnetic ergoregion instability of rapidly-rotating Kerr-like exotic
compact objects with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, δ ≪ M ,
where rH is the would-be event horizon of the corresponding Kerr black hole, as a function
of δ/M , for l = m = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.5 Imaginary part of the fundamental l = m = 1 electromagnetic quasinormal mode frequencies
of a Kerr-like exotic compact object with a partially-reflecting (|R|2 < 1) surface at
r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding Kerr
black hole and δ/M = 10−5, as a function of a/M , for quasi-Dirichlet and quasi-Neumann
boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.6 Fit of the maximum value of the imaginary part of the fundamental l = m = 1 electro-
magnetic QNM frequency of a Kerr-like ECO with reflectivity R in the range [−0.985,−1]
(quasi-Dirichlet boundary conditions) to the polynomial (4.22), for different values of δ/M . 62
5.7 Real and imaginary parts of the fundamental l = m = 1 electromagnetic quasinormal mode
frequencies of a superspinar with a perfectly-reflecting (|R|2 = 1) surface at r = r0 > 0, as
a function of a/M , for both Dirichlet and Neumann boundary conditions. . . . . . . . . . 63
5.8 Timescale of the electromagnetic ergoregion instability of superspinars with a perfectly-
reflecting (|R|2 = 1) surface at r = r0 > 0, as a function of r0, for l = m = 1. . . . . . . . 63
5.9 Imaginary part of the fundamental l = m = 1 electromagnetic quasinormal mode frequencies
of a superspinar featuring a surface with reflectivity R at r = r0 > 0, as a function of a/M . 64
List of tables
1.1 Classic black hole solutions of Einstein field equations. . . . . . . . . . . . . . . . . . . . . 3
1.2 Inferred mass and angular momentum of the final black hole candidates for all gravitational-
wave observations ever reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Exotic compact object models proposed over the last decades. . . . . . . . . . . . . . . . 8
3.1 Group and phase velocities of the quasi-monochromatic waves (3.5) and (3.6). . . . . . . . 23
3.2 Nature of the quasi-monochromatic waves (3.5) and (3.6) according to the value of sign(ωϖ). 23
4.1 Fundamental l = m = 1 scalar quasinormal mode frequencies of a Kerr-like exotic compact
object with a perfectly-reflecting (R = −1) surface at r = r0 ≡ rH + δ, 0 < δ ≪ M , where
rH is the would-be event horizon of the corresponding Kerr black hole. . . . . . . . . . . . 33
4.2 Numerical value of the coefficients of the polynomials (4.15)−(4.16), which fit the real
and imaginary parts of the fundamental l = m = 1 scalar quasinormal mode frequency
of a perfectly reflecting (|R|2 = 1) Kerr-like exotic compact object, for both Dirichlet
and Neumann boundary conditions. The fits were performed for values of the rotation
parameter a/M over the range [0,0.25]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Numerical value of the coefficients of the second-order polynomial (4.21) which fits the
timescale of the scalar ergoregion instability of a perfectly-reflecting (|R|2 = 1) Kerr-like
exotic compact object, for l = m = 1 and both Dirichlet and Neumann boundary conditions. 37
4.4 Numerical value of the coefficients of the second-order polynomial (4.22) which fits the
maximum value of the imaginary part of the fundamental l = m = 1 scalar quasinormal
mode frequency of a Kerr-like exotic compact object with reflectivity R in the range
[−0.9980,−1] (quasi-Dirichlet boundary conditions), for different values of δ/M . . . . . . . 38
4.5 l = m = 1 scalar quasinormal mode frequencies corresponding to the resonance peaks
displayed in Figure 4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.1 Numerical value of the coefficients of the second-order polynomial (4.21) which fits the
timescale of the electromagnetic ergoregion instability of a perfectly-reflecting (|R|2 = 1)
Kerr-like exotic compact object, for l = m = 1 and both Dirichlet and Neumann boundary
conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xiv List of tables
5.2 Numerical value of the coefficients of the second-order polynomial (4.22) which fits the
maximum value of the imaginary part of the fundamental electromagnetic l = m = 1
quasinormal mode frequency of a Kerr-like exotic compact object with reflectivity R in the
range [−0.985,−1] (quasi-Dirichlet boundary conditions), for different values of δ/M . . . . 62
Acronyms
BC Boundary Condition
DBC Dirichlet Boundary Condition
NBC Neumann Boundary Condition
RBC Robin Boundary Condition
BH Black Hole
ECO Exotic Compact Object
EFE Einstein Field Equations
EMFE Einstein-Maxwell Field Equations
EW Electromagnetic Wave
GR General Relativity
GW Gravitational Wave
LIGO Laser Interferometer Gravitational-Wave Observatory
ODE Ordinary Differential Equation
PDE Partial Differential Equation
QNM Quasinormal Mode
ZAMO Zero Angular Momentum Observer
Chapter 1
Introduction
The first direct observation of GWs [1] opened a new window onto the cosmos, particularly to the most
energetic astrophysical events in the universe. Unlike EWs, GWs interact weakly with matter [2,3], thus
enabling the unhindered access to GW sources, such as binary systems made up of two compact objects.
Among the many questions precision GW astronomy can tackle a fundamental one is whether it can
probe the true nature of BH candidates [4–6]. Although the very definition of BH [7,8], which lies on the
mathematical concept of event horizon, precludes any observational proof of its occurrence in Nature, a
conclusive evidence of the existence of BHs might be coded in the late-time GW ringdown signal from
compact binary coalescences [2,4–6]. Nevertheless, current observations do not rule out small deviations
from the BH paradigm and, therefore, the existence of other ECOs rather than BHs [5,6]. These are
loosely defined in the literature as hypothetical objects more massive than neutron stars, sufficiently dim
not to have been observed by state-of-the-art EW telescopes and detectors yet and, in most cases, without
event horizon [6]. There has been a renewed interest in these exotic alternatives over the last decades,
namely because some ECOs can mimic the physical behavior of BHs, even in the absence of an event
horizon.
An especially simple ECO model replaces the event horizon of a classic four-dimensional BH solution
of GR by a reflective surface near the would-be event [5,9]. In this thesis the phenomenology of these
horizonless reflecting ECOs will be addressed. Although rotating horizonless ECOs are prone to instabilities
[10–12], partial absorption at the reflecting ECO surface seems to mitigate a class of instabilities [13] − at
least in the simple model to be analyzed herein. A comprehensive analysis of scalar and electromagnetic
perturbations in this setup shall provide a holistic perspective on the effect of reflecting BCs on the
physical properties of horizonless ECOs.
This introductory chapter aims to provide a brief overview of the classic BH solutions of GR (Section 1.1)
and a summary of the physical processes driving the formation and evolution of stellar-mass BHs or similar
objects (Section 1.2), as well as to highlight the intrinsic non-detectability of BHs (Section 1.3) and, as a
result, the importance of also studying ECOs (Section 1.4). At the end of the chapter (Section 1.5) is the
thesis scope and outline.
2 Introduction
1.1 Black holes in general relativity
One of the most fundamental and groundbreaking predictions of Albert Einstein’s geometric theory of
gravity [14] is the existence of BHs. A BH is a region of spacetime where gravity is so strong that its
surroundings are warped in such a way that not even light can escape from it [15]. The geometric features
of spacetime around BHs are mathematically encoded in the metric tensor gµν , which satisfies the EFE
[14–16],
Rµν − 12Rgµν = 8πG
c4 Tµν , (1.1)
where Rµν is the Ricci tensor, R is the Ricci scalar, G is Newton’s gravitational constant, c is the speed of
light in vacuum and Tµν is the stress–energy tensor. While Rµν and R, which depend on gµν , are measures
of curvature, Tµν encodes the flux of four-momentum across a spacetime surface [16]. Apart from equation
(1.1), geometrized units (G = c = 1) are consistently used throughout the text. Additionally, the metric
signature (−, +, +, +) is adopted.
The simplest non-trivial exact solution of the EFE is the Schwarzschild metric [17]. This vacuum
solution (Tµν = 0) describes a static spherically symmetric spacetime around a mass M . In Schwarzschild
coordinates (t, r, θ, ϕ), the line element is [15–17]
ds2 = −(
1 − 2Mr
)dt2 +
(1 − 2M
r
)−1dr2 + r2 (
dθ2 + sin2 θ dϕ2). (1.2)
If the mass M is concentrated at r = 0, the solution is valid for r > 0 and is called Schwarzschild BH.
In this case, the line element (1.2) has two peculiarities. On the one hand, the metric component grr
diverges everywhere on the hypersurface r = 2M , thus being a singularity. However, this singularity is
just the result of a deficiency in the coordinate system adopted. For that reason, it is called coordinate
singularity and can be removed by an appropriate change of coordinates [18]. The hypersurface r = 2M
is known as the Schwarzschild event horizon and the value rS ≡ 2M defines the Schwarzschild radius.
On the other hand, the singularity at the origin (r = 0) turns out to be irremovable and is therefore an
intrinsic singularity. In fact, the Kretschmann scalar [19], whose value at each point is the same in all
coordinate systems, blows up on the hypersurface r = 0.
According to Birkhoff’s theorem (1923) [20], the only spherically symmetric solution of the EFE in
vacuum is the Schwarzschild metric. This means that the exterior geometry of a neutral spherical object
is always the Schwarzschild geometry, whether the object is static or not. The converse is not necessarily
true, i.e. a static vacuum solution of the EFE is not perforce spherically symmetric (it can be cylindrically
symmetric [21], for instance). Nevertheless, Israel theorem (1967) [22] states that any asymptotically-flat
static vacuum solution of the EFE which is regular on and outside an event horizon must belong to the
1-parameter Schwarzschild family, defined by M. This theorem is also known as the uniqueness theorem
of the Schwarzschild metric [23].
If one takes Tµν to be the electromagnetic stress–energy tensor [15,24],
TEMµν = 1
4π
[FµρFν
ρ − 14gµνFρσF
ρσ
], (1.3)
1.1 Black holes in general relativity 3
where Fµν = 2A[ν;µ] is the electromagnetic-field strength tensor and Aµ is the electromagnetic four-
potential, and plugs it into (1.1), one finds how curvature dictates the dynamics of the electromagnetic
field and, the other way around, how the electromagnetic field generates curvature. Maxwell’s equations
[15,24] in free space, compactly written in the form ∇µFµν = 0 and ∇[µFνσ] = 0, together with EFE with
Tµν given by (1.3) are called the source-free EMFE.
The most general asymptotically-flat, stationary solution of the EMFE is the Kerr-Newman metric
[25]. This electro-vacuum solution(Tµν = TEM
µν
)describes a stationary axisymmetric spacetime around
a central body of mass M , intrinsic angular momentum J and electric charge Q. In Boyer-Linquist
coordinates (t, r, θ, ϕ), the line element reads [15,24,26]
ds2 = − ∆Σ
[dt− a sin2 θ dϕ
]2 + sin2 θ
Σ[(r2 + a2) dϕ− a dt
]2 + Σ∆ dr2 + Σ dθ2, (1.4)
where Σ ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2Mr+ a2 +Q2 and a ≡ J/M is the angular momentum per unit mass.
Using the same coordinates, the electromagnetic four-potential is [24]
Aµ = Qr
Σ(−1, 0, 0, a sin2 θ
). (1.5)
The Kerr-Newman metric (1.4) actually contains the classic set of solutions of the EFE in GR:
the Schwarzschild, the Kerr, the Reissner-Nordstrom and the Kerr-Newman metrics [16]. While the
Schwarzschild and the Reissner-Nordstrom solutions describe static spacetimes, the Kerr and the Kerr-
Newman solutions refer to stationary spacetimes. Thus, they all are equilibrium solutions. Table 1.1
summarizes the main features of each of the aforementioned metrics.
BH solutionJ Q
Event horizon Uniqueness theorem= 0 = 0 = 0 = 0
Schwarzschild r = 2M Israel theorem
Kerr r = M +√M2 − a2 Carter-Robinson theorem
Reissner-Nordstrom r = M +√M2 −Q2 Generalization of Israel theorem
Kerr-Newman r = M +√M2 − a2 −Q2 Generalization of Carter-Robinson theorem
Table 1.1 Classic BH solutions of EFE.
For reasons that will become clear in the next section, the Kerr metric [27] is widely considered the
most relevant exact solution of the EFE in astrophysics. It characterizes the geometry of a stationary
spacetime around an axisymmetric central body of mass M and intrinsic angular momentum J . In
Boyer-Lindquist coordinates (t, r, θ, ϕ), the Kerr line element has the form (1.4) with ∆ = r2 − 2Mr + a2
(Q = 0). It has coordinate singularities at ∆ = 0, which solves for r± ≡ M ±√M2 − a2. These roots
define the outer (r+) and the inner (r−) horizons [24]. The former is the event horizon of a Kerr BH,
whereas the latter is the Cauchy horizon.
4 Introduction
A key feature of the Kerr metric is the existence of the ergoregion [28], the spacetime region between
the event horizon and the ergosphere, the hypersurface r+E = M +
√M2 − a2 cos2 θ ≥ r+. If a particle
enters the ergoregion of a Kerr BH and splits into two particles, it is possible that one of them has negative
energy and falls into the event horizon and the other one escapes to infinity carrying more energy than
the original particle. This phenomenon, known as the Penrose process [29], comes from the transfer of
rotational energy from the BH to the escaping particle.
There is no analogue Birkhoff’s theorem for the Kerr solution [28], i.e. an axisymmetric solution of
the EFE in vacuum is not necessarily stationary. On the other hand, Carter-Robinson theorem [30,23]
states that an asymptotically-flat stationary axisymmetric vacuum solution of the EFE which is regular
on and outside an event horizon must belong to the 2-parameter Kerr family, defined by M, J. In fact,
it has been shown by Hawking [16,31] that the axisymmetry condition is unnecessary: the geometry of a
stationary spacetime external to the event horizon of a rotating BH is axisymmetric.
1.2 Black holes as an endpoint of stellar evolution
Einstein’s theory of gravity predicts the existence of BHs and it should therefore be asked whether
they exist or not. Although astronomical observations support the existence of BHs [32], they have not
been proven yet. Astrophysicists can only list BH candidates, i.e. objects which resemble BHs but are not
proven as so.
Assuming that they do exist, the following question should then be raised: how do astrophysical BHs
form? They are believed to be one of the endpoints of stellar evolution. When a star’s nuclear fuel is
exhausted, the star undergoes gravitational collapse and evolves into one out of three possible final states:
white dwarf, neutron star or BH [16,33]. The fate of the star is mainly determined by its mass, M .
A low-mass star gradually radiates their stored thermal energy and cools off. The gravitational pull
outweighs the total pressure and then the star begins contracting. The matter inside the star is so squeezed
that electrons form a degenerate gas, which gives rise to a quantum-mechanical pressure that supports
the star against gravity [34]. This final state is called a white dwarf. Nevertheless, if the star is more
massive than about 1.4 M⊙, with M⊙ being the mass of the Sun, electron degeneracy pressure cannot
prevent gravity from keeping collapsing the star. For that reason, the star cannot be a white dwarf. This
theoretical result is known as Chandrasekhar limit [35].
When M ≳ 1.4M⊙, the star’s core matter is compressed further than the characteristic radius of
white dwarfs. Owing to this shrinking process, electrons start scattering off protons, producing neutrons
and neutrinos. The energy released during this process is carried away by neutrinos. Due to the loss of
electrons, electron degeneracy pressure decreases drastically. Further compression leads to the formation
of a degenerate gas of neutrons, whose pressure balances the star’s gravitational self-interaction, along
with strong interaction forces [34]. This is another possible endpoint of stellar evolution and has been
named as neutron star.
If M ≳ 3M⊙, i.e. the star’s mass is greater than the Tolman–Oppenheimer–Volkoff limit [36,37], its
core keeps collapsing and a stellar-mass BH is formed. BHs are commonly classified according to their
1.3 Observing astrophysical black holes 5
mass M as follows: primordial BHs (M ≲M⊙); stellar-mass BHs (M ∼ 3 − 70M⊙); intermediate BHs
(M ∼ 103 M⊙); and supermassive BHs (M ∼ 105 − 1010 M⊙).
From a theoretical point of view, Price’s law [38,39] suggests that the multipole moments which cannot
be radiated away during gravitational collapse are fully defined by three parameters (the end-product’s
degrees of freedom): mass, charge and angular momentum. However, astrophysical BHs may be considered
neutral objects, since their net charge is almost zero. Thus, Kerr’s solution to the EFE seems to be one of
the most general descriptions of BH candidates.
1.3 Observing astrophysical black holes
As described in the previous section, if a star is so massive its core cannot support itself against the
pull of gravity, it will keep collapsing and a BH is formed. The gravitational field becomes so strong
it traps everything within a certain region of spacetime, enclosed by a surface known as event horizon
[7,8,16,15]. This surface can be loosely defined as a boundary through which no information can be sent
to distant observers.
Despite this puzzling BH property, there is nothing special about near-horizon physics [5]. According
to GR, if an external observer shoots a bullet into a BH’s event horizon, he sees the infalling object slowing
down such that it never reaches the event horizon. Nevertheless, the bullet actually reaches and crosses
the event horizon in a finite proper time [40].
Event horizons are 3-dimensional null hypersurfaces, which can be described by 2-dimensional spacelike
surfaces at any particular instant of time [8]. Such a spacelike surface is often referred to as the boundary or
surface of a BH. Event horizons are not physically detectable by observers working in finite-size telescopes
or detectors. Indeed, it is only possible to locate an event horizon if the whole future history of spacetime
(i.e. the entire future of null infinity) is known. The global nature of event horizons, known as teleological
property, prevents any (local) experiment whatsoever to determine its position [8].
Despite this hopeless scenario, in which no tangible direct proof of BHs exists [5], astronomers and
astrophysicists can still look for strong gravity effects which are expected in the surrounding environment
of BH candidates. Around astrophysical objects such as white dwarfs, neutron stars or BH candidates is
commonly an accretion disk, a structure of material swirling onto a compact central object [15]. Accretion
disks are formed in close binary systems, i.e. systems in which the distance separating two gravitationally
bound compact objects is comparable to their size. In such a system mass streams from one compact
object to the other, around which an accretion disk is settled.
Matter in accretion disks emits radiation at different wavelengths. Observatories across the electromag-
netic spectrum thus provide indirect information about the properties of each BH candidate [32]. There
has been a steady progress on improving the angular resolution capabilities of radio telescopes in pursuit
of finer and finer images. International collaborations such as the Space Very Long Baseline Interferometry
mission and, more recently, the Event Horizon Telescope, in particular, have been committed to imaging
SgrA∗ features [41,42]. On the other hand, X-rays spectroscopy offers the possibility of probing the
neighborhood of BH candidates down to just a few gravitational radii from the event horizon.
6 Introduction
As well as EWs, a close binary system emits GWs and therefore looses angular momentum and energy
over time. As a result, its components move close to one another faster and faster until they coalesce.
After the coalescence of two compact objects in a binary system, the remnant undergoes a ringdown stage,
whose waveform is dominated by its QNMs [4]. If the final object settles down into a Kerr BH, then the
QNM spectrum is fully defined by the its mass and angular momentum.
Coalescence events produce strong enough GWs to be detected by Earth’s ground-based observatories,
such as LIGO and the interferometric GW antenna Virgo. On September 14, 2015 the two detectors of
LIGO have directly observed for the first time a GW signal, emitted by a compact binary made up of
two BH candidates more than a billion years ago [1]. Since then, LIGO collaboration has identified four
more similar compact binary coalescence events [43–46], two of which were also detected by the Virgo
collaboration. For future reference, the inferred mass and angular momentum of the final BH candidate
left behind each reported compact binary merger are summarized in Table 1.2.
Event M/M⊙ a
GW150914 62.0+4.0−4.0 0.67+0.05
−0.07
GW151226 20.8+6.1−1.7 0.74+0.06
−0.06
GW170104 48.7+5.7−4.6 0.64+0.09
−0.20
GW170814 53.2+3.2−2.5 0.70+0.07
−0.05
GW170608 18.0+4.8−0.9 0.69+0.04
−0.05
Table 1.2 Inferred mass and angular momentum of the final BH candidates for all GW observations everreported.
GW astronomy opens a new window on the universe and will unveil spacetime features in the vicinity
of compact objects, testing both GR and BH physics predictions. However, ringdown signal detections
may not provide a conclusive proof of the existence of BHs [4].
1.4 Exotic compact objects
Notwithstanding the wealth of theoretical research on the mathematical theory of BHs, event horizon’s
non-detectability poses serious problems for BH physicists. In addition, its teleogical property seems to
be broken by the introduction of quantum effects, such as Hawking radiation [47], i.e. particle-antiparticle
radiation emission near the event horizon. This mechanism lowers the mass and energy of BHs and
triggers BH evaporation. The process is intimately linked to the information loss paradox [48]. On the
other hand, the weak cosmic censorship conjecture [29] claims that there are no naked singularities besides
the Big Bang singularity. That is to say that every singularity is enclosed by an event horizon. Thus,
denying the existence of event horizons seems to open the window to naked singularities. It is still not
clear whether a quantum version of GR can solve these open questions or not.
On the experimental side, present GW observations are not precise enough to (indirectly) probe the
true nature of BH candidates. It has actually been argued that ringdown signals from binary BH candidate
coalescences are signatures of the presence of light rings rather than of event horizons [4]. In other words,
1.5 Thesis scope and outline 7
even if the BH candidate remnant formed at the end of the inspiral phase does not have an event horizon,
it can be so similar to a BH that its QNMs correspond to those of a Kerr BH. Such objects are usually
known as BH mimickers.
The aforementioned theoretical and experimental difficulties have been one of the strongest motivations
behind ECO models. Their phenomenology has been widely addressed in search of alternatives to the BH
paradigm.
Just like Schwarzschild BHs, spherically-symmetric ECOs are characterized by unstable circular null
geodesics: high-frequency EWs or GWs can follow circular orbits, defining a surface known as photosphere
[5]. The photosphere controls the optical perception of BHs and ECOs by distant observers, i.e. determines
their so-called shadows. The photosphere properties together with the compactness parameter M/r0,
with M being the mass and r0 the effective radius of the object, are commonly used to classify ECOs into
two categories [6]:
• Ultracompact objects (UCOs): feature a photosphere and are very similar to BHs with respect
to geodesic motion; M/r0 > 1/3 in the static limit.
• Clean-photosphere objects (ClePhOs): feature a surface whose proper distance to the photo-
sphere is such that the time it takes for light to travel from the photosphere to the surface is longer
than the characteristic time scale for null geodesic motion (∼ M), thus having a ‘clean photosphere’;
M/r0 > 1/3 in the static limit.
Table 1.3 summarizes the main ECO models which appeared over the last decades. An exhaustive
review including references (if any) on the formation, stability and electromagnetic and gravitational
signatures of such objects can be found in [5].
1.5 Thesis scope and outline
Following [13,9], this thesis will focus on classical phenomenological aspects of two very simple models
of ECOs described by Kerr spacetime. The first model introduces a surface with reflective properties at a
microscopic or Planck distance (lP = 1.616229 × 10−35 m) from the would-be event horizon of a Kerr BH.
Thus, the background geometry is given by the line element (1.4) with Q = 0 for r > r0, where
r0 ≡ rH + δ, 0 < δ ≪ M, (1.6)
is the location of the surface and rH = M +√M2 − a2 is the event horizon of a Kerr BH. Hereafter, the
objects described by this model will be referred to as Kerr-like ECOs.
For r0 in (1.6) to be real, the inequality a ≤ M (Kerr bound) must be satisfied. Nevertheless, in
general, the aforementioned restriction is not strictly necessary. In the case when the rotation parameter
exceeds the Kerr bound, i.e. a > M , the radial coordinate r (in Boyer-Lindquist coordinates) can take
any real value. However, to avoid naked singularities, the condition r > 0 must be assumed. Kerr-like
compact objects with the aforementioned features are called superspinars (Table 1.3). These ECOs are
the second model this thesis will focus on.
8 Introduction
ECO Year Description
Boson star 1968 [49] Macroscopic Bose-Einstein condensate of massive, complex scalarparticles; Heisenberg uncertainty principle prevents gravitationalcollapse.
Anisotropicstar
1974 [50] Relativistic sphere with locally anisotropic equations of state.
Wormhole 1988 [51] Regular geometry which in general is characterized by deviationsfrom the Kerr metric.
Gravastar 2004 [52] Gravitational Bose-Einstein condensate star; the presence of athin layer of perfect fluid between an interior de Sitter condensatephase and a Schwarzschild exterior prevents the formation of anevent horizon.
Fuzzball 2005 [53] Made up of a ball of strings which ends outside the would-be eventhorizon; avoids the information loss paradox.
Black star 2008 [54] Slowly free-falling matter whose collapse is delayed ad infinitumby vacuum polarization.
Superspinar 2009 [55] BH-like object whose rotation parameter exceeds its mass (Kerrbound violation).
Proca star 2016 [56] Macroscopic Bose-Einstein condensate of massive, complex vectorparticles; Heisenberg uncertainty principle prevents gravitationalcollapse.
Collapsedpolymer
2017 [57] Bound state of highly excited strings.
2 − 2 hole 2017 [58] Horizonless matter-sourced solution of classical quadratic gravitywhich closely matches the exterior Schwarzschild solution down toabout a Planck length of the would-be event horizon.
AdS bubble 2017 [59] AdS interior surrounded by string theory branes.
Table 1.3 Exotic compact object models proposed over the last decades.
Given that Kerr-like ECOs and superspinars are described by Kerr solution, Chapter 2 covers in brief
the main mathematical and physical features of Kerr spacetime and provides an overview of two energy
extraction processes which occur in the scattering of particles and field perturbations off Kerr BHs: the
Penrose process and superradiant scattering (or superradiance), respectively.
Chapter 3 offers a practical introduction to first-order BH perturbation theory, introduces the concept
of QNMs and presents a simple numerical method to compute their corresponding frequencies.
Both Chapter 2 and Chapter 3 should be regarded as a basic toolkit for the reader to keep ready to
hand while going through Chapter 4 and Chapter 5. These are devoted to classical phenomenological
aspects of scalar and electromagnetic field perturbations of Kerr-like ECOs and superspinars, respectively.
A QNM analysis reveals that they are prone to ergoregion instabilities, which develop due to the possible
existence of negative-energy physical states inside the ergoregion. The heuristic approach therein presented
highlights the role of absorption and/or over-reflection in their stability.
Finally, a concise overview of the work is sketched in Chapter 6, together with some closing remarks
on future prospects.
Chapter 2
The Kerr metric
This chapter is devoted to the most relevant properties of Kerr spacetime with interest in the study of
field perturbations of ECOs. For a detailed reading regarding the mathematical properties of Kerr metric,
the careful reader should go through [28,60]. Section 2.5, Section 2.6 and Section 2.7 are particularly
important for understanding the existence of ergoregion instabilities.
2.1 The Kerr metric in the Boyer-Lindquist form
The most general asymptotically-flat stationary solution of EFE in vacuum which is regular on and
outside an event horizon is the Kerr metric [27]. This vacuum solution (Tµν = 0) describes a stationary
axisymmetric spacetime around a central body of mass M and intrinsic angular momentum J (as measured
from spatial infinity [61,7]). In Boyer-Linquist coordinates (t, r, θ, ϕ), the line element reads [15,24,26]
ds2 = − ∆Σ
[dt− a sin2 θ dϕ
]2 + sin2 θ
Σ[(r2 + a2) dϕ− a dt
]2 + Σ∆ dr2 + Σ dθ2, (2.1)
where
Σ ≡ r2 + a2 cos2 θ, (2.2)
∆ ≡ r2 − 2Mr + a2, (2.3)
and a ≡ J/M is the angular momentum per unit mass (or rotation parameter). The Kerr metric is fully
defined by two parameters: M and J or, equivalently, M and a. The line element (2.1) is written in the
form
ds2 = −[e(0)]2 + [e(1)]2 + [e(2)]2 + [e(3)]2, (2.4)
with the 1-forms e(a), a = 0, 1, 2, 3, forming an orthonormal basis. The corresponding dual basis e(a),
a = 0, 1, 2, 3, is obtained by means of the orthonormalization condition e(a) · e(b) = η(a)(b) = (−1, 1, 1, 1).
10 The Kerr metric
e(0) =√
∆Σ
(dt− a sin2 θdϕ
)e(1) =
√Σ∆ dr
e(2) =√
Σdθ
e(3) = sin θ√Σ
[(r2 + a2)
dϕ− adt]
e(0) = 1√Σ∆
[(r2 + a2)
∂t + a ∂ϕ
]e(1) =
√∆Σ ∂r
e(2) = 1√Σ∂θ
e(3) = 1√Σ sin θ
(a sin2 θ ∂t + ∂ϕ
)(2.5)
In the dual basis, the inverse line element is written as
∂2s = −[e(0)]2 + [e(1)]2 + [e(2)]2 + [e(3)]2, (2.6)
which yields
∂2s = − 1
Σ∆[(r2 + a2)
∂t + a ∂ϕ
]2 + ∆Σ ∂2
r + 1Σ∂
2θ + 1
Σ sin2 θ
[∂ϕ + a sin2 θ ∂t
]2 (2.7)
for Kerr spacetime. This result could have been obtained by standard matrix inversion of gµν . For that
purpose, one would have to compute the determinant g of the matrix (gµν) [28],
g ≡ det(gµν) = −Σ2 sin2 θ. (2.8)
The line element (2.1) becomes the Schwarzschild line element (1.2) in the limit a → 0 (M > 0). On
the other hand, if M = 0, but J = 0, the line element (2.1) reduces to the line element of flat spacetime
in oblate spheroidal coordinates [28],
ds2 = − dt2 + dx2 + dy2 + dz2, (2.9)
where x =√r2 + a2 sin θ cosϕ, y =
√r2 + a2 sin θ sinϕ and z = r cos θ, which yields spherical coordinates
in the limit J → 0. The approximate form of the Kerr line element (2.1) at large distances is [28]
ds2 = −[1 − 2M
r+ O
(r−3)]
dt2 −[
4aM sin2 θ
r+ O
(r−3)]
dϕ dt
+[1 + 2M
r+ O
(r−2)] [
dr2 + r2 (dθ2 + sin2 θdϕ2)]
. (2.10)
This result can be derived solving EFE in vacuum in the weak-field approximation [60]. It is straightforward
to check that the line element (2.10) reduces to the line element of flat spacetime in spherical coordinates
in the limit r → +∞, which evinces its asymptotically-flat nature.
Reading the metric components gµν from (2.1), one finds that the metric tensor written in Boyer-
Lindquist coordinates is singular for Σ = 0 and for ∆ = 0. To find out whether the roots of (2.2) and (2.3)
are curvature or coordinate singularities, one must compute the curvature invariants. The only nontrivial
quadratic curvature invariant, the Kretschmann scalar, is [28]
RµνσλRµνσλ = 48M2
Σ6
(r2 − a2 cos2 θ
) (Σ2 − 16r2a2 cos2 θ
). (2.11)
2.2 Symmetries 11
The scalar blows up on the hypersurface Σ = 0, but remains finite on ∆ = 0. Indeed,
r = 0, θ = π
2 (2.12)
is a curvature singularity and the roots of ∆ = 0 are coordinate singularities. The two values of r at which
the second-order polynomial ∆ vanishes are
r± = M ±√M2 − a2, (2.13)
which are only real when a2 ≤ M2. The hypersurfaces r = r+ and r = r− are the outer and inner event
horizons, respectively. Note that they coincide when a2 = M2. In the case when a2 > M2, r± are complex
and, therefore, are not coordinate singularities, meaning that the line element (2.1) is singular only at
(2.12).
2.2 Symmetries
2.2.1 Discrete symmetries
The line element (2.1) is not invariant under the time reversal transformation
T : t → −t, (2.14)
meaning that the spacetime is not static. Since the Kerr metric describes the geometry around a rotating
central body, one expects that transformation (2.14) holds an object which rotates in the opposite direction
[28]. In fact, Kerr spacetime is invariant under the simultaneous transformation1
T : t → −t
Pϕ : ϕ → −ϕor, equivalently,
T : t → −t
Pa : a → −a.
2.2.2 Continuous symmetries
The continuous symmetries of a manifold manifest themselves in isometries. An isometry is a coordinate
transformation xµ → x′µ which leaves the metric tensor gµν form-invariant, i.e. g′µν(y) = gµν(y) for
all coordinates yµ, where g′µν is the transformed metric. An infinitesimal coordinate transformation
xµ → x′µ = xµ + ϵξµ, where ϵ is a small arbitrary constant and ξµ is a vector field, is an isometry if ξµ
satisfies the Killing’s equations ξ(µ;ν) = 0, in which case ξµ is called a Killing vector field [40].
Computing Killing vector fields is often a laborious task. However, some continuous symmetries of a
manifold can be easily identified, and their corresponding Killing vectors readily guessed, just by looking
at its line element.
1PX is the parity operator, corresponding to the transformation X → −X.
12 The Kerr metric
In the case of the Kerr line element (2.1), since no metric coefficients depend on t nor on ϕ, the
Kerr spacetime is stationary, i.e. it does not depend explicitly on time, and axisymmetric, respectively.
Therefore, the Kerr metric admits the Killing vector fields [24]
ξt ≡ ∂t = (1, 0, 0, 0) (2.15)
ξϕ ≡ ∂ϕ = (0, 0, 0, 1), (2.16)
written in Boyer-Lindquist coordinates. ξt is the time translation Killing vector field and ξϕ is the
azimuthal Killing vector field. It can be shown that ξt and ξϕ are the only linearly independent Killing
vectors of Kerr spacetime. In other words, any Killing vector field of the metric (2.1) is a linear combination
of ξt and ξϕ [28].
2.3 Curvature singularity and maximal analytical extension
A clear understanding of the nature of Kerr spacetime when a2 > M2, namely of the curvature
singularity (2.12), requires casting the Kerr metric (2.1), written in Boyer-Lindquist coordinates (t, r, θ, ϕ),
in the Kerr-Schild form,
gµν = ηµν + lµlν , (2.17)
where ηµν is the metric of flat spacetime and l is a null vector with respect to ηµν , i.e. ηµν lµlν = 0. To do
so, the Boyer-Lindquist coordinates t and ϕ are replaced by the new variables [7]
dt = dt+[1 − r2 + a2
∆
], dϕ = dϕ− a
∆dr, (2.18)
which bring the Kerr line element (2.1) to the form [7,28,60]
ds2 = −dt2 + dx2 + dy2 + dz2
+ 2Mr3
r4 + a2z2
[dt+ r
r2 + a2 (x dx+ y dy) + a
r2 + a2 (y dx− x dy) + z
rdz
]2, (2.19)
where the substitutions
x = (r cos ϕ+ a sin ϕ) sin θ, y = (r sin ϕ− a cos ϕ) sin θ, z = r cos θ, (2.20)
were performed to cast it in the Kerr-Schild form (2.17), with
lµ =√
2Mr3
r4 + a2z2
(1, rx+ ay
r2 + a2 ,ry − ax
r2 + a2 ,z
r
). (2.21)
2.3 Curvature singularity and maximal analytical extension 13
It follows from relations (2.20) that r is implicitly defined in terms of x, y and z,
r4 − r2(x2 + y2 + z2 − a2) − a2z2 = 0. (2.22)
To better capture the geometrical meaning of the above equation, one plugs the last relation in (2.20)
into (2.22) and obtains
x2 + y2 = (r2 + a2) sin2 θ, z = r cos θ (2.23)
which shows that the surfaces of non-zero constant r are confocal ellipsoids in the (x, y, z) plane. These
ellipsoids degenerate when r = 0 into the disc
x2 + y2 ≤ a2, z = 0, (2.24)
whose boundary, i.e. the ring x2 + y2 = a2, z = 0, is the curvature singularity identified in Section 2.1.
Although the Kretschmann scalar diverges on the ring, all curvature invariants remain finite on the open
disc x2 + y2 < a2, z = 0, meaning that r, as defined in (2.22), can be analytically continued and, thus,
take positive and negative values [7,28].
The maximal analytic extension of the solution [26] is obtained by attaching a new plane, say (x′, y′, z′),
in such a way that each point on the top of the open disc x2 + y2 < a2, z = 0 in the (x, y, z) plane is
identified with the point on the bottom of the corresponding open disc in the (x′, y′, z′) plane, and vice
versa (Figure 2.1). The (x′, y′, z′) plane is described by the line element (2.19), but with negative values
of r.
Figure 2.1 Maximal extension of Kerr solution for a2 > M2. Source: [7].
The case when a2 < M2 is harder to tackle due to the coordinates singularities at r = r+ and r = r−.
To analytically extend the solution (2.1) across these hypersurfaces, one introduces the Kerr coordinates
14 The Kerr metric
(u+, r, θ, ϕ+) [28], defined by
du+ = dt+ r2 + a2
∆ dr, dϕ+ = dϕ+ a
∆dr, (2.25)
yielding [28,27]
ds2 = −(
1 − 2Mr
Σ
)du2
+ + 2du+dr + Σdθ2 + (r2 + a2)2 − ∆a2 sin2 θ
Σ sin2 θdϕ2+
− 2a sin2 θdrdϕ+ − 4aMr sin2 θ
Σ du+dϕ+, (2.26)
which is clearly regular at r = r+ and r = r−. Note that the Kerr line element written in Kerr coordinates
still features the ring singularity (2.12). A similar analytic extension is obtained by introducing the
coordinates (u−, r, θ, ϕ−), with
du− = dt− r2 + a2
∆ dr, dϕ− = dϕ− a
∆dr, (2.27)
which bring the Kerr line element (2.1) into the form (2.26), but with u+ and ϕ+ replaced by −u−
and −ϕ−, respectively. With these analytic extensions at hand, one can build up the maximal analytic
extension of Kerr solution when a2 < M2. The Carter-Penrose diagram of Kerr spacetime along the axis
of symmetry (θ = 0) for a2 < M2 is depicted in the left panel of Figure 2.2, where three different types of
regions are identified: regions I (r+ < r < +∞) refer to the asymptotically-flat spacetime regions exterior
to the outer event horizon; regions II (r− < r < r+) contain closed trapped surfaces; and regions III
(−∞ < r < r−) feature the ring singularity (2.12). In the extreme case (a2 = M2), regions II are absent
(right panel of Figure 2.2).
Figure 2.2 Carter-Penrose of the maximal analytical extension of Kerr spacetime along the axis ofsymmetry (θ = 0) for a2 < M2 (left) and a2 = M2 (right). Source: [7].
2.4 Zero angular momentum observer (ZAMO) and frame dragging 15
2.4 Zero angular momentum observer and frame dragging
In general relativity timelike geodesics are of interest as they are the four-trajectories of physical
observers. A special sort of observers, known as Zero Angular Momentum Observers (ZAMOs), is
particularly relevant when defining physical quantities.
Definition. A zero angular momentum observer (ZAMO) is a timelike geodesic in the equatorial plane
(θ = π/2, θ = 0, in Boyer-Lindquist coordinates), defined by the parametric equation xµ = xµ(τ) and
with the tangent vector uµ = xµ, where τ is an affine parameter and the dot denotes differentiation with
respect to τ , with zero angular momentum per unit mass with respect to infinity.
The angular velocity of a ZAMO is given by [28]
ΩZAMO ≡ ϕ
t= − gϕt
gϕϕ= 2Jr
(r2 + a2)2 − a2∆ sin2 θ(2.28)
As one would expected, the angular velocity vanishes at spatial infinity (r → +∞). Furthermore, it follows
from the readily verifiable inequality (r2 + a2)2 > a2∆ sin2 θ that
sign(ΩZAMO) = sign(J). (2.29)
The foregoing relation shows that ZAMOs always co-rotate with the background geometry, i.e. the four-
trajectory of such observers is dragged by the gravitational field of the compact object. This phenomenon
is known as frame dragging [16].
2.5 Ergoregion
A timelike curve C• with parametric equation xµ = xµ(t), where t is the Boyer-Lindquist time
coordinate, defines a static observer if its tangent vector Tµ• is proportional to ξt, i.e.
Tµ• = dxµ
dt = (1, 0, 0, 0) (say). (2.30)
Since the curve C• is timelike, the norm of Tµ• satisfies the condition
gµνTµ• T
ν• = gtt = −
(1 − 2Mr
Σ
)< 0, (2.31)
which solves for r < r−E(θ) ∨ r > r+
E(θ), where
r±E(θ) ≡ M ±
√M2 − a2 cos2 θ. (2.32)
16 The Kerr metric
The hypersurfaces r = r±E are known as infinite redshift surfaces. Therefore, the tangent vector Tµ
• is null
on the hypersurfaces r = r±E and becomes spacelike in the spacetime region
r−E(θ) < r < r+
E(θ), (2.33)
where observers are never static. Note that r−E ≤ r− < r+ ≤ r+
E . When a2 < M2 (a2 > M2), the
spacetime region r+ < r < r+E (r−
E < r < r+E) is known as ergoregion and its outer boundary, r = r+
E ,
on which the metric component gtt vanishes, is called the stationary limit surface or ergosphere. The
ergosphere is timelike except at θ = 0, π, where it coincides with r = r+, thus being null.
Considering a constant time slice (dt = 0), the proper volume of the ergoregion is given by [62]
VE =∫ 2π
0dϕ
∫ π
θ−0
dθ∫ r+
0
r−0
dr√grrgθθgϕϕ, (2.34)
where the limits of integration θ−0 , r−
0 and r+0 are defined according to the value of a. θ−
0 = 0, r−0 = r+
and r+0 = r+
E for a2 < M2, whereas θ−0 = arcos(a/M), r−
0 = r−E and r+
0 = r+E for a2 > M2. Figure 2.3
shows the proper volume of the ergoregion as a function of the dimensionless parameter |a/M |. The
proper volumes increases (decreases) monotonically as |a/M | increases when |a/M | < 1 (|a/M | > 1) and
diverges logarithmically as |a/M | approaches 1±.
0 1 2 3 4 50
50
100
150
200
|a/M |
VE/M
3
Figure 2.3 Proper volume of the ergoregion of Kerr spacetime as a function of |a/M |.
On the other hand, a timelike curve C with the parametric equation xµ = xµ(t), where t is the Boyer-
Lindquist time coordinate, defines a stationary observer if its tangent vector Tµ is a linear combination of
ξt and ξϕ, i.e.
Tµ = dxµ
dt = ξt + Ωξϕ = (1, 0, 0,Ω) (say). (2.35)
2.6 Penrose process 17
Since the curve C is timelike, the norm of Tµ satisfies the condition
gµνTµ T
ν = gtt + 2gtϕΩ + gϕϕΩ2 < 0, (2.36)
whose solution depends on the value of Ω. The condition gµνTµ T
ν = 0 solves for
Ω± =−gtϕ ±
√g2
tϕ − gttgϕϕ
gϕϕ. (2.37)
Since g2tϕ − gttgϕϕ = ∆ sin2 θ, condition (2.36) is only satisfied when ∆ > 0, i.e. in the spacetime region
r > r+. When ∆ < 0, i.e. in the spacetime region r− < r < r+, there cannot be stationary observers. For
∆ > 0, gϕϕ > 0 and, as a result, inequality (2.36) holds for
Ω− < Ω < Ω+, (2.38)
which is the allowed range for the angular velocity of a stationary observer.
The tangent vector Tµ becomes null on the hypersurface r = r+ (∆ = 0) and, therefore, Ω+ = Ω−.
ΩH ≡ Ω+ = Ω− is the angular velocity of the event horizon. Note that
ΩH = ΩZAMO (2.39)
at r = r+: the only possible stationary null curve on the event horizon has the ZAMO angular velocity.
2.6 Penrose process
The existence of the ergoregion makes possible the extraction of angular momentum and energy from
Kerr BHs [63,64]. This physical effect, theorized by Penrose [29] and therefore known as Penrose process,
lies in the possible existence of negative-energy particles inside the ergoregion. The energy of a massive
particle with four-momentum p is commonly defined as
E = −p · ξt (2.40)
where the minus sign follows from the fact that p and ξt are timelike at infinity. Inside the ergoregion
(r+ < r < r+E), the time translation Killing vector field ξt is spacelike, meaning that the particle can have
negative energy.
Consider that a massive particle (0) with energy E(0) = −p(0) ·ξt > 0 is shot from infinity to a rotating
BH. Suppose that, once inside the ergoregion, particle (0) splits into particles (1) and (2) and, thus, at
the instant the decay occurs
p(0) = p(1) + p(2), (2.41)
18 The Kerr metric
where p(1) and p(2) stand for the four-momentum of particle (1) and (2), respectively. Contracting with
ξt, one gets
E(0) = E(1) + E(2), (2.42)
with E(1) = −p(1) · ξt and E(2) = −p(2) · ξt. As particles (1) and (2) are created inside the ergoregion,
E(1) and E(2) can be either positive or negative. It can be shown that it is possible for particle (0) to fall
into the ergoregion and split into particles (1) and (2) in such a way that particle (1) (say) falls into the
outer event horizon with E(1) < 0 and particle (2) escapes to infinity with E(2) > E(0), i.e. with a larger
energy than particle (0). This mechanism extracts energy from a rotating BH by decreasing its angular
momentum.
2.7 Superradiance
Although the Penrose process is not likely to be relevant from an astrophysical viewpoint [60], it
sheds some light on the nature of Kerr spacetime, namely the practical implications of the existence of
an ergoregion. A much more relevant mechanism is superradiance or superradiant scattering [63], the
wave analogue of the Penrose process. The condition for the occurrence of this radiation enhancement
mechanism follows naturally from the classical laws of BH mechanics [65], namely the first and second
ones.
The first law of BH mechanics states that, when a Kerr BH (say) is perturbed, the change in its mass
M to first order is
δM = κ
8π δAH + ΩHδJ, (2.43)
where dAH and dJ are the changes in the area of the event horizon and in the angular momentum of the
BH, respectively, and
κ = r+ −M
2Mr+(2.44)
is the surface gravity.
On the other hand, the second law of BH mechanics asserts that, in any classical process, the area of a
BH’s event horizon never decreases, i.e.
dAH ≥ 0. (2.45)
Consider a bosonic field perturbation ψ• in Kerr spacetime. The index • recalls that the field
perturbation can be a scalar or a component of either a vector or a tensor. From the stationary and
2.7 Superradiance 19
axisymmetric nature of Kerr spacetime, one can decompose the field perturbation into modes of the form
ψ• ∝ e−i(ωt−mϕ), (2.46)
where ω is the mode frequency and m is the azimuthal number. The ratio of angular momentum flux to
energy flux of the field perturbation is
δJ
δM= m
ω. (2.47)
It follows immediately from (2.43) and (2.45) that
δM − ΩHδJ ≥ 0. (2.48)
Plugging (2.47) into (2.48), one obtais
(1 − ΩH
m
ω
)δM ≥ 0, (2.49)
which shows that there is extraction of rotational energy from the BH, i.e. δM < 0, only if
ω < mΩH . (2.50)
Thus, the scattered field perturbation has greater amplitude and, thus, carries more energy than its
incident counterpart.
Chapter 3
Quasinormal modes
The dynamics of physical fields around black holes and other compact objects has been studied
extensively since Regge and Wheeler’s pioneer work on the stability of Schwarzschild BHs under small
linear perturbations [66]. BH perturbation theory is a powerful tool to address problems in BH physics,
such as the generation and propagation of GWs by compact binary mergers and their remnants or the
scattering and absorption of fields by compact objects.
The current chapter presents a practical approach to first-order BH perturbation theory for asymptotically-
flat spacetimes in Section 3.1 and introduces the definition of QNM in Section 3.2, a key concept in this
field. A simple numerical method for computing QNMs is described in Section 3.3.1.
3.1 Black-hole perturbation theory
In the framework of BH perturbation theory, the propagation of fundamental field perturbations in
BH spacetimes can in general be reduced to a second-order PDE of the form [67]
[− ∂2
∂t2+ ∂2
∂r2∗
− V (r)]
Ψ = 0, (3.1)
where r∗ is some suitable spatial coordinate, commonly called tortoise coordinate, which tends to −∞ at
the event horizon (or would-be event horizon)1 and to +∞ at infinity. Furthermore, Ψ is some field mode
and V is a time-independent effective potential. The field mode can be expressed in terms of a continuous
Fourier transform,
Ψ(r∗, t) = 12π
∫ +∞
−∞dω e−iωt Ψ(r∗, ω), (3.2)
where Ψ is a field mode in the frequency domain (with frequency ω). The Fourier decomposition of the
field modes reduces the second-order PDE (3.1) to a second-order ODE,
[d2
dr2∗
+ ω2 − V (r)]
Ψ = 0. (3.3)
1Although not strictly necessary, this condition is common in the literature and will be adopted hereafter for the sake ofsimplicity.
22 Quasinormal modes
In asymptotically-flat spacetimes, the effective potential satisfies the relations
V (r) ∼
µ2, r∗ → +∞
ω2 −ϖ2, r∗ → −∞, (3.4)
where µ is the mass parameter of the field perturbation and ϖ is a linear function of ω, namely ∂ϖ/∂ω = 1
(say)2. In the following, only the case when µ = 0 is considered, as the next chapters will solely focus on
the massless case.
Given the asymptotic behavior (3.4) of the effective potential and, thus, of ODE (3.3), one can define
two sets of solutions with asymptotics [68]
Ψ+(r) ∼
e−iωr∗ +A+e+iωr∗ , r∗ → +∞
B+e−iϖr∗ r∗ → −∞
(3.5)
Ψ−(r) ∼
B−e+iωr∗ r∗ → +∞
e+iϖr∗ +A−e−iϖr∗ , r∗ → −∞
(3.6)
The field mode Ψ+ (Ψ−) is commonly referred to as ‘in’ (‘up’) mode.
Before proceeding, it is important to introduce two fundamental concepts in wave theory: the group
velocity (vg) and the phase velocity (vp). For that purpose, consider the one-dimensional wave-packet
φ(x, t) = 12π
∫ +∞
−∞dω C(ω) e−i(ωt−kx), (3.7)
where ω ∈ R is the frequency, C is some amplitude which depends on the frequency and k = k(ω) defines
the dispersion relation. The group and phase velocities are respectively defined as
vg =(∂k
∂ω
)−1, vp = k
ω. (3.8)
While the former dictates the direction of the energy flow as seen locally, the latter determines how energy
flows as seen by an observer at infinity [69].
In most applications in physics, one considers wave-packets with frequencies in an infinitesimal interval
around some frequency ω such that (3.7) can be replaced by a quasi-monochromatic wave of frequency ω,
φ(x, t) ≃ C(ω)2π e−i(ωt−kx), (3.9)
where k ≡ k(ω), with group velocity
vg =(∂k
∂ω
)−1∣∣∣∣∣ω=ω
. (3.10)
2The condition ∂ϖ/∂ω = 1 is introduced merely to compute the group and phase velocities of the field modes (3.5) and(3.6).
3.1 Black-hole perturbation theory 23
The waves in (3.5) and (3.6) should be regarded as quasi-monochromatic waves. Table 3.1 summarizes
the group and phase velocities of those waves. Waves with positive (negative) group velocities are called
outoing (ingoing) waves. Additionally, waves with positive (negative) phase velocities are called reflected
(incident) waves3 (Table 3.2). As expected, sign(vg) = sign(vp) at infinity (r∗ → +∞). On the other hand,
near the would-be event horizon (r∗ → −∞), the ingoing wave e−iϖr∗ is directed inwards (outwards) if
sign(ωϖ) = +1 (−1). Similarly, the outgoing wave e+iϖr∗ is directed outwards (inwards) if sign(ωϖ) = +1
(−1).
Wave vg vp
e−iωr∗ −1 −1
e+iωr∗ +1 +1
e−iϖr∗ −1 −ϖ/ω
e+iϖr∗ +1 +ϖ/ω
Table 3.1 Group and phase velocities of thequasi-monochromatic waves (3.5) and (3.6).
Wave ωϖ > 0 ωϖ < 0
e−iωr∗ Incident Incident
e+iωr∗ Reflected Reflected
e−iϖr∗ Incident Reflected
e+iϖr∗ Reflected Incident
Table 3.2 Nature of the quasi-monochromatic waves(3.5) and (3.6) according to the value of sign(ωϖ).
Ψ+, Ψ∗+, Ψ− and Ψ∗
− are linearly independent solutions of equation (3.3). From the constancy of some
Wronskians4 of equation (3.3), one can write the following relations between the coefficients A± and B±:
W[Ψ+, Ψ∗+]|r∗=+∞ = W[Ψ+, Ψ∗
+]|r∗=−∞ → ω(1 − |A+|2) = ϖ|B+|2 (3.11)
W[Ψ−, Ψ∗−]|r∗=+∞ = W[Ψ−, Ψ∗
−]|r∗=−∞ → ϖ(1 − |A−|2) = ω|B−|2 (3.12)
W[Ψ+, Ψ−]|r∗=+∞ = W[Ψ+, Ψ−]|r∗=−∞ → ωB− = ϖB+ (3.13)
W[Ψ+, Ψ∗−]|r∗=+∞ = W[Ψ+, Ψ∗
−]|r∗=−∞ → ωA+B∗− = −ϖA∗
−B+, (3.14)
where ω and ϖ were considered real.
Equating W[Ψ∗+, Ψ∗
−]|r∗=+∞ to W[Ψ∗+, Ψ∗
−]|r∗=−∞ and W[Ψ∗+, Ψ−]|r∗=+∞ to W[Ψ∗
+, Ψ−]|r∗=−∞, one
obtains the complex conjugate version of relations (3.13) and (3.14), respectively. Using relations
(3.11)−(3.14), it is straightforward to show that |A−| = |A+|.
If follows from relations (3.11)−(3.12) that
|A±|2 < 1, if sign(ωϖ) = +1, (3.15)
|A±|2 > 1, if sign(ωϖ) = −1. (3.16)
In light of Table 3.1 and Table 3.2, one concludes that the outgoing (reflected) wave at r∗ → +∞,
A+e+iωr∗ , carries more energy than the ingoing (incident) wave, e−iωr∗ , whenever |A+|2 > 1, i.e. for
ω > 0 and ϖ < 0 or ω < 0 and ϖ > 0. Thus, the field modes satisfying the foregoing conditions are
superradiant modes. On the other hand, the outgoing wave at r∗ → −∞, e+iϖr∗ , carries more energy3It is common to find a distinction in the literature between waves with negative phase velocities at r∗ → −∞ and
r∗ → +∞. The former are usually called transmitted waves, while the latter incident waves.4The Wronskian of two differentiable functions f and g is W[f, g] = fg′ − f ′g. If f and g are two linearly independent
solutions of the ODE y′′(x) + p(x)y(x) = 0, then W′[f, g] = fg′′ − f ′′g = 0, meaning that W [f, g] is constant.
24 Quasinormal modes
than the ingoing wave, A−e−iϖr∗ , as seen by an observer at infinity, whenever |A−|2 < 1, i.e. for ω > 0
and ϖ > 0 or ω < 0 and ϖ < 0.
3.2 Quasinormal modes
Once physical BCs at r∗ → ±∞ are set, equation (3.3) defines an eigenvalue problem. Namely, if one
requires purely outgoing waves at infinity,
Ψ(ω, r∗) ∼ e+iωr∗ as r∗ → +∞, (3.17)
the eigenvalues, the characteristic frequencies ω, are called QNM frequencies and the fields Ψ QNMs
[67,70]. The set of all eigenfrequencies is often referred to as QNM spectrum. The QNM frequencies ω
are in general complex, i.e. ω = ωR + iωI , where ωR ≡ Re(ω) and ωI ≡ Im(ω). The sign of ωI defines
the stability of the corresponding mode. According to the convention for the Fourier transform (3.2), if:
ωI < 0, the mode is stable and τdam ≡ 1/|ωI | defines the damping e-folding timescale; ωI > 0, the mode is
unstable and τins ≡ 1/ωI defines the instability e-folding timescale; ωI = 0, the mode is marginally stable.
The BC to be imposed at r∗ → −∞ depends on the nature of the compact object under study. For
instance, in the case of BHs, for which the hypersurface r = rH describes the event horizon, the QNMs
satisfy the BC [67,70]
Ψ(ω, r∗) ∼ e−iϖr∗ as r∗ → −∞, (3.18)
meaning physical solutions behave as purely ingoing waves at the event horizon.
The solutions to equation (3.3) for the ECOs described in Section 1.5 must be a superposition of
ingoing and outgoing plane waves at the reflecting surface. Thus, one is interested in the solution with
asymptotics (3.6). For Kerr-like ECOS, the reflecting surface is not at r = rH , but at r = r0 ≡ rH + δ.
However, given that δ ≪ M , one can consider the reflecting surface to be at r∗ → −∞. In particular, one
will focus on those solutions Ψ− which satisfy the BC [13]
Ψ−(ω, r∗0) = (1 + R)A−e
−iϖr∗0 , (3.19)
where r∗0 ≡ r∗(r0) < 0 and R is a complex parameter encoding the reflective features of the surface at
r = r0 (hereafter referred to as reflectivity)5. Note that |R| = 1 defines perfectly-reflecting BCs, whereas
R = 0 refers to a perfectly-absorbing BC (i.e. the BH case) [13,9,71]. The BC (3.19) is equivalent to
requiring that the asymptotic coefficient A− satisfies the relation [13]
RA−e−i2ϖr∗
0 = 1. (3.20)
5For the sake of simplicity, the reflectivity R will be considered model-independent, i.e. the underlying physicalmechanisms giving rise to the surface’s reflective properties (e.g. frictional dissipation) shall be ignored and a heuristicapproach adopted. Furthermore, although R may depend on ω and/or δ [71], this thesis will only focus on constant-valued(ω− and δ−independent) reflectivities.
3.2 Quasinormal modes 25
It is straightforward to show that R = −1 and R = +1 correspond to DBCs and NBCs at r = r0, i.e.
Ψ−(ω, r∗0) = 0, Ψ′
−(ω, r∗0) = 0, (3.21)
respectively, where the prime denotes differentiation with respect to the tortoise coordinate r∗. Explicitly,
e+iϖr∗0 ±A−e
−iϖr∗0 = 0, (3.22)
where the plus (minus) sign corresponds to the DBC (NBC). In general, a perfectly-reflecting BC is
defined by a RBC [72],
cos(ξ)Ψ−(ω, r∗0) + sin(ξ)Ψ′
−(ω, r∗0) = 0, (3.23)
where ξ ∈ [0, π). Plugging relations Ψ−(ω, r∗0) = (1+R)A−e
−iϖr∗0 and Ψ′
−(ω, r∗0) = −iϖ(1−R)A−e
−iϖr∗0
into (3.23) and solving for R, one gets
R = −cos(ξ) − iϖ sin(ξ)cos(ξ) + iϖ sin(ξ) , (3.24)
whose absolute value is, in fact, equal to 1. Note that ξ = 0 corresponds to a DBC (R = −1), whereas
ξ = π/2 refers to a NBC (R = 1).
If the coefficient A− is represented in the form A− = |A−|eiΦ [68], where Φ = argA−, equation (3.22)
solves for6
ϖ = − 12r∗
0[pπ − Φ + i ln |A−|] , p =
2n+ 1, for DBCs
2n, for NBCs, (3.25)
with n ∈ Z.
The potential for scalar (s = 0), electromagnetic (s = ±1) and gravitational (s± 2) field perturbations
in Kerr spacetime, firstly derived by Teukolsky [69], can be reduced to an effective potential with the
asymptotic behavior (3.4) and ϖ = ω − mΩH [69,73,74]. Thus, if follows from relation (3.25) that the
real and imaginary parts of the QNM frequency read
ωR = mΩH − 12r∗
0[pπ − Φ] , ωI = − 1
2r∗0
ln |A−|. (3.26)
Computing the imaginary part of the frequency requires finding the value of |A−| or, alternatively, of |A+|,
since |A−| = |A+|. An important quantity in the context of superradiant scattering of bosonic waves is
the amplification factor [63], defined as
Z = |A+|2 − 1. (3.27)
6Recall that the principal value of the logarithm of z ∈ C is defined as ln z = ln |z| + i arg z.
26 Quasinormal modes
Starobinskii and Churilov showed that the amplification factors of the superradiant scattering of a
low-frequency neutral bosonic wave of spin s off a Kerr BH have the form [75]
Z = 4Qβsl
l∏n=1
(1 + 4Q2
n2
)[ωR(r+ − r−)]2l+1
, (3.28)
where
βsl =[
(l − s)!(l + s)!(2l)!(2l + 1)!!
]2, Q = 2Mr+
r+ − r−(mΩ − ωR). (3.29)
This analytic expression, valid when MωR ≪ 1, offers an expression for |A−|. Assuming ω ∼ mΩH , then
Q2 ≪ 1 and one obtains7
ωI = βsl
r∗0
2Mr+
r+ − r−[ωR(r+ − r−)]2l+1 (ωR −mΩ). (3.30)
3.3 Methods for computing quasinormal modes
There is a plethora of numerical and semianalytical approaches for computing the eigenfrequencies
of BHs and other compact objects. These include Wentzel–Kramers–Brillouin (WKB) approximations,
Leaver’s continued-fraction technique for asymptotically-flat spacetimes, the matrix-valued series method
for asymptotically Anti de Sitter spacetimes, the direct-integration shooting method, the Breit-Wigner
resonance method, monodromy techniques, etc.. A pedagogical introduction to these advanced methods
in BH perturbation theory can be found in [76] (see also references therein).
The numerical results to be presented in Chapter 4 and Chapter 5 were obtained using the direct-
integration shooting method sketched in the next section.
3.3.1 Direct-integration shooting method
The direct-integration shooting method comprises two steps: the integration of the second-order
ODE (3.3) from infinity inwards up to the reflecting surface at r = r0, followed by a one-parameter
shooting method. A non-exhaustive qualitative description of the algorithm is presented below, with some
comments on consistency checks and numerical stability tests.
1. The integration of equation (3.3) from infinity inwards up to the reflecting surface is performed
using a convenient ansatz. On the one hand, QNMs behave as purely outgoing waves at infinity.
On the other hand, field modes propagating in asymptotically-flat spacetimes cannot grow faster
than r. Thus, a suitable ansatz ψ for the field Ψ is
ψ(r) = e+iωr∗
N∑n=0
c(n)∞
rn, r∗ = r∗(r) (3.31)
7ln |A−| = ln(1 + Z) ≃ Z, since Z ≪ 1.
3.3 Methods for computing quasinormal modes 27
where the coefficients c(n)∞ depend on scattering problem parameters (e.g. M , a, Q,. . .) and N is
the number of terms of the partial sum. In order to find explicit expressions for the coefficients one
first needs to write equation (3.3) in the asymptotic form
[d2
dr2 + ω2 − V (r)]
Ψ = 0, (3.32)
where it was used the fact that r∗ → r as r → +∞. Inserting the ansatz (3.31) (with the
substitution r∗ → r) into equation (3.32) and equating coefficients order by order, it is possible to
write c(1)∞ , . . . , c
(N)∞ in terms of c(0)
∞ . The latter is usually set to 1. The choice of N should be a
trade-off between computational time and accuracy.
2. Once the coefficients c(n)∞ are defined, one assigns a guess value to ω = ωR + iωI and integrates
ODE (3.3) from r = r∞ to r = r0 so that the solution satisfies the BCs
Ψ(r∞) = ψ(r∞) and dΨdr
∣∣∣∣r=r∞
= dψdr
∣∣∣∣r=r∞
, (3.33)
where r∞ stands for the numerical value of infinity.
3. Finally, the previous step is repeated for different guess values of ω until the solution satisfies
the desired BC (3.20). The two crucial parameters for consistent results are N and r∞. If the
algorithm is numerically stable, variations in N and/or r∞ do yield similar results. An additional
consistency test is to check relation (3.12) or similar.
Chapter 4
Scalar perturbations
of exotic compact objects
The present chapter focuses on scalar field perturbations of Kerr-like ECOs and superspinars. The
equation of motion of a free massless scalar field in Kerr spacetime is derived in Section 4.1 and reduced
to a radial equation and an angular equation. The radial equation is then transformed so that the
resulting effective potential exhibits the asymptotic behavior (3.4). Following the approach drawn in
Chapter 3, Section 4.2 is devoted to an extensive analysis of the scalar QNM spectra of Kerr-like ECOs
and superspinars. Section 4.3, on the other hand, addresses the superradiant scattering of scalar field
perturbations off Kerr-like ECOs. A summary of the numerical results can be found in Section 4.4.
4.1 Klein-Gordon equation on Kerr spacetime
The action for a free, uncharged, massless scalar field Φ in curved spacetime is [15,24]
S[Φ] = −12
∫d4x
√−g [(∇µΦ) (∇µΦ)] , (4.1)
which yields the equation of motion
Φ = 0, (4.2)
where ≡ ∇µ∇µ stands for the D’Alembert operator. Equation (4.2), known as Klein-Gordon equation,
describes the dynamics of the field Φ in any spacetime geometry, since it is not written in any particular
coordinate system. In the case when the background geometry is given by Kerr solution, it follows from the
existence of the Killing vectors ξt and ξϕ (Section 2.2) that one can separate the t− and ϕ−dependence of
the field Φ, which in turn can be expressed as a superposition of modes with different complex frequencies
30 Scalar perturbations of exotic compact objects
ω and periods in ϕ, i.e.1
Φ = e−iωtF0(r, θ)e+imϕ. (4.3)
Plugging the field mode (4.3) into the equation of motion (4.2), one gets
[−ω2gtt + 2mωgtϕ −m2gϕϕ
]Σ + 1
F0∂r (Σgrr∂rF0) + 1
F0 sin θ∂θ
(Σ sin θ gθθ∂θF0
)= 0. (4.4)
Equation (4.4) is a second-order linear PDE in the independent variables r and θ. Given the explicit
expressions for the metric components gµν in (2.1), one unexpectedly finds that equation (4.4) is separable.
Therefore, the function F0 can be written as a product of two functions, i.e.
F0(r, θ) ≡ R0(r)S0(θ), (4.5)
where R0(r) and S0(θ) are called radial and angular functions, respectively. Introducing this separation of
variables, equation (4.4) reduces to
[(r2 + a2)2
∆ − a2 sin2 θ
]ω2 +
[a2
∆ − 1sin2 θ
]m2 + 2
[1 − r2 + a2
∆
]amω
+ 1R0
ddr
(∆dR0
dr
)+ 1S0
1sin θ
ddθ
(sin θdS0
dθ
)= 0. (4.6)
Applying now the trigonometric identity sin2 θ = (1 − cos2 θ) to the second term in (4.6) and then moving
the θ−dependent terms to the left-hand side and the r-dependent and constant terms to the right-hand
side, one can equate each side of the equation to the separation constant −0Em
l and get two second-order
linear ODEs [24,69]: the radial equation,
ddr
(∆dR0
dr
)− V T
0 R0 = 0, V T0 = λ0 − K2
∆ , (4.7)
where K ≡ ω(r2 + a2) − am and λ0 ≡ 0Em
l + a2ω2 − 2amω, and the angular equation,
1sin θ
ddθ
(sin θdS0
dθ
)−
[m2
sin2 θ− a2ω2 cos2 θ − 0E
ml
]S0 = 0. (4.8)
When |aω| ≪ 1, the separation constant admits the power series expansion [77–79]
0Em
l =+∞∑n=0
f(n)0lm(aω)n with f
(0)0lm = l(l + 1), f
(1)0lm = 0, f
(2)0lm = h0(l + 1) − h0(l) − 1, . . . (4.9)
where h0(l) = 2l(l2 −m2)/(4l2 −1). The solutions to the angular equation (4.8) are called scalar spheroidal
harmonics. When aω = 0, they are simply the spherical harmonics Ylm(θ, ϕ) [77–79].
1Equation (4.3) should be a continuous-frequency Fourier transform and a sum over all possible values of m ofe−iωtF0(r, θ)e+imϕ. However, for the sake of simplicity, Φ will hereafter stand for a field (or Fourier) mode.
4.2 Quasinormal modes 31
Introducing the tortoise coordinate [69] defined by
dr∗
dr = r2 + a2
∆ (4.10)
together with the new radial function
Y0(r) = (r2 + a2)R0, (4.11)
the radial equation (4.7) becomes [24,69]
d2Y0
dr2∗
− V T0 Y0 = 0, V T
0 = G20 + dG0
dr∗− K2 − ∆λs
(r2 + a2)2 , (4.12)
where
G0(r) = r∆(r2 + a2)2 . (4.13)
The effective potential (ω2 + V T0 ) has the asymptotic behavior (3.4) with µ = 0 and ϖ = ω − mΩH .
Therefore, the approach sketched in Chapter 3 for computing the QNM frequencies can be used in a
straightforward way.
4.2 Quasinormal modes
The scalar QNM spectrum of Kerr-like ECOs and superspinars was obtained integrating the radial
equation (4.12) by means of the direct-integration shooting method described in Chapter 3. The numerical
integration was performed using the first three terms of the power series expansion (4.9) for the separation
constant20E
ml and the integration parameters N = 10 and r∞/M = 400 (Section 3.3.1). It was checked
that the scalar QNM frequencies of Schwarzschild and Kerr BHs are recovered when setting R = 0. All
physical quantities are normalized to the mass parameter M . The guess value to the QNM frequency was
chosen according to the numerical results reported in [13].
4.2.1 Schwarzschild-like exotic compact objects
Before addressing the scalar QNMs of Kerr-like ECOs, it is worth looking at the non-rotating case and
inferring the dependence of the QNM frequencies on the distance δ/M .
The real and imaginary parts of the fundamental (n = 0)3 |l| = 1, 2 QNM frequencies of Schwarzschild-
like ECOs as a function of δ/M are depicted in Figure 4.1, for both DBCs (R = −1) and NBCs (R = 1).
Note that, when a = 0, the potential V T0 in (4.12) only depends on l and, thus, there is azimuthal
degeneracy: the QNMs with m = −l, . . . , l are degenerate, i.e. have the same frequency.2The power series expansion (4.9) is a good approximation of 0Em
l when |aω| ≲ 1. This inequality is satisfied by thefundamental l = m = 1 QNM frequencies when a/M ≲ O(0.1) or a/M ≳ O(1.0), as it will be shown in the next section.Considering only the first three terms of the series expansion (4.9) suffices to compute the eigenfrequencies with greataccuracy. However, it is worth pointing out that the numerical results are more sensitive to the number of terms of theexpansion when a/M approaches 1±.
3n is the overtone number.
32 Scalar perturbations of exotic compact objects
10-7 10-6 10-5 10-4 10-3
0.05
0.10
0.15
0.20
0.25
0.30
δ/M
MωR
Dirichlet l=1
Dirichlet l=2
Neumann l=1
Neumann l=2
10-7 10-6 10-5 10-4 10-3
10-10
10-8
10-6
10-4
10-2
δ/M
-M
ωI
Figure 4.1 Real (left) and imaginary (right) parts of the fundamental (n = 0) |l| = 1, 2 scalar QNMfrequencies of a Schwarzschild-like ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ,0 < δ ≪ M , where rH is the would-be event horizon of the corresponding Schwarzschild BH, as a functionof δ/M , for both DBCs and NBCs.
Figure 4.1 shows that the spectra for DBCs and NBCs are qualitatively similar, displaying the same
dependence on δ/M . ωR (ωI) is a monotonically increasing (decreasing) function of δ/M . In particular,
ωR is positive and ωI is negative. It follows from the latter that Schwarzschild-like ECOs are stable
against scalar field perturbations, with damping time τdam = 1/|ωI | (Chapter 3). Since the imaginary
part of the frequency tends monotonically to 0− as δ/M decreases, the damping time diverges in the limit
δ/M → 0 and, therefore, the QNMs become extremely long lived.
4.2.2 Kerr-like exotic compact objects
The fact that the QNM frequencies of Schwarzschild-like ECOs become purely real as δ/M approaches
0 suggests that scalar QNMs of Kerr-like ECOs may be unstable. The introduction of rotation breaks
azimuthal degeneracy, thus splitting the frequencies of QNMs with different azimuthal numbers m [13].
In the small-rotation limit (χ ≡ a/M ≪ 1), the real and imaginary parts of the QNM frequency can be
written as [62]
ωR,I ∼ ω(0)R,I +mω
(1)R,Iχ+ O(χ2), (4.14)
where ω(0)R and ω
(0)I are the real and imaginary parts of the frequency in the absence of rotation (a = 0),
respectively, and ω(1)R,I are first-order corrections which depend on l and δ/M . The coefficient of the
dimensionless parameter χ in (4.14) is proportional to m, meaning that the first-order correction ω(1)I
controls the stability of the QNM as δ/M approaches 0. Since ω(0)I → 0 in that limit (Figure 4.1), ω(1)
I
can make the mode unstable (ωI > 0). This is indeed the case for highly-spinning, perfectly-reflecting
Kerr-like ECOs, as it will be shown in the next section.
4.2 Quasinormal modes 33
Total reflection (|R|2 = 1)
Table 4.1 lists the fundamental (n = 0) l = m = 1 scalar QNM frequency of perfectly-reflecting
(|R|2 = 1) Kerr-like ECOs with different characteristic parameters a, δ, for DBCs4 (R = −1). From a
quick inspection of Table 4.1, one can draw the following conclusions:
• ωR and ωI are negative (positive) in the slow-rotation (fast-rotation) regime;
• ωR and ωI always increase with increasing rotation parameter a in the range [0, 0.9[M and both
appear to have the same sign regardless the value of a.
As a consequence, both ωR and ωI change sign from negative to positive at some critical value of the
rotation parameter, ac (say), and, thus, the QNMs turn from stable to unstable at a = ac. Such critical
value decreases monotonically as δ/M approaches 0. Similar results hold for NBCs.
a/M(ωR, 104ωI)
δ/M = 10−7 δ/M = 10−5 δ/M = 10−3
0.0 (−0.104718, −0.050573) (−0.148527, −0.521523) (−0.240067, −20.631736)
0.1 (−0.079630, −0.017440) (−0.123798, −0.235257) (−0.217841, −12.789579)
0.2 (−0.053645, −0.004035) (−0.097963, −0.087099) (−0.194173, −7.121307)
0.3 (−0.026286, −0.000345) (−0.070529, −0.023158) (−0.168466, −3.429338)
0.4 † (−0.040843, −0.003031) (−0.139942, −1.340715)
0.5 (+0.035255, +0.000606) (−0.007976, −0.000014) (−0.107508, −0.376153)
0.6 (+0.071631, +0.004957) (+0.029514, +0.000615) (−0.069471, −0.055322)
0.7 (+0.114473, +0.018803) (+0.074135, +0.008863) (−0.022865, −0.000902)
0.8 (+0.168413, +0.050741) (+0.131001, +0.040899) (+0.038461, +0.002485)
0.9 (+0.246078, +0.106546) (+0.214180, +0.119942) (+0.131810, +0.065206)
Table 4.1 Fundamental (n = 0) l = m = 1 scalar QNM frequencies of a Kerr-like ECO with a perfectly-reflecting (R = −1) surface at r = r0 = rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon of thecorresponding Kerr BH. The shaded rows represent the experimental band for the rotation parameter ofthe end product of compact-binary coalescences, based on the currently available data from GW detections[1,43–46].† Not presented due to lack of numerical accuracy.
The foregoing observations are better understood by looking at Figure 4.2, which shows the QNM
spectrum for both DBCs and NBCs. Note that the bottom panels are plots of the absolute value of ωI . In
accordance with Table 4.1, the left (right) arms of the interpolating functions refer to negative (positive)
frequencies.
The shaded rows in Table 4.1 and the shaded regions in Figure 4.2 refer to the experimental band
for the rotation parameter of remnants of the compact-binary mergers detected so far by GW detectors
[1,43–46] (cf. Table 1.2). It is clear that for a perfectly-reflecting Kerr-like ECO to be a plausible candidate
4It was checked that the real part of the frequency changes sign upon the transformation m → −m, while the imaginarypart yields the same numerical value.
34 Scalar perturbations of exotic compact objects
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
a/M
MωR
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.0 0.2 0.4 0.6 0.8 1.0
-0.1
0.0
0.1
0.2
0.3
0.4
a/M
MωR
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.0 0.2 0.4 0.6 0.8 1.0
10-10
10-8
10-6
10-4
a/M
M|ωI|
0.0 0.2 0.4 0.6 0.8 1.0
10-11
10-9
10-7
10-5
a/M
M|ωI|
Figure 4.2 Real (top) and imaginary (bottom) part of the fundamental (n = 0) l = m = 1 scalar QNMfrequencies of a Kerr-like ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ, δ ≪ M ,where rH is the would-be event horizon of the corresponding Kerr BH, as a function of δ/M , for the DBCs(left) and NBCs (right). The left (right) arms of the interpolating functions of the imaginary part referto negative (positive) frequencies. The shaded regions represent the experimental band for the rotationparameter of the end-product of compact-binary coalescences, based on the currently available data fromGW detections [1,43–46].
for such remnants it must be stable and, thus, have a characteristic distance δ/M larger than 10−4 (10−3)
for DBCs (NBCs).
Following the slow-rotation approximation (4.14) for the QNM frequencies, the data in Figure 4.2 was
fitted to the polynomials (m = 1)
ωR = ω(0)R + ω
(1)R χ+ O(χ2) (4.15)
ωI = ω(0)I + ω
(1)I χ+ ω
(2)I χ2 + O(χ3) (4.16)
for values of χ over the range [0, 0.25]. The coefficients ω(0)R,I , ω(1)
R,I and ω(2)I are listed in Table 4.2 for both
BCs.
As already mentioned, within numerical accuracy, both ωR and ωI vanish for some critical value of
the rotation parameter, ac. Thus, perfectly-reflecting Kerr-like ECOs admit zero-frequency scalar QNMs
[80]. In particular, ωR, ωI < 0 when a < ac and ωR, ωI > 0 when a > ac. Scalar QNMs turn from stable
4.2 Quasinormal modes 35
δ/MDBCs NBCs
ω(0)R ω
(1)R 103ω
(0)I 103ω
(1)I 103ω
(2)I ω
(0)R ω
(1)R 104ω
(0)I 104ω
(1)I 104ω
(2)I
10−7 −0.105 0.257 −0.005 0.040 −0.087 −0.053 0.257 −0.002 0.017 −0.045
10−6 −0.124 0.257 −0.014 0.104 −0.210 −0.063 0.257 −0.004 0.045 −0.115
10−5 −0.149 0.255 −0.051 0.340 −0.627 −0.077 0.256 −0.014 0.134 −0.329
10−4 −0.186 0.250 −0.265 1.504 −2.428 −0.098 0.256 −0.061 0.512 −1.133
10−3 −0.241 0.233 −2.062 8.889 −10.684 −0.133 0.252 −0.453 3.124 −5.930
Table 4.2 Numerical value of the coefficients of the polynomials (4.15)−(4.16), which fit the real andimaginary parts of the fundamental (n = 0) l = m = 1 scalar QNM frequency of a perfectly reflecting(|R|2 = 1) Kerr-like ECO, for both DBCs and NBCs. The fits were performed for values of the rotationparameter a/M over the range [0, 0.25].
to unstable at a = ac, for which ω = 0. Such critical value depends on δ/M , as shown in Figure 4.2. An
exact analytical relation between δ/M and ac can be found by setting ω to 0 in the radial equation (4.7).
The general solution of the ODE reads [9]
R0(x) = cPPiνl (2x+ 1) + cQQ
iνl (2x+ 1), (4.17)
where x ≡ (r−r+)/(r+ −r−) and ν ≡ 2mac/(r+ −r−), P iνl and Qiν
l are the associated Legendre functions
of the first and second kinds, respectively, and cP , cQ ∈ R. The asymptotically-flat nature of Kerr
spacetime requires that the field modes remain finite at infinity. Given that P iνl (x) diverges as x → +∞
(r → +∞) [81], one must impose the condition cP = 0. As a result, perfectly-reflecting BCs at the surface
r = r0 hold Qiνl (2x+ 1)
∣∣x=x0
= 0 for DBCsd
dxQiνl (2x+ 1)
∣∣x=x0
= 0 for NBCs, (4.18)
where x0 ≡ δ/(r+ − r−). Since δ ≪ M , one can write the associated Legendre function of the second kind
in the approximate form [9]
Qiνl ∼ e−πν
2Γ(iν)
[x−i ν
2 + Γ(−iν)Γ(l + 1 + iν)Γ(iν)Γ(l + 1 − iν) x+i ν
2
](4.19)
and, therefore, obtain the compact analytical expression
ln(
δ
r+ − r−
)∼ −π(p+ 1)
ν+ i
νln
[Γ(1 − iν)Γ(l + 1 + iν)Γ(1 + iν)Γ(l + 1 − iν)
], (4.20)
where p is an odd (even) integer for DBCs (NBCs). Figure 4.3 shows the critical value ac as a function of
δ/M for both BCs. The shaded regions represent the instability domain.
Another interesting observation regarding the spectra in Figure 4.2 is that, while ωR is a monotonically
increasing function of the rotation parameter a in the range [0, 1[M , ωI has a maximum value in the
fast-rotation regime. This behavior is better illustrated in Figure 4.4, a detailed view of the spectra
36 Scalar perturbations of exotic compact objects
-7 -6 -5 -4 -30.2
0.3
0.4
0.5
0.6
0.7
0.8
log10(δ/M)
ac/M
Dirichlet l=m=1
Neumann l=m=1
Figure 4.3 Critical value of the rotation parameter above which the fundamental (n = 0) scalar l = m = 1QNM frequency of a perfectly-reflecting (|R|2 = 1) Kerr-like ECO is unstable, for both DBCs and NBCs.The shaded regions refer to the domain of the ergoregion instability.
for a ∈ [0.80, 1.00[M . In general, for the values of δ/M under investigation, the maximum value of ωI
occurs for a ∈ [0.95, 1.00[M and is greater for less compact objects, i.e. as δ/M increases. Furthermore,
the QNMs appear to be unstable in the extreme case. Although the numerical integration of the radial
equation (4.7) is not feasible when a = M , the interpolating functions for ωI in Figure 4.2 are positive.
On one hand, this suggests that the instability might not be totally quenched for Kerr-like ECOs with
a = M . On the other hand, the fact that ωR approaches the superradiance threshold (ωR → mΩH) as
a/M → 1 bears some resemblance to the zero-damped modes (ωI = 0) reported in [82] for extremal Kerr
BHs.
0.80 0.85 0.90 0.95 1.00
0.05
0.10
0.15
0.20
0.25
a/M
104M
ωI
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.80 0.85 0.90 0.95 1.000.05
0.10
0.15
0.20
0.25
0.30
0.35
a/M
104M
ωI
Figure 4.4 Detailed view of the imaginary part of the fundamental (n = 0) scalar l = m = 1 QNMfrequencies of a Kerr-like ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH +δ, 0 < δ ≪ M ,where rH is the would-be event horizon of the corresponding Kerr BH, as a function of the rotationparameter a/M in the range [0.8, 1], for both DBCs (left) and NBCs (right).
One is interested in computing the timescale of the ergoregion instability. Given that the imaginary part
of the frequency displays a maximum value, a suitable choice for the instability timescale is τins ≡ 1/max[ωI ],
whose values are plotted in Figure 4.5 for the explored range of δ/M . The instability timescale is well
4.2 Quasinormal modes 37
fitted by a general second-order polynomial in log10(δ/M) (plotted in Figure 4.5),
τins ∼ M
10M⊙
2∑m=0
τ(m)ins [log10(δ/M)]m [s], (4.21)
whose coefficients τ (m)ins are listed in Table 4.3 for both BCs. The instability timescale is greater for more
compact objects. In particular, for fixed δ/M , it is greater for objects with δ/M > 10−6 (< 10−6) which
satisfy DBCs (NBCs) rather than NBCs (DBCs) at r = r0. Typical values are of the order of 1−10 s for
compact objects with mass M = 10−100M⊙ and both BCs.
-7 -6 -5 -4 -31
2
3
4
5
log10(δ/M)
τins(10M
⊙/M
)[s]
Dirichlet l=m=1
Neumann l=m=1
Figure 4.5 Timescale of the scalar ergoregion in-stability of rapidly-rotating Kerr-like ECOs with aperfectly-reflecting (|R|2 = 1) surface at r = r0 ≡rH + δ, 0 < δ ≪ M , where rH is the would-be eventhorizon of the corresponding Kerr BH, as a functionof δ/M , for l = m = 1 and both DBCs and NBCs.
τ(m)ins DBC NBC
τ(0)ins 0.03340 0.05623
τ(1)ins −0.10331 −0.11122
τ(2)ins 1.44978 0.56792
Table 4.3 Numerical value of the coefficientsof the second-order polynomial (4.21) which fitsthe timescale of the scalar ergoregion instabilityof a perfectly-reflecting (|R|2 = 1) Kerr-likeECO, for l = m = 1 and both DBCs and NBCs.
Partial reflection (|R|2 < 1)
The main point of the last section is that perfectly-reflecting (|R|2 = 1) Kerr-like ECOs are unstable
against scalar field perturbations when rapidly spinning. This instability finds its origin in the possible
existence of negative-energy physical states inside the ergoregion. In general, the absence of an event
horizon turns horizonless rotating ECOs unstable. The event horizon of Kerr BHs, which can be regarded
as a perfectly-absorbing surface (R = 0), prevents the falling into lower and lower negative-energy states.
Then, it is worth asking whether the introduction of some absorption at the reflective surface r = r0 can
quench or even shut down the ergoregion instability presented in Figure 4.2. Figure 4.6 shows the effect of
small absorption coefficients on the stability of Kerr-like ECOS, for δ/M = 10−5 and both quasi-DBCs
(R ≳ −1) and quasi-NBCs (R ≲ 1). Note that the greater the absorption coefficient (1 − |R|2), the lower
the maximum value of the imaginary part of the frequency. In fact, the introduction of an absorption
coefficient of approximately 0.4% (or, equivalently, a reflectivity of |R|2 = 0.996) completely quenches the
instability for any spin value. When |R|2 < 0.996, the setup is stable (ωI > 0) whatever the value of a.
38 Scalar perturbations of exotic compact objects
0.80 0.85 0.90 0.95 1.00
-0.3
-0.2
-0.1
0.0
0.1
0.2
a/M
104M
ωI
ℛ=-1.0000
ℛ=-0.9995
ℛ=-0.9990
ℛ=-0.9985
ℛ=-0.9980
0.80 0.85 0.90 0.95 1.00
-0.3
-0.2
-0.1
0.0
0.1
0.2
a/M
104M
ωI
ℛ=1.0000
ℛ=0.9995
ℛ=0.9990
ℛ=0.9985
ℛ=0.9980
Figure 4.6 Imaginary part of the fundamental (n = 0) l = m = 1 scalar QNM frequencies of a Kerr-likeECO with a partially-reflecting (|R|2 < 1) surface at r = r0 ≡ rH + δ, where rH is the would-be eventhorizon of the corresponding Kerr BH and δ/M = 10−5, as a function of a/M , for quasi-DBCs (left) andquasi-NBCs (right). Absorption at the surface quenches or even shuts down the ergoregion instability.
The maximum value of the imaginary part of the frequency is well fitted by a general second-order
polynomial in R (plotted in Figure 4.7),
Max(MωI) ∼2∑
k=0a(k)Rk, (4.22)
whose coefficients a(k) are listed in Table 4.4 for values of R in the range [−0.9980,−1] (quasi-DBCs).
0.9980 0.9985 0.9990 0.9995 1.0000-0.5
0.0
0.5
1.0
1.5
2.0
2.5
|ℛ|
105Max(M
ωI)
δ/M=10-3
δ/M=10-4
δ/M=10-5
δ/M=10-6
δ/M=10-7
Figure 4.7 Fit of the maximum value of the imag-inary part of the fundamental (n = 0) l = m = 1scalar QNM frequency of a Kerr-like ECO with re-flectivity R in the range [−0.9980,−1] (quasi-DBCs)to the polynomial (4.22), for different values of δ/M .
δ/M a(0) a(1) a(2)
10−7 0.69571 1.39987 0.70417
10−6 0.78566 1.58100 0.79535
10−5 0.91735 1.84597 0.92864
10−4 1.11362 2.24057 1.12696
10−3 1.40146 2.81878 1.41734
Table 4.4 Numerical value of the coefficientsof the second-order polynomial (4.22) which fitsthe maximum value of the imaginary part of thefundamental (n = 0) l = m = 1 scalar QNMfrequency of a Kerr-like ECO with reflectivityR in the range [−0.9980,−1] (quasi-DBCs), fordifferent values of δ/M .
Figure 4.7 shows that the maximum reflectivity for a Kerr-like ECO to be stable against scalar field
perturbations is about |R| = 0.998 for the explored range of δ/M . The same results hold for NBCs.
4.2 Quasinormal modes 39
4.2.3 Superspinars
Superspinars may be regarded as Kerr-like ECOs violating the Kerr bound. Nevertheless, Kerr
spacetime does not feature an event horizon when a2 > M2 and then the reflective surface does not need
to lie outside the would-be event horizon. In fact, r0, the location of the reflective surface, can take
any real value as one can work with the maximal analytic extension of Kerr spacetime (Section 2.3).
However, since the region r < 0 contains closed timelike curves and a naked singularity at r = 0 forms
when a2 > M2, one requires the reflecting surface to be located at r = r0 > 0 [13,62].
Although Kerr-like ECOs and superspinars differ in the domain of r0, their phenomenology is quite
similar and somewhat symmetric. In the last section it was shown that perfectly-reflecting Kerr-like ECOs
are prone to ergoregion instabilities when fast spinning. One could argue on that account that the violation
of the Kerr bound would strengthen the instability, thus ruling out perfectly-reflecting superspinars as
viable astrophysical compact objects. On other hand, the instability is strongly quenched as a → M ,
suggesting that perfectly-reflecting superspinars with a ≳M may be stable. In fact, it turns out that the
instability is present (absent) when a/M ∼ O(1) (a/M > O(1)), as it will be show in the next section.
Total reflection (|R|2 = 1)
The fundamental (n = 0) l = m = 1 scalar QNM frequencies of perfectly-reflecting (|R|2 = 1)
superspinars with different characteristic parameters a, r0 are plotted in Figure 4.8. The left (right)
arms of the interpolating functions in the bottom panels refer to positive (negative) frequencies. The
spectra resemble those in Figure 4.2, but flipped horizontally. In fact,
• ωR and ωI are positive (negative) when a/M ∼ O(1) (a/M > O(1));
• ωR and ωI always decrease with increasing rotation parameter a for a/M > O(1).
Contrarily to Kerr-like ECOs, superspinars turn from unstable to stable as a increases. The onset of
ergoregion instabilities, ac, decreases monotonically as r0 increases for DBCs, whereas it displays a
maximum value for NBCs. This behavior is illustrated in Figure 4.9, a plot of the critical value ac
as a function of r0 for both BCs. The shaded regions represent the instability domain. In general, a
perfectly-reflecting superspinar is stable against scalar perturbations provided that a ≳ 1.3M . The fact
that the ergoregion instability is not present for sufficiently large values of a agrees with the decrease in
the ergoregion volume as a increases (Figure 2.3). Furthermore, Figure 4.9 suggests that there is a critical
value of r0 above which perfectly-reflecting superspinars are stable regardless the value of a. This should
coincide with the ergosphere: if the reflective surface is located at r = r0 ≥ r+E , the superspinar’s exterior
geometry does not contain an ergoregion (VE = 0), meaning negative-energy physical states cannot exist,
hence the absence of instability.
Similar to Kerr-like ECOs, the imaginary part of the QNM frequencies has a maximum value near
a = M . The corresponding timescale of the ergoregion instability for the explored range of r0 is presented
in Figure 4.10. The plot shows that typical values are of the same order of magnitude as those referring
to Kerr-like ECOs, i.e. 1−10 s for compact objects with mass M = 10−100M⊙ and both BCs.
40 Scalar perturbations of exotic compact objects
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
a/M
MωR
r0/M=0.1
r0/M=0.3
r0/M=0.5
r0/M=0.7
r0/M=0.9
1.00 1.05 1.10 1.15 1.20 1.25 1.30
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
a/M
MωR
r0/M=0.1
r0/M=0.3
r0/M=0.5
r0/M=0.7
r0/M=0.9
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.1410-11
10-9
10-7
10-5
10-3
a/M
M|ωI|
1.00 1.05 1.10 1.15 1.20 1.25 1.3010-11
10-9
10-7
10-5
10-3
a/M
M|ωI|
Figure 4.8 Real (top) and imaginary (bottom) parts of the fundamental (n = 0) l = m = 1 scalar QNMfrequencies of a superspinar with a perfectly-reflecting (|R|2 = 1) surface at r = r0 > 0, as a functionof a/M , for DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions of theimaginary part refer to positive (negative) frequencies.
Partial- and over-reflection (|R|2 ≶ 1)
The argument put forward in Section 4.2.2 on how to quench the ergoregion instability of highly-
spinning Kerr-like ECOs would in principle apply also to superspinars. On the contrary, the introduction
of some absorption (|R|2 < 1) at the reflective surface r = r0 appears to enhance rather than attenuate
the instability, as shown in Figure 4.11. On the other hand, if the absolute value of the reflectivity
slightly exceeds 1 (|R|2 ≳ 1), the imaginary part of the QNM frequency decreases by comparison with the
perfectly-reflecting case. In other words, the effect of a negative absorption coefficient on the stability
of superspinars is quite similar to that of a positive absorption coefficient on the stability of Kerr-like
ECOs. In general, superspinars featuring surfaces with absorption coefficients smaller than −0.4% (or,
equivalently, |R| ≳ 1.002) are stable against scalar field perturbations for any spin value.
4.3 Superradiant scattering 41
0.0 0.2 0.4 0.6 0.8 1.0
1.05
1.10
1.15
1.20
1.25
r0/M
ac/M
Dirichlet l=m=1
Neumann l=m=1
Figure 4.9 Critical value of the rotation parame-ter under which the fundamental (n = 0) l = m = 1scalar QNM frequency of a perfectly-reflecting(|R|2 = 1) superspinar is unstable, for both DBCsand NBCs. The shaded regions refer to the domainof the ergoregion instability.
0.2 0.4 0.6 0.8
0.8
1.0
1.2
1.4
1.6
r0/M
τins(10M
⊙/M
)[s]
Dirichlet l=m=1
Neumann l=m=1
Figure 4.10 Timescale of the scalar ergoregion in-stability of superspinars with a perfectly-reflecting(|R|2 = 1) surface at r = r0 > 0, as a function ofr0, for l = m = 1.
1.00 1.01 1.02 1.03 1.04 1.05
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
a/M
104M
ωI
ℛ=-0.9995
ℛ=-1.0000
ℛ=-1.0005
ℛ=-1.0010
ℛ=-1.0020
1.00 1.02 1.04 1.06 1.08 1.10
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
a/M
104M
ωI
ℛ=0.9995
ℛ=1.0000
ℛ=1.0005
ℛ=1.0010
ℛ=1.0020
Figure 4.11 Imaginary part of the fundamental (n = 0) l = m = 1 scalar QNM frequencies of asuperspinar featuring a surface with reflectivity R at r = r0 => 0, as a function of a/M . The introductionof absorption (|R|2 < 1) enhances the ergoregion instability, whereas over-reflecting (|R|2 > 1) BCsmitigate it.
4.3 Superradiant scattering
Following the results discussed in Section 4.2 and also those reported in [71], this section addresses
the superradiant scattering of scalar field perturbations off Kerr-like ECOs with different reflectivities.
The approach presented herein aims in particular to enlighten the role of absorption in the amplification
factors of Kerr-like ECOs. For that purpose, radial equation (4.12) is solved using a direct-integration
method, whose step 1. coincide with that outlined in Section 3.3.1 for the direct-integration shooting
method. Step 2. consists of integrating the differential equation from r = r0 to r = r∞ so that the
solution satisfies BCs similar to those in (3.33) but at r = r0 rather than at r = r∞. With the solution at
42 Scalar perturbations of exotic compact objects
hand, one then extracts the amplitude of the outgoing wave e+iωr∗ to compute the amplification factor Z
as defined in (3.27).
Figure 4.12 displays the amplification factors for superradiant (0 < ω < mΩH) l = m = 1 scalar field
perturbations scattered off Kerr-like ECOs with a/M = 0.90, δ/M = 10−5 and different reflectivities
R. The numerical results for the BH case (R = 0), in agreement with those reported in [63], show
that the amplification factor increases as ω increases, except when ω → mΩH ∼ 0.313/M , i.e. near
the superradiant threshold. The introduction of a partially-reflecting BC at r = r0 does not alter the
domain of the superradiant regime. However, when |R|2 is nonzero, some resonances become noticeable
around frequencies which match the real part of QNM frequencies of the Kerr-like ECO (Table 4.5).
Furthermore, the peaks become steeper and narrower as |R|2 increases. Like in classical mechanics, the
scalar field perturbation extracts more rotational energy when its frequency coincides with the object’s
proper frequencies of vibration.
0.01 0.02 0.05 0.10 0.2010-5
10-4
0.001
0.010
0.100
1
MωR
Z011(%
)
ℛ=0.00
ℛ=-0.25
ℛ=-0.50
ℛ=-0.75
Figure 4.12 Amplification factors for superradiant (0 <ω < mΩH) l = m = 1 scalar field perturbations scatteredoff Kerr-like ECOs with a/M = 0.9 and featuring a surfacewith reflectivity R at r = r0 ≡ rH + δ, where rH is thewould-be event horizon of the corresponding Kerr BH andδ/M = 10−5.
R ωR ωI
0.03239 −0.01894
−0.25 0.12043 −0.01997
0.21400 −0.02142
0.03246 −0.00947
−0.50 0.12053 −0.00998
0.21414 −0.01069
0.03248 −0.00393
−0.75 0.12056 −0.00414
0.21417 −0.00443
Table 4.5 Scalar l = m = 1 QNMfrequencies corresponding to the reso-nance peaks displayed in Figure 4.12.
The very same problem can be solved analytically in the low-frequency regime (Mω ≪ 1) using
matching-asymptotic techniques [11]. For that, the spacetime region outside the reflective surface r = r0
is split into a region near the would-be event horizon, where r − rH ≪ 1/ω, and a region far from it, i.e.
at infinity, where r − rH ≫ M . One starts looking for asymptotic solutions to the radial equation in each
spacetime region, imposing the desired BC at r = r0, and then matches them in the overlapping region,
where M ≪ r − rH ≪ 1/ω. In the following, besides Mω ≪ 1, the assumption a ≪ M for slowly-rotating
objects is also considered.
4.3 Superradiant scattering 43
The spacetime curvature induced by Kerr-like ECOs vanishes at infinity, where the approximations
M ∼ 0 and a ∼ 0 hold. With these assumptions, the radial equation (4.7) reduces to the wave equation
for a massless scalar field of angular frequency ω and angular momentum l in Minkowski spacetime,
d2R0
dr2 +[ω2 − l(l + 1)
r2
]R0 = 0, (4.23)
where R0(r) ≡ rR0. The general solution of this ODE is a linear combination of Bessel functions of the
first kind [81],
R0(r) = 1√r
[αJl+1/2(ωr) + βJ−l−1/2(ωr)
], (4.24)
where, in general, α, β ∈ C. The large-r behavior of the asymptotic solution (4.24) is
R0(r) ∼√
2πω
1r
[α sin(ωr − lπ/2) + β cos(ωr + lπ/2)] , (4.25)
which can be written as a superposition of ingoing and outgoing waves,
R0(r) ∼ eilπ/2√
2πω1r
[(β − αei(l+1/2)π
)e+iωr +
(αeiπ/2 + eilπβ
)e−iωr
]. (4.26)
It is worth pointing out that, in the case of QNMs, the absence of ingoing waves at infinity requires the
BC β = αei(l−1/2)π [11]. As for the scattering problem, both terms in (4.26) are considered, with the
first (second) corresponding to a reflected (an incident) wave. The reflected and incident energy fluxes at
infinity are proportional to the quantities
|O|2 ∝ |α|2 + |β|2 + 2(−1)l Im(αβ), (4.27)
|I|2 ∝ |α|2 + |β|2 − 2(−1)l Im(αβ), (4.28)
respectively. The main goal of the asymptotic matching is to find expressions for α and β in terms of M ,
a, l, m, δ and R to compute the amplification factors (3.27), where A+ = A− = |O|2/|I|2.
The small-r behavior of the asymptotic solution (4.24) is
R0(r) ∼ α(ω/2)l+1/2
Γ(l + 3/2) rl + β(ω/2)−l−1/2
Γ[−l + 1/2] r−l−1. (4.29)
Near the would-be event horizon, the radial equation (4.7) reduces to
ddr
(∆dR
dr
)+
[r4
+(ω −mΩ)2
∆ − l(l + 1)]R = 0. (4.30)
Introducing the radial coordinate z = (r− r+)/(r− r−) and the definition R0(z) = ziω(1 − z)l+1Q0(z) for
the radial function, where ω = (ω −mΩ)r2+/(r+ − r−), one can bring the radial equation into the form
z(z − 1)d2Q0
dz2 + [c− (2b+ c)z] dQ0
dz − abQ0 = 0, (4.31)
44 Scalar perturbations of exotic compact objects
with a = l+ 1 + 2iω, b = l+ 1 and c = 1 + 2iω. Equation (4.31) is a standard hypergeometric ODE, whose
most general solution is a superposition of hypergeometric functions. In terms of the radial function R0,
the solution reads
R0(z) = A z−2iωF (a− c+ 1, b− c+ 1, 2 − c; z) +B F (a, b, c; z), (4.32)
with A,B ∈ C. The small-z (r ∼ rH) behavior of the asymptotic solution (4.32) is
R0(r) ∼ Az−iω +Bz+iω ∼ Ae−i(ω−mΩ) r+2M r∗ +Be+i(ω−mΩ) r+
2M r∗ , (4.33)
where r∗ is the tortoise coordinate defined in (4.10). For slowly-rotating objects (a ≪ M), equation (4.33)
can be written as
R0(r) ∼ Ae−i(ω−mΩ)r∗ +Be+i(ω−mΩ)r∗ . (4.34)
Therefore, the BC (3.20) to be imposed at the reflecting surface r = r0 is
B
A= R ≡ R e−2i(ω−mΩ)r∗
0 , (4.35)
where, as defined in Chapter 3, r∗0 ≡ r∗(r0) < 0.
On the other hand, the large-z (large-r) behavior of the asymptotic solution (4.32) is
R0(r) ∼Γ(2l + 1)Γ(l + 1)
[A
Γ(1 − 2iω)l + 1 − 2iω +B
Γ(1 + 2iω)Γ[l + 1 + 2iω]
] (r
r+ − r−
)l
+Γ(−2l − 1)Γ(−l)
[A
Γ(1 − 2iω)Γ(−l − 2iω) +B
Γ(1 + 2iω)Γ(−l + 2iω)
] (r
r+ − r−
)−l−1, (4.36)
which exhibits the same dependence on r as the small-r behavior of the asymptotic solution in the far
region (4.29). Matching the two solutions, it is straightforward to show that
α = Γ(l + 3/2)(ω/2)l+1/2(r+ − r−)l
Γ(2l + 1)Γ(l + 1)
[Γ(1 − 2iω)
Γ(l + 1 − 2iω) + R Γ(1 + 2iω)Γ(l + 1 + 2iω)
](4.37)
β = Γ(−l + 1/2)(ω/2)−l−1/2(r+ − r−)−l
Γ(−2l − 1)Γ(−l)
[Γ(1 − 2iω)
Γ(−l − 2iω) + R Γ(1 + 2iω)Γ(−l + 2iω)
]. (4.38)
Plugging the expressions for α and β into (4.28) and (??), one obtains an approximate analytical
expression for the amplification factors for the scattering of low-frequency scalar field perturbations off
slowly-rotating Kerr-like ECOs. A comparison between this approximation and the numerical results
displayed in Figure 4.12 is presented in Figure 4.13. As one would expect, the agreement between the
low-frequency approximation and the numerical results is better when MωR ≪ 1 . Nevertheless, the
curves corresponding to the analytical results do reproduce the shape of the resonance peaks, which occur
when 0.03 ≲Mω ≲ 0.313.
4.4 Summary 45
0.01 0.02 0.05 0.10 0.2010-5
10-4
0.001
0.010
0.100
1
MωR
Z011(%
)ℛ=0.00
Numerical
Analytical
0.01 0.02 0.05 0.10 0.2010-5
10-4
0.001
0.010
0.100
1
MωR
Z011(%
)
ℛ=-0.25
0.01 0.02 0.05 0.10 0.2010-5
10-4
0.001
0.010
0.100
1
MωR
Z011(%
)
ℛ=-0.50
0.01 0.02 0.05 0.10 0.2010-5
10-4
0.001
0.010
0.100
1
MωR
Z011(%
)
ℛ=-0.75
Figure 4.13 Numerical and analytical values for the amplification factors of superradiant (0 < ω < mΩH)l = m = 1 scalar field perturbations scattered off Kerr-like ECOs with a/M = 0.9 and featuring a surfacewith reflectivity R at r = r0 ≡ rH + δ, where rH is the would-be event horizon of the corresponding KerrBH and δ/M = 10−5. The agreement between numerical and analytical results is better when Mω ≪ 1.
4.4 Summary
This chapter addressed scalar field perturbations of Kerr-like ECOs and superspinars with different
degrees of compactness M/r0 and reflectivities R. The analysis led to two highlighting conclusions.
First, when the object’s surface is perfectly-reflecting (|R|2 = 1), an instability develops when the
object is spinning at a rate either above or below some critical value of the rotation parameter. Despite
the dependence of the instability domain on the compactness of the object, it generally occurs for orders
of magnitude of a/M between 0.1 and 1. The instability is intimately linked to the ergoregion, where
negative-energy physical states can form. These cannot be absorbed by the object’s surface and, therefore,
cause the exponential growth of field perturbations.
Second, the ergoregion instability of a Kerr-like ECO (superspinar) is either attenuated or neutralized
when its surface is not perfectly-reflecting but partially-absorbing (over-reflecting): an absorption coefficient
greater (smaller) than approximately 0.4% (−0.4%) prevents unstable QNMs to develop. From a dynamical
46 Scalar perturbations of exotic compact objects
point of view, this precludes the object from falling into more negative-energy physical states, which would
be energetically favorable in the case of total reflection.
The results presented herein are in agreement with those reported in [13], in which the authors
conjecture that the phenomenological aspects drawn above for scalar field perturbations of Kerr-like ECOs
and superspinars would also apply to the electromagnetic case. While this turns out to be true, as it will
be shown in Chapter 5, the minimum absolute value of the absorption coefficient required to power off
any effect of ergoregion instabilities appears to be much smaller than that put forward in [13].
Chapter 5
Electromagnetic perturbations
of exotic compact objects
The current chapter addresses electromagnetic field perturbations of Kerr-like ECOs and superspinars.
Deriving Maxwell’s equations on Kerr spacetime requires some familiarity with the tetrad formalism,
namely the Newman-Penrose formalism. Those less familiar with this approach to GR should read over
the original paper by Newman and Penrose [83] and Chandrasekhar’s monograph The Mathematical
Theory of Black Holes [64]. The derivation itself is outlined in Section 5.2 and reduces to a radial equation
and an angular equation. Section 5.3 introduces the definition of both the electric and magnetic fields in
the tetrad formalism and presents their components as measured by a ZAMO. These are relevant physical
quantities to define perfectly-reflecting BCs for the electromagnetic field. This is done in Section 5.4. In
Section 5.5, as in Chapter 4, the radial equation is transformed so that the resulting effective potential
exhibits the asymptotic behavior (3.4). Finally, following once more the approach drawn in Chapter 3,
Section 5.6 is devoted to an extensive analysis of the electromagnetic QNM spectra of Kerr-like ECOs and
superspinars. A summary of the numerical results can be found in Section 5.7.
5.1 The Newman-Penrose formalism
The Newman-Penrose formalism is a tretad formalism with null basis vectors commonly defined
as e(1), e(2), e(3), e(4) = l,n,m, m. The four vectors of the null tetrad satisfy the orthogonality
conditions
l · m = l · m = n · m = n · m = 0, (5.1)
and are normalized so that
l · n = 1, m · m = −1. (5.2)
48 Electromagnetic perturbations of exotic compact objects
These normalization conditions have two main advantages: on the one hand, the Ricci rotation coefficients
are antisymmetric in their first two indices; on the other hand, raising and lowering tetrad indices is
equivalent to operating with directional derivatives.
The fundamental matrix η(µ)(ν) and its inverse are given by
η(µ)(ν) = η(µ)(ν) =
0 1 0 0
1 0 0 0
0 0 0 −1
0 0 −1 0
. (5.3)
It follows that the co-tetrad is e(1), e(2), e(3), e(4) = n, l,−m,−m.
In this tetrad representation, the Ricci rotation coefficients, or spin coefficients, are given by
− κ = lµ;νmµlν − ρ = lµ;νm
µmν − ε = 12 (lµ;νn
µlν −mµ;νmµlν)
− σ = lµ;νmµmν µ = nµ;νm
µmν − γ = 12 (lµ;νn
µnν −mµ;νmµnν)
λ = nµ;νmµmν − τ = lµ;νm
µnν − α = 12 (lµ;νn
µmν −mµ;νmµmν)
ν = nµ;νmµnν π = nµ;νm
µlν − β = 12 (lµ;νn
µmν −mµ;νmµmν) . (5.4)
It is useful to introduce the following notation for the directional derivatives
D ≡ l = lµ∂µ, ∆ ≡ n = nµ∂µ, δ ≡ m = mµ∂µ, δ ≡ m = mµ∂µ. (5.5)
5.2 Maxwell’s equations
The action for classical electrodynamics in curved spacetime is [24]
S[A] = − 116π
∫V
√−g d4x FµνF
µν +∫
V
√−g dx4 AµJ
µ, (5.6)
where Fµν = 2A[ν;µ]. Fµν is the electromagnetic-field tensor, Aµ is the electromagnetic four-potential
and Jµ is the four-current. In the absence of sources (Jµ = 0), the equations of motion describing the
electromagnetic four-potential Aµ are the source-free Maxwell’s equations, which can be written in the
form
Fµν;ν = 0, F[µν;λ] = 0. (5.7)
In the Newman-Penrose formalism, these equations are expressed in terms of the spin coefficients (5.4),
the directional derivatives (5.5) and the three complex scalars
ϕ0 = Fµν lµmν , ϕ1 = 1
2Fµν(lµnν + mµmν), ϕ2 = Fµνmµnν , (5.8)
5.2 Maxwell’s equations 49
which are projections of the electromagnetic-field tensor onto the null tetrad. In this tetrad representation,
Maxwell’s equations (5.7) become [64]
Dϕ1 − δϕ0 = (π − 2α)ϕ0 + 2ρϕ1 − κϕ2
δϕ1 − ∆ϕ0 = (µ− 2γ)ϕ0 + 2τϕ1 − σϕ2
Dϕ2 − δϕ1 = −λϕ0 + 2πϕ1 + (ρ− 2ε)ϕ2
δϕ2 − ∆ϕ1 = −νϕ0 + 2µϕ1 + (τ − 2β)ϕ2.
(5.9)
The coupled equations (5.9) describe the dynamics of electromagnetic perturbations of any spacetime
geometry (in the absence of sources), as they are not written in any particular coordinate system. Once
the background geometry is specified and the null tetrad is chosen, one can write down the explicit form
of Maxwell’s equations.
5.2.1 Maxwell’s equations on Kerr spacetime
Before specializing Maxwell’s equations for Kerr spacetime, one can simplify equations (5.9) with
the aid of the Goldberg-Sachs theorem [84]. An immediate consequence of this theorem is that the spin
coefficients κ, σ, ν, λ vanish for any metric tensor which is algebraically special and of Petrov type D1
[64,85], such as the Kerr metric. As a result, Maxwell’s equations in vacuum for Kerr background reduce
to
Dϕ1 − δϕ0 = (π − 2α)ϕ0 + 2ρϕ1, (5.10)
δϕ1 − ∆ϕ0 = (µ− 2γ)ϕ0 + 2τϕ1, (5.11)
Dϕ2 − δϕ1 = 2πϕ1 + (ρ− 2ϵ)ϕ2, (5.12)
δϕ2 − ∆ϕ1 = 2µϕ1 + (τ − 2β)ϕ2. (5.13)
Acting on equation (5.10) with2 (δ− β−α∗ − 2τ + π∗) and on equation (5.11) with (D− ε+ ε∗ − 2ρ− ρ∗)
and subtracting one equation from the other, the terms in ϕ1 vanish by the identity3 [69]
[D − (p+ 1)ϵ+ ϵ∗ +qρ− ρ∗] (δ − pβ + qτ) − [δ − (p+ 1)β − α∗ + π∗ + qr] (D − pϵ+ qρ) = 0, (5.14)
with p = 0 and q = −2, one obtains a decoupled equation for ϕ0 [69],
[(D − ϵ+ ϵ∗ − 2ρ− ρ∗) (∆ + µ− 2γ) − (δ − β − α∗ − 2τ + π∗)(δ + π − 2α)]ϕ0 = 0. (5.15)
Similarly, acting on equation (5.12) with (δ + α+ β∗ + 2π − τ∗) and on equation (5.13) with (∆ + γ −
γ∗ + 2µ+ µ∗), subtracting one equation from the other and using the identity (5.14) with the substitution
1Rigorous definitions can be found in [64].2The complex conjugate of the spin coefficients will be denoted by a superscript asterisk instead of an over-bar, because
ρ will be introduced later so that ρ = ρ∗.3The identity (5.14), where p and q are arbitrary, holds whenever κ, σ, ν, λ vanish.
50 Electromagnetic perturbations of exotic compact objects
l → n and m → m and p = 0 and q = −2, one obtains a decoupled equation for ϕ2,
[(∆ + γ − γ∗ + 2µ+ µ∗) (D − ρ+ 2ϵ) − ( δ + α+ β∗ + 2π − τ∗)(δ − τ + 2β)]ϕ2 = 0. (5.16)
The decoupled equations (5.15) and (5.16) are valid for any background geometry for which κ = σ = ν =
λ = 0 hold, such as the Schwarzschild and Kerr spacetimes [64].
To write the explicit form of Maxwell’s equations on Kerr spacetime, one introduces the Kinnersley
tetrad [85]
D = r2 + a2
∆ ∂t + ∂r + a
∆∂ϕ
∆ = 12Σ
[(r2 + a2)
∂t − ∆∂r + a ∂ϕ
]δ = 1
ρ√
2
[ia sin θ ∂t + ∂θ + i
sin θ∂ϕ
],
(5.17)
written in Boyer-Lindquist coordinates, where ρ ≡ r + ia cos θ (and ρ∗ = r − ia cos θ). Additionally, one
can set the spin coefficient ϵ to 0 due to the freedom of making a null rotation [85]. The non-vanishing
spin coefficients are given by [24]
ρ = −(r − ia cos θ)−1
µ = 12ρ
[1 +M(ρ+ ρ∗) + (a|ρ| sin θ)2
]τ = − ia√
2|ρ|2 sin θ
π = ia√2ρ2 sin θ
γ = 12Mρ
[ρ− iaρ∗ cos θ + (a|ρ| sin θ)2
]α = 1
2ρ[iaρ sin θ − 1
2 cot θ]
β = −ρ∗ cot θ2√
2.
From the stationary and axisymmetric nature of the Kerr spacetime (Section 2.2), one expects that
any field perturbation can be expressed as a superposition of modes with different complex frequencies ω
and different periods in ϕ. In others words, each mode of the field perturbation is expected to exhibit a
dependence on t and ϕ given by
ϕi ∝ e−i(ωt−mϕ), (5.18)
with i = 1, . . . 3. Given the ansatz (5.18), one can immediately perform the substitutions ∂t → −iω and
∂ϕ → im. As a result, the Kinnersley tetrad (5.17) becomes [64]
D = D0, ∆ = − ∆2ρ2 D†
0, δ = 1ρ√
2L†
0, δ∗ = 1ρ∗
√2
L0, (5.19)
where the derivative operators Dn, D†n, Ln, L†
n (n ∈ Z) are given by
Dn = ∂r − iK
∆ + 2nr −M
∆ ,
D†n = ∂r + iK
∆ + 2nr −M
∆ ,
Ln = ∂θ −Q+ n cot θ,
L†n = ∂θ +Q+ n cot θ,
(5.20)
5.2 Maxwell’s equations 51
and K = (r2 +a2)ω−am and Q = aω sin θ−m cosec θ. While Dn and D†n are purely radial operators, Ln
and L†n are purely angular operators. Note that D†
n(ω,m) = Dn(−ω,−m) and L†n(ω,m) = Ln(−ω,−m).
Additionally, although D†n = (Dn)∗, L†
n = (Ln)∗. Finally, using the tetrad (5.19) together with the
foregoing notation, one can rewrite equations (5.15) and (5.16) in the form
[∆
(D1 + 1
ρ∗
) (D†
1 − 1ρ∗
)+
(L†
0 + ia sin θρ∗
) (L1 − ia sin θ
ρ∗
)]Φ0 = 0, (5.21)[
∆(
D†0 + 1
ρ∗
) (D†
0 − 1ρ∗
)+
(L0 + ia sin θ
ρ∗
) (L†
1 − ia sin θρ∗
)]Φ2 = 0, (5.22)
where Φ0 = ϕ0 and Φ2 = 2ρ−2ϕ2. Equations (5.21) and (5.22) are second-order linear PDEs in two
independent variables, r and θ. Unlike the t− and ϕ−dependence, one would not expect that the
coordinates r and θ could be separated as well. Remarkably, it turns out that the dependence on r and θ
does separate. In fact, with the aid of the identities [64]
∆(
D1 + 1ρ∗
) (D†
1 − 1ρ∗
)= ∆D1D†
1 + 2iKρ∗ ,(
L†0 + ia sin θ
ρ∗
) (L1 − ia sin θ
ρ∗
)= L†
0L1 − 2ia sin θρ∗ Q,
(5.23)
equation (5.21) takes the simpler form
(∆D1D†1 + L†
0L1 + 2iωρ) Φ0 = 0. (5.24)
In a similar way, using the identities [64]
∆(
D†0 + 1
ρ∗
) (D0 − 1
ρ∗
)= ∆D0D†
0 − 2iKρ∗ ,(
L0 + ia sin θρ∗
) (L†
1 − ia sin θρ∗
)= L0L†
1 + 2ia sin θρ∗ Q,
(5.25)
one can convert equation (5.22) into the form
(∆D†0D0 + L0L†
1 − 2iωρ) Φ2 = 0. (5.26)
It is now obvious that equations (5.24) and (5.26) are separable. In other words, taking into account the
ansatz (5.18), one can perform the substitution
Φ0 = e−iωtR+1(r)S+1(θ)e+imϕ, Φ2 = e−iωtR−1(r)S−1(θ)e+imϕ, (5.27)
where R±1 and S±1 are called, respectively, radial and angular functions, and reduce the PDEs (5.21)
and (5.22), respectively, to a set of ODEs for R+1 and S+1,
(∆D1D†1 + 2iωr)R+1 = λ+1R+1 (5.28)
(L†0L1 − 2aω cos θ)S+1 = −λ+1S+1 (5.29)
52 Electromagnetic perturbations of exotic compact objects
and for R−1 and S−1,
(∆D†0D0 − 2iωr)R−1 = λ−1R−1 (5.30)
(L0L†1 + 2aω cos θ)S−1 = −λ−1S−1 (5.31)
λ±1 are separation constants. The angular equations (5.29) and (5.31) have two regular singular points at
θ = 0 and θ = π. The requirement of regularity at θ = 0 and θ = π defines the same Sturm-Liouville
eigenvalue problem for both separation constants λ±1, meaning λ+1 = λ−1 ≡ λ|±1|. In fact, replacing θ
by (π − θ) in equation (5.31), for instance, one gets the same operator acting on S−1 as the one acting on
S+1 in equation (5.29), since Ln(π − θ) = −L†n(θ). As a result, any solution S+1(θ) to equation (5.29) is
also a solution to equation (5.31), provided that θ is replaced by (π − θ). This means that, if the angular
functions S±1 are normalized to unity,
∫ π
0(S±1)2 sin θ dθ = 1, (5.32)
then S+1(θ) = S−1(π − θ).
As for the radial equations (5.28) and (5.30), it is straightforward to check that ∆R+1 and R−1
satisfy complex-conjugate equations4. Indeed, using the readily verifiable relations ∆Dn+1 = Dn∆ and
∆D†n+1 = D†
n∆, equation (5.28) can be written in the form
(∆D0D†0 + 2iωr)∆R+1 = λ|±1|∆R+1. (5.33)
Furthermore, the radial functions R+1 and R−1 and the angular functions S+1 and S−1 are related via
the Teukolsky-Starobinsky identities (see Appendix A) [60,64,86],
∆D0D0R−1 = B∆R+1
L†0L†
1S−1 = BS+1,
∆D†0D†
0∆R+1 = BR−1
L0L1S+1 = BS−1
(5.34)
where B =√λ2
|±1| − 4a2ω2 + 4amω. The Teukolsky-Starobinski identities are algebraic relations between
the radial or angular functions and their first derivatives, as one can reduce any second derivative of these
functions to a linear combination of radial or angular functions and their corresponding first derivatives,
with the aid of equations (5.28)−(5.31).
Equations (5.28)−(5.31) were firstly derived by Teukolsky in 1973 [69] and therefore are usually
called Teukolsky equations. Plugging the explicit form of the derivative operators (5.20) into equations
(5.28)−(5.31), one gets
∆−s ddr
[∆s+1 dRs
dr
]− V T
s Rs = 0, V Ts = 2is(r −M)K −K2
∆ − 4isωr + λ|s| − s(s+ 1), (5.35)
4That does not necessarily mean that ∆R+1 is a complex constant multiple of R−1.
5.3 Electric and magnetic fields in the ZAMO frame 53
and
1sin θ
ddθ
[sin θdSs
dθ
]−
[m2 + s2 + 2ms cos θ
sin2 θ− a2ω2 cos2 θ + 2aωs cos θ − sE
ml
]Ss = 0, (5.36)
where sEm
l ≡ λ|s| + 2amω − a2ω2, with the definition λ|s| = λs + s(s + 1), and s is the spin-weight
parameter, which takes the values ±1 in the case of electromagnetic field perturbations. Surprisingly,
equations (5.35) and (5.36) also encode the dynamics of (massless) scalar (s = 0)5, neutrino (s = ±1/2)
and gravitational (s = ±2) field perturbations on Kerr spacetime (in the absence of sources).
It follows from the symmetries of the angular equation (5.36) that sEm
l satisfy the relations6 [77]
sEm
l(aω) = −sEm
l(aω), sE
ml(−aω) = sE
−ml(aω). (5.37)
In the case when aω ≪ 1, perturbation theory holds [77–79]
sEm
l =+∞∑n=0
f(n)slm(aω)n with f
(0)slm = l(l + 1), f
(1)slm = − 2ms2
l(l + 1) , f(2)slm = hs(l + 1) − hs(l) − 1, . . . ,
(5.38)
where
hs(l) = (l2 − s2)[l2 − 1
4 (α+ β)2] [l2 − 1
4 (α− β)2]2l3
(l2 − 1
4) , (5.39)
with (α+β) = 2 max|m|, |s| and (α−β) = 2ms/max|m|, |s|. This first-order expansion in aω suffices
to perform the numerical integration of radial equation (5.35) with great accuracy (see Section 5.6 below).
Introducing the tortoise coordinate (4.10) and the new radial function
Ys(r) = (r2 + a2)∆ s2Rs, (5.40)
the radial equation (5.35) becomes
d2Ys
dr2∗
−[G2
s + dGs
dr∗+ 2is(r −M)K −K2 − ∆(4irωs− λ)
(r2 + a2)2
]Ys = 0, (5.41)
where λs ≡ λ|s| − s(s+ 1) and
Gs(r) = s(r −M)r2 + a2 + r∆
(r2 + a2)2 . (5.42)
5.3 Electric and magnetic fields in the ZAMO frame
To define physically motivated BCs for Teukolsky radial equation (5.41) with s = ±1, it is useful to
derive expressions for E(α) and B(α), i.e. for the electric and the magnetic components of the electromagnetic
5If one takes s = 0 in equations (5.35) and (5.36), equations (4.7) and (4.8) are recovered, respectively. Moreover, notethat λ|0| = λ0 (Section 4.1).
6The first relation in (5.37) can be inferred from the very definition of sEml, whose terms do not depend on sign(s).
54 Electromagnetic perturbations of exotic compact objects
field perturbations, respectively. In the ZAMO frame (Section 2.4), characterized by the tetrad [87]
e(t) = 1√AΣ∆
[A∂
∂t+ 2Mar
∂
∂ϕ
]e(θ) = 1√
Σ∂
∂θ
e(r) =√
∆Σ∂
∂r
e(ϕ) =√
Σ√A sin θ
∂
∂ϕ
where A ≡ (r2 + a2)2 − a2∆ sin2 θ, E(α) and B(α) are defined by
E(α) = Fµν e(α)µe(t)
ν , B(α) = −12ϵµνλτF
λτ e(α)µe(t)
ν ,
with ϵµνλτ ≡√
−g [µνλτ ], where [µνλτ ] is the four-dimensional Kronecker delta. Explicitly [87],
E(r) =[ia sin θ√
2Aρ
(ϕ2 − 1
2∆ρ2ϕ0
)+ c.c.
]+ 2r
2 + a2√A
Re(ϕ1),
E(θ) =[r2 + a2
ρ∗Σ
√∆2A
(12ϕ0 − ϕ2
∆ρ2
)+ c.c
]− 2a sin θ
√∆A
Im(ϕ1),
E(ϕ) = −i ρ√
∆2
(ϕ0
2 + ϕ2
∆ρ2
)+ c.c.,
B(r) =[a sin θ√
2Aρ
(ϕ2 − 1
2∆ρ2ϕ0
)+ c.c.
]+ 2r
2 + a2√A
Im(ϕ1),
B(θ) = −
[ir2 + a2
ρ∗Σ
√∆2A
(12ϕ0 − ϕ2
∆ρ2
)+ c.c
]+ 2a sin θ
√∆A
Re(ϕ1),
B(ϕ) = −ρ√
∆2
(ϕ0
2 + ϕ2
∆ρ2
)+ c.c..
5.4 Perfectly-reflecting boundary conditions
EWs are totally reflected when their Poynting vector is perpendicular to the surface of a perfect con-
ductor at the point of incidence, where the tangential component of the electric field and the perpendicular
component of the magnetic field vanish. Thus, one requires that
E(θ) = 0, E(ϕ) = 0, B(r) = 0, (5.43)
at r = r0. The foregoing BCs can be expressed in terms of the field quantities ϕi, i = 1, . . . 3,
a√
∆ sin θ B(r) − (r2 + a2) E(θ) = 0 → Re(ρΦ0) = Re(ρ∗Φ∗2)/∆
a√
∆ sin θ B(r) + (r2 + a2) E(θ) = 0 → Im(ρΦ0) = Im(ρ∗Φ∗2)/∆
E(ϕ) = 0 → Im(ϕ1) = 0
(5.44)
where, as before, Φ0 = ϕ0 and Φ2 = 2ρ−2ϕ2. The first two conditions in (5.44) can be reduced to the form
|Φ0|2 = |Φ2|2
∆2 at r = r0. (5.45)
5.5 Detweiler transformation 55
Given the field decompositions (5.27) and the normalization conditions (5.32), this can be further simplified
to
∆2|R+1|2 = |R−1|2 or ∆R+1 = eiΥR−1 at r = r0, (5.46)
where Υ ∈ [0, 2π). Making use of the radial equation (5.35) and the Teukolsky-Starobinski identities
(5.34), one can express BC (5.46) in terms of R+1 and its derivative or R−1 and its derivative only, D†0∆R+1 = Λ∆R+1
D0R−1 = ΛR−1, Λ =
λ|s| + 2iωr − eiΥB
2iK , at r = r0. (5.47)
Υ = 0 refers to axial modes, whereas Υ = π refers to polar modes [63].
5.5 Detweiler transformation
Solutions to (5.35) have the asymptotic behavior [24]
Ys(r) ∼
r−seiωr∗
r+se−iωr∗or Rs(r) ∼
r−2s−1eiωr∗
r−1e−iωr∗as r → +∞ (5.48)
Ys(r) ∼
∆+s/2e+iϖr∗
∆−s/2e−iϖr∗or Rs(r) ∼
eiϖr∗
∆−se−iϖr∗as r → r0, (5.49)
where ϖ ≡ ω −mΩH . The radial function Y±1 for the electromagnetic (s = ±1) and for the gravitational
(s = ±2) cases do not exhibit the same sort of asymptotic behavior at r∗ → ±∞ as the radial function
for the scalar (s = 0) case, thus turning impossible to apply the approach sketched in Chapter 3 for the
computation of QNMs in a straightforward way. A possible solution to overcome the unsuitable form
of the aforementioned radial functions is to transform the radial equation (5.35) in such a way that its
solutions behaves asymptotically as the radial function Y0 for the scalar case. The formalism to do so was
developed by Detweiler for electromagnetic perturbations [73] and by Chandrasekhar and Detweiler for
gravitational perturbations [74] and introduces the new radial function
Xs = ∆ s2 (r2 + a2) 1
2
[α(r)Rs + β(r)∆s+1 dRs
dr
], (5.50)
which satisfies the second-order linear ODE
d2Xs
dr2∗
− V Ds Xs = 0, V D
s (r, ω) = U∆(r2 + a2)2 +G2 + dG
dr∗, (5.51)
where
U(r) = V Ts + β−1∆−s
[2dα
dr + ∆s+1 dβdr
]. (5.52)
56 Electromagnetic perturbations of exotic compact objects
The functions α(r) and β(r) are chosen so that the potential V Ds is purely real. This thesis will
only focus on the electromagnetic (s = ±1) case. Given that ∆R+1 and R−1 satisfy complex-conjugate
equations, it suffices to consider the case when s = −1 only. The radial function X−1 is defined by (5.50)
with
α(r) = α∆ + 1√2B[Re(α∆) + 1]
, β(r) = β∆√2B[Re(α∆) + 1]
, (5.53)
where
α(r) = 2K2 − ∆(2iωr + λ−1)B∆2 , β(r) = 2iK
B∆ . (5.54)
The explicit form of V D−1 can be found in [88] (see Appendix A therein)7. The effective potential (ω2 +V D
−1)
has the asymptotic behavior (3.4) with µ = 0 and ϖ = ω −mΩH . Thus, the radial function X−1 can be
written as a linear combination of the standard ‘in’ (3.5) and ‘up’ (3.6) modes, hereafter called X+−1 and
X−−1, respectively.
In addition to the radial equation (5.35), the BC (5.47) must also be written in terms of the new radial
function X−1. Such transformation can be performed using a near-horizon expansion for X−−1, since the
proper distance between the reflecting surface and the would-be event horizon is small (δ/M ≪ 1)8. This
means that the BC (5.47) is to be expressed in terms of the asymptotic coefficient A−. For that purpose,
a near-horizon expansion for the ‘up’ mode of the radial function R−1 is needed. This is given by [9,89]
R−−1 ∼ A−∆e−iϖr∗ + B−e
+iϖr∗ , as r → r0, (5.55)
where A− = A(0)− + A(1)
− η + . . . and B− = B(0)− + B(1)
− η + . . ., with η ≡ r − r+.
One should use expansions for A−∆ and B− with the same truncation order in η. Using the
approximation ∆ = (r+ − r−)η, which is valid near the would-be event horizon, and keeping only terms
up to the first order in η, the asymptotic coefficients read
A− = A(0)− , B− = B(0)
− + B(1)− η. (5.56)
Following [89], A(0)− and B(0)
− are related to the asymptotic coefficients of X−−1 as r → r0 by
A(0)− = −ϖ
B
[2(r2
+ + a2)] 1
2 ,B(0)
−
A(0)−
= − iB
4K+KA−, (5.57)
where K+ ≡ K(r+) and K ≡ iK+ + (r+ − r−)/2. Plugging the asymptotic expansion (5.55) into the
radial equation (5.35) and using the relations (5.57), one obtains
B(1)− =
[ima
M(r+ − r−) + 2ωr+ − iλs
4Mϖr+
]B(0)
− . (5.58)
7When finding numerical solutions to the ODE (5.51) using Mathematica the explicit form of V D−1 should be used. An
implicit definition, in terms of U , α and β, will not work properly.8Recall that the ‘up’ mode X−
−1 is actually a QNM (Chapter 3).
5.6 Quasinormal modes 57
Inserting the asymptotic expansion (5.55) now into (5.47) and with the above relations at hand, one finds
the BC to be imposed on the radial function X−1 at r = r0,
e+iϖr∗0 ±A−e
−iϖr∗0 = 0, (5.59)
where r∗0 ≡ r∗(r0) < 0 and the plus (minus) sign corresponds to axial (polar) modes. Remarkably, the
non-trivial BC (5.47) for the radial function R−1 was reduced to DBCs and NBCs on the radial function
X−1. The transformation (5.50) allows one to make use of the general formalism introduced in Chapter 3.
Furthermore, from a numerical point of view, Detweiler’s transformation turns easier the integration of
the radial equation (5.35) for electromagnetic (s = ±1) perturbations, thanks to the simplicity of the form
of the BCs (5.59).
5.6 Quasinormal modes
The electromagnetic QNM spectrum of Kerr-like ECOs and superspinars was obtained integrating the
radial equation (5.51) by means of the direct-integration shooting method described in Chapter 3. The
numerical integration was performed using the first two terms of the power series expansion (5.38) for
the separation constant ±1Em
land the integration parameters N = 10 and r∞/M = 400 (Section 3.3.1).
It was checked that the electromagnetic QNM frequencies of Schwarzschild and Kerr BHs are recovered
when setting R = 0. All physical quantities are normalized to the mass parameter M . The guess value to
the QNM frequency was chosen according to the numerical results presented in Section 4.2.
The results for the electromagnetic QNMs are qualitatively similar to those presented in Section 4.2,
thus showing that the most relevant phenomenological features of Kerr-like ECOs and superspinars are
already present in the scalar case. Therefore, for the sake of comparison, this section closely follows the
structure of Section 4.2.
5.6.1 Schwarzschild-like exotic compact objects
The real and imaginary parts of the fundamental (n = 0) electromagnetic QNM frequencies of
Schwarzschild-like ECOs as a function of the distance δ/M are depicted in Figure 5.1, for |l| = 1, 2 and
both DBCs (R = −1) and NBCs (R = 1). When a = 0, the potential V D−1 in (5.51) only depends on l
and, as a result, there is azimuthal degeneracy.
As one would expect, the spectra in Figure 5.1 are very similar to those presented in Figure 4.1 for
the scalar case: ωR (ωI) is positive (negative) and a monotonically increasing (decreasing) function of
δ/M . Since ωI < 0, Schwarzschild-like ECOs are stable against electromagnetic perturbations. Recall
that, in the absence of rotation (a = 0), the spacetime does not feature an ergoregion and so there are no
negative-energy physical states which can trigger an ergoregion instability.
Once again, the imaginary part of the frequency appears to vanish in the limit δ/M → 0. Following
the same argument evoked in Section 4.2, namely the form of the frequencies in the small-rotation limit
(4.14), the foregoing behavior suggests that the electromagnetic QNMs may turn unstable as soon as
58 Electromagnetic perturbations of exotic compact objects
rotation is turned on. Similar to the scalar case, the ergoregion instability is present in highly-spinning,
perfectly-reflecting Kerr-like ECOs and can be quenched or shut down by some absorption at the reflecting
surface, as it will be shown in the next section.
10-7 10-6 10-5 10-4 10-3
0.05
0.10
0.15
0.20
0.25
0.30
δ/M
MωR
Dirichlet l=1
Dirichlet l=2
Neumann l=1
Neumann l=2
10-7 10-6 10-5 10-4 10-3
10-10
10-8
10-6
10-4
10-2
δ/M-MωI
Figure 5.1 Real (left) and imaginary (right) parts of the fundamental (n = 0) |l| = 1, 2 electromagneticQNM frequencies of a Schwarzschild-type ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡rH + δ, 0 < δ ≪ M , where rH is the would-be event horizon of the corresponding Schwarzschild BH, as afunction of δ/M , for both DBCs and NBCs.
5.6.2 Kerr-like exotic compact objects
Total reflection (|R|2 = 1)
Figure 5.2 displays the fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of perfectly-
reflecting (|R|2 = 1) Kerr-like ECOs with different characteristic parameters a, δ, for both DBCs
(R = −1) and NBCs (R = 1). Like in Figure 4.2, the left (right) arms of the interpolating functions in
the bottom panels of Figure 5.2 refer to negative (positive) frequencies. Recall that the shaded regions
in Figure 5.2 represent the experimental band for the rotation parameter of remnants of the compact-
binary mergers detected so far by GW detectors [1,43–46]. For the range of δ/M under investigation,
only perfectly reflecting Kerr-like ECOs with characteristic distances approximately larger than about
10−4 (10−3) and satisfying DBCs (NBCs) at r = r0 can be stable, hence a plausible candidate for the
end-product of a compact-binary merger.
The similitude between scalar and electromagnetic QNMs is evident. The conclusions outlined in
Section 4.2 for the scalar case also hold for the electromagnetic case. In fact,
• ωR and ωI are negative (positive) in the slow-rotation (fast-rotation) regime;
• ωR and ωI always increase with increasing rotation parameter a in the range [0, 0.9[M and both
appear to have the same sign regardless the value of a.
It then follows that both ωR and ωI changes sign from negative to positive at some critical value of the
rotation parameter, ac (say), and, thus, the QNMs turn from stable to unstable at a = ac. In other words,
perfectly-reflecting Kerr-like ECOs admit zero-frequency electromagnetic QNMs.
5.6 Quasinormal modes 59
0.0 0.2 0.4 0.6 0.8 1.0
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
a/M
MωR
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.0 0.2 0.4 0.6 0.8 1.0
-0.1
0.0
0.1
0.2
0.3
0.4
a/M
MωR
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.0 0.2 0.4 0.6 0.8 1.0
10-10
10-8
10-6
10-4
a/M
M|ωI|
0.0 0.2 0.4 0.6 0.8 1.0
10-11
10-9
10-7
10-5
a/M
M|ωI|
Figure 5.2 Real (top) and imaginary (bottom) part of the fundamental (n = 0) l = m = 1 electromagneticQNM frequencies of a Kerr-like ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ,0 < δ ≪ M , where rH is the would-be event horizon of the corresponding Kerr BH, as a function ofa/M , for both DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions of theimaginary part refer to negative (positive) frequencies. The shaded regions represent the experimentalband for the rotation parameter of the end product of compact-binary coalescences, based on the currentlyavailable data from GW detections [1,43–46].
In comparing the spectra in Figure 4.2 and Figure 5.2, the critical values ac for both scalar and
electromagnetic QNMs appear to be the same. Surprisingly, it can be shown that these zero-frequency
modes are linked via Teukolsky’s radial equation (5.35) in the case when ω = 0 [9]. In the static limit, the
BC (5.47) reduces to
dR−1
dr + i
[am
∆ − (1 − eiΥ) l(l + 1)2am
]R−1 = 0 at r = r0. (5.60)
Recall that Υ = 0 refers to axial modes, whereas Υ = π refers to polar modes. Setting ω to 0 in the radial
equation (5.35) for s = 0 and s = −1, one can find several relations between the radial functions R0 and
R−1 and their first derivatives. One such relation is
R0 = −i am
l(l + 1)
[dR−1
dr + iam
∆ R−1
]. (5.61)
60 Electromagnetic perturbations of exotic compact objects
The expression enclosed by square brackets in (5.61) has the exact same form of the BC (5.47) for axial
modes. BC (5.60) with Υ = 0 holds if R0(r0) = 0, i.e. the axial modes of perfectly-reflecting Kerr-like
ECOs are generated by a scalar radial function satisfying a DBC.
Taking the derivative of (5.61), multiplying it by ∆ and using the radial equation (5.35) for s = −1 to
get rid off the second derivative of R−1, one gets
∆dR0
dr = a2m2
l(l + 1)
[dR−1
dr + i
(am
∆ − l(l + 1)am
)R−1
]. (5.62)
The expression enclosed by square brackets in (5.61) has the exact same form of BC (5.47) for polar
modes. BC (5.60) with Υ = π holds if R′0(r0) = 0, where the prime denotes differentiation with respect to
r. Although the QNM frequencies presented in Section 4.2 for NBCs were computed using the modified
radial function Y0, defined in (5.40), it turns out that the conditions R′0(r0) = 0 and Y ′
0(r0) = 0 yield
similar results as long as δ ≪ 1. Therefore, the polar modes of perfectly-reflecting Kerr-like ECOs are
generated by a scalar radial function satisfying a NBC.
The domain of the ergoregion instability for the electromagnetic case, thus, coincides with that for the
scalar case (Figure 4.3).
Similar to the scalar case, ωI displays a maximum value in the large-spin regime, as better illustrated
in Figure 5.3, a detailed view of the spectra for a/M ∈ [0.80, 1.00[. In general, for the values of δ/M
under investigation, the maximum value of ωI occurs for a/M ∈ [0.95, 1.00[ and is greater for less compact
objects, i.e. as δ increases. Also, the QNMs appear to be unstable in the extreme case.
0.80 0.85 0.90 0.95 1.00
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
a/M
104M
ωI
δ/M=10-7
δ/M=10-6
δ/M=10-5
δ/M=10-4
δ/M=10-3
0.80 0.85 0.90 0.95 1.00
0.5
1.0
1.5
2.0
2.5
a/M
104M
ωI
Figure 5.3 Detailed view of the imaginary part of the fundamental (n = 0) l = m = 1 electromagneticQNM frequencies of a Kerr-like ECO with a perfectly-reflecting (|R|2 = 1) surface at r = r0 ≡ rH + δ,0 < δ ≪ M , where rH is the would-be event horizon of the corresponding Kerr BH, as a function of a/Min the range [0.8, 1[, for both DBCs (left) and NBCs (right).
The instability timescale as defined in Section 4.2 is plotted in Figure 5.4 for the explored range of
δ/M , together with the fit to the general second-order polynomial (4.21) whose coefficients τ (m)ins are listed
in Table 5.1 for both BCs. For fixed δ/M , it is greater for objects with δ/M ≳ 10−6 (≲ 10−6) which
satisfy DBCs (NBCs) rather than NBCs (DBCs) at r = r0. Typical values are of the order of 0.1−1.0 s
for compact objects with mass M = 10−100M⊙ and both BCs.
5.6 Quasinormal modes 61
-7 -6 -5 -4 -30.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
log10(δ/M)
τins(10M
⊙/M
)[s]
Dirichlet l=1
Neumann l=1
Figure 5.4 Timescale of the electromagnetic ergore-gion instability of rapidly-rotating Kerr-like ECOswith a perfectly-reflecting (|R|2 = 1) surface atr = r0 ≡ rH + δ, δ ≪ M , where rH is the would-beevent horizon of the corresponding Kerr BH, as afunction of δ/M , for l = m = 1.
τ(m)ins DBC NBC
τ(0)ins 0.21781 0.10261
τ(1)ins −0.02102 −0.02044
τ(2)ins 0.00421 0.00690
Table 5.1 Numerical value of the coefficientsof the second-order polynomial (4.21) which fitsthe timescale of the electromagnetic ergoregioninstability of a perfectly-reflecting (|R|2 = 1)Kerr-like ECO, for l = m = 1 and both DBCsand NBCs.
0.80 0.85 0.90 0.95 1.00
-2
-1
0
1
a/M
104M
ωI
ℛ=-1.000
ℛ=-0.995
ℛ=-0.990
ℛ=-0.985
0.80 0.85 0.90 0.95 1.00
-2
-1
0
1
a/M
104M
ωI
ℛ=1.000
ℛ=0.995
ℛ=0.990
ℛ=0.985
Figure 5.5 Imaginary part of the fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of aKerr-like ECO with a partially-reflecting (|R|2 < 1) surface at r = r0 ≡ rH + δ, where rH is the would-beevent horizon of the corresponding Kerr BH and δ/M = 10−5, as a function of a/M , for quasi-DBCs (left)and quasi-NBCs (right). Absorption at the surface quenches or even shuts down the ergoregion instability.
Partial reflection (|R|2 < 1)
Up to now, no significant differences between scalar and electromagnetic QNMs of ECOs were found.
Therefore, it seems highly likely that both respond similarly to the introduction of absorbing BCs at
r = r0. Figure 5.5 shows the effect of small absorption rates on the stability of Kerr-like ECOS, for
δ/M = 10−5 and both quasi-DBCs (R ≳ −1) and quasi-NBCs (R ≲ 1) BCs. Note that the greater the
absorption rate (1 − |R|2), the lower the maximum value of the imaginary part of the frequency. In
fact, the introduction of an absorption coefficient of approximately 3% (or, equivalently, a reflectivity of
|R|2 = 0.97) completely quenches the instability for any spin value. When |R|2 < 0.97, the setup is stable
(ωI > 0) whatever the value of a.
62 Electromagnetic perturbations of exotic compact objects
The maximum value of the imaginary part of the frequency is well fitted by the general second-order
polynomial (4.22). The fit is plotted in Figure 5.6 and the corresponding coefficients are listed in Table 5.2
for values of R in the range [−0.985,−1] (quasi-DBCs).
0.985 0.990 0.995 1.000
0.0
0.5
1.0
1.5
|ℛ|
104Max(M
ωI)
δ/M=10-3
δ/M=10-4
δ/M=10-5
δ/M=10-6
δ/M=10-7
Figure 5.6 Fit of the maximum value of the imag-inary part of the fundamental (n = 0) l = m = 1electromagnetic QNM frequency of a Kerr-like ECOwith reflectivity R in the range [−0.985,−1] (quasi-DBCs) to the polynomial (4.22), for different valuesof δ/M .
δ/M a(0) a(1) a(2)
10−7 0.08824 0.18353 0.09537
10−6 0.10791 0.20759 0.10791
10−5 0.11586 0.24098 0.12524
10−4 0.14463 0.30017 0.15567
10−3 0.12954 0.27145 0.14207
Table 5.2 Numerical value of the coefficientsof the second-order polynomial (4.22) which fitsthe maximum value of the imaginary part ofthe fundamental (n = 0) electromagnetic l =m = 1 QNM frequency of a Kerr-like ECO withreflectivity R in the range [−0.985,−1] (quasi-DBCs), for different values of δ/M .
5.6.3 Superspinars
Total reflection (|R|2 = 1)
The fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of perfectly-reflecting (|R|2 = 1)
superspinars with different characteristic parameters a, r0 are plotted in Figure 5.7. The left (right) arms
of the interpolating functions in the bottom panels refer to positive (negative) frequencies. The spectra
display the exact same qualitative features of those in Figure 4.8. The ergoregion instability domain for
both scalar and electromagnetic QNMs coincide and is depicted in Figure 4.3. This follows from the fact
that relations (5.61) and (5.62) are valid for any value of a. Thus, the axial modes of perfectly-reflecting
superspinars are generated by scalar QNMs satisfying DBCs, whereas their polar modes result from scalar
QNMs satisfying NBCs.
The instability time scale as defined in Section 4.2 is plotted in Figure 5.8 for the explored range of r0.
Typical values are of the same order of magnitude as those referring to Kerr-like ECOs, but about ten
times lower than the corresponding values for scalar QNMs.
Partial and over-reflection (|R|2 ≶ 1)
Unsurprisingly, Figure 5.9 shows that imposing partially- and over-reflecting BCs on electromagnetic
field perturbations at the surface r = r0 does yield QNM spectra similiar to those reported in Figure 4.8:
when |R|2 ≲ 1 (|R|2 ≳ 1), the instability is enhanced (quenched). An absorption coefficient smaller than
−3.0% (or, equivalently, |R| ≳ 1.015) appears to guarantee the stability of superspinars for any spin value.
5.6 Quasinormal modes 63
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
a/M
MωR
r0/M=0.1
r0/M=0.3
r0/M=0.5
r0/M=0.7
r0/M=0.9
1.00 1.05 1.10 1.15 1.20 1.25 1.30-0.1
0.0
0.1
0.2
0.3
0.4
0.5
a/M
MωR
r0/M=0.1
r0/M=0.3
r0/M=0.5
r0/M=0.7
r0/M=0.9
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.1410-11
10-9
10-7
10-5
10-3
a/M
M|ωI|
1.00 1.05 1.10 1.15 1.20 1.25 1.3010-11
10-9
10-7
10-5
10-3
a/M
M|ωI|
Figure 5.7 Real (top) and imaginary (bottom) parts of the fundamental (n = 0) l = m = 1 electromagneticQNM frequencies of a superspinar with a perfectly-reflecting (|R|2 = 1) surface at r = r0 > 0, as a functionof a/M , for both DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions ofthe imaginary part refer to positive (negative) frequencies.
0.0 0.2 0.4 0.6 0.8 1.0
0.10
0.15
0.20
0.25
r0/M
τins(10M
⊙/M
)[s]
Dirichlet l=1
Neumann l=1
Figure 5.8 Timescale of the electromagnetic ergoregion instability of superspinars with a perfectly-reflecting (|R|2 = 1) surface at r = r0 > 0, as a function of r0, for l = m = 1.
64 Electromagnetic perturbations of exotic compact objects
1.00 1.01 1.02 1.03 1.04 1.05
-2
-1
0
1
2
3
4
a/M
104M
ωI
ℛ=-0.995
ℛ=-1.000
ℛ=-1.005
ℛ=-1.010
ℛ=-1.015
1.00 1.02 1.04 1.06 1.08 1.10
-2
0
2
4
6
8
a/M
104M
ωI
ℛ=0.995
ℛ=1.000
ℛ=1.005
ℛ=1.010
ℛ=1.015
Figure 5.9 Imaginary part of the fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of asuperspinar featuring a surface with reflectivity R at r = r0 > 0, as a function of a/M . The introductionof absorption (|R|2 < 1) enhances the ergoregion instability, whereas over-reflecting (|R|2 > 1) BCsmitigate it.
5.7 Summary
This chapter addressed electromagnetic field perturbations of Kerr-like ECOs and superspinars with
different degrees of compactness M/r0 and reflectivities R. The results presented herein confirm the
predictions put forth in [13] on the stability of perfectly-reflecting (|R|2 = 1) Kerr-like ECOs and on how
to quench or shut down the ergoregion instability developed by fast-spinning Kerr-like ECOs. Absorbing
BCs (|R|2 < 1) at the object’s surface has an effect similar to the one presented in Chapter 4 for scalar field
perturbations: a small absorption coefficient of about 3.0% completely wrecks unstable QNMs which are
present when |R|2 = 1, which is in agreement with heuristic arguments drawn in [9]. As for superspinars,
the instability is quenched when over-reflecting BCs are considered: an absorption coefficient smaller than
about −3.0% is sufficient to destroy the instability.
Chapter 6
Conclusion and Future Work
The recent GW detections from compact binary coalescences heralded the dawn of a brand new field in
astronomy and astrophysics. The newborn era of precision GW physics is expected to probe strong-field
gravity spacetime regions in the vicinity of compact objects and, most importantly, to provide strongest
evidence of event horizons. While current EW and GW observations do support the existence of black
holes, some other exotic alternatives are not excluded yet − not even those which do not feature an event
horizon. Thus, there has been a remarkable theoretical effort to propose new models of ECOs, most of
them quantum-inspired, and evaluate their phenomenology.
Following this trend, the present thesis aimed to explore classical phenomenological aspects of scalar
and electromagnetic field perturbations of two simple models of ECOs built from Kerr solution, named
herein as Kerr-like ECOs and superspinars. Both objects feature an ergoregion and are endowed with
a surface with reflective properties rather than an event or stringy horizon, i.e. a perfectly-absorbing
surface. As first shown by Friedman [10], asymptotically-flat stationary solutions to EFE possessing
an ergoregion but not an event horizon may develop instabilities when linearly interacting with scalar
and electromagnetic field perturbations, especially when rapidly rotating. Although Kerr-like ECOs and
superspinars have the key ingredients to trigger ergoregion instabilities, it turns out that only those whose
surface is perfectly- or quasi-perfectly-reflecting admit unstable scalar and electromagnetic QNMs. If
the absorption coefficient of Kerr-like ECOs is greater than about 3.0%, such configurations become
stable, which hints at the possibility of Kerr-like ECOs being plausible astrophysical objects, at least from
the point of view of the superradiant instability being mitigated. To the author’s best knowledge, the
numerical results presented herein regarding the effect of reflective properties on the interaction between
both Kerr-like ECOs and superspinars and electromagnetic field perturbations have never been discussed
in the literature.
Extensions of the work presented herein are manifold. A first step in broadening the work scope
could be to study Kerr-like ECOs and superspinars satisfying RBCs at the reflective surface and also to
apply the framework to Reissner-Nordstrom- and Kerr-Newman-like ECOs. However, perhaps the most
significant line of research concerns gravitational perturbations of perfectly-, partially- and over-reflecting
Kerr-like ECOs and superspinars. Given that the phenomenology of scalar perturbations bear close
66 Conclusion and Future Work
resemblance to that of electromagnetic perturbations, it is worth asking whether such similarity also
extends to gravitational perturbations or not. It would be interesting to assess the minimum value of the
absorption coefficient needed to shut down ergoregion instabilities triggered by gravitational perturbations
and compare it to the heuristic prediction drawn in [9]. However, since the canonical energy-momentum
tensor for the gravitational field vanishes identically, one cannot derive the form of perfectly-reflecting BCs
near the would-be event horizon. Despite this subtlety, it is tempting to argue that perfectly-reflecting
BCs, whatever form they take for gravitational perturbations, reduce to DBCs and NBCs on Detweiler’s
radial function, similarly to the scalar and electromagnetic cases. Tracing back DBCs and NBCs on
Detweiler’s radial function to their form in terms of Teukolsky’s radial function and its first derivative could
in particular provide some physical insight into the nature of the gravitational field’s energy-momentum
tensor.
A second line of research is related to the reflectivity coefficient R introduced in Chapter 3. For
the sake of simplicity, this parameter was assumed to be not only frequency-independent, but also
model-independent. A possible future extension may lift the first assumption and consider Kerr-like
ECOs and superspinars with frequency-dependent reflectivities. As for the second, while, from a purely
theoretical perspective, a suitable choice of the object’s reflectivity appears to fully neutralize any
ergoregion instabilities that would develop in the case of total reflection, whether the values of reflectivity
needed for stability can be achieved in naturally-occurring ECOs or not surely depends on the object’s
interior structure, whose intrinsic features and interactions define in turn the reflective properties of its
surface. To the author’s best knowledge, satisfying models for the interior structure of Kerr-like ECOs
and superspinars have never been reported and, therefore, the nature of the mechanisms giving rise to
their reflective properties remains an open question.
At last, the method of matched asymptotic expansions used to derive an approximate analytical
expression for the amplification factors of the superradiant scattering of low-frequency scalar perturbations
off slowly-rotating Kerr-like ECOs could also be applied to the electromagnetic case. Furthermore,
following [71], it would be interesting to compute the emission cross section and corresponding spectrum
for Kerr-like ECOs and superspinars.
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Appendix A
Teukolsky-Starobinsky identities
The Teukolsky-Starobinsky identities were first derived by Teukolsky [69] and Starobinsky [90] in the
context of field perturbations of rotating compact objects. In this appendix, a partial derivation of these
identities is provided. A more detailed derivation can be found in [64].
A.1 Definitions and operator identities
In the Newman-Penrose formalism, when the field quantities have harmonic dependence on t and ϕ,
i.e. e−i(ωt−mϕ), the directional derivatives (5.5) can be written in terms of the differential operators
Dn = ∂r − iK
∆ + 2nr −M
∆ ,
D†n = ∂r + iK
∆ + 2nr −M
∆ ,
Ln = ∂θ −Q+ n cot θ,
L†n = ∂θ +Q+ n cot θ
(A.1)
where n ∈ Z, K = (r2 + a2)ω − am and Q = aω sin θ −m cosec θ.
The differential operators (A.1) satisfy the following readily verifiable identities:
∆(D†n − Dn) = 2iK, ∆Dn+1 = Dn∆, ∆D†
n+1 = D†n∆, (A.2)
L†n − Ln = 2Q, sin θ Ln+1 = Ln sin θ, sin θ L†
n+1 = L†n sin θ, (A.3)∫ π
0g(Lnf) sin θ dθ = −
∫ π
0f(L†
−n+1g) sin θ dθ, (A.4)
Ln+1Ln+2 . . .Ln+m(f cos θ) = cos θ Ln+1 . . .Ln+mf −m sin θ Ln+2 . . .Ln+mf, (A.5)
where f is a smooth function of θ. The foregoing relations will be used in the derivation of the Teukolsky-
Starobinsky identities.
72 Teukolsky-Starobinsky identities
A.2 Teukolsky-Starobinsky identities for spin-1 fields
The radial and the angular functions of the fields Φ0 and Φ2, as introduced in Chapter 5, satisfy the
ordinary differential equations (∆D1D†1 + 2iωr)R+1 = λR+1
(L†0L1 − 2aω cos θ)S+1 = −λS+1
(A.6)
(A.7)
(∆D†0D0 − 2iωr)R−1 = λR−1
(L0L†1 + 2aω cos θ)S−1 = −λS−1
(A.8)
(A.9)
Here, the angular functions S±1 are normalized to unity,
∫ π
0(S±1)2 sin θ dθ = 1. (A.10)
The Teukolsky-Starobinsky identities encode the relation between the radial functions R+1 and R−1 and
between the angular functions S+1 and S−1.
Acting on λR−1 with D0D0 and using equation (A.8) and the elementary identities (A.2), one gets
D0D0(∆D†0D0 − 2iωr)R−1 = (∆D1D†
1 + 2iωr)D0D0R−1. (A.11)
The operator acting on D0D0R−1 is the same as the one acting on R+1 in equation (A.6), meaning that
∆D0D0R−1 ∝ ∆R+1. (A.12)
Similarly, acting on λ∆R+1 with ∆D†0D†
0 and using the equation (A.6) and the elementary identities
(A.2), one obtains
∆D†0D†
0(∆D0D†0 + 2iωr)∆R+1 = (∆D†
0D0 − 2iωr)∆D†0D†
0∆R+1 (A.13)
The operator acting on ∆D†0D†
0∆R+1 is the same as the one acting on R−1 in equation (A.8), meaning
∆D†0D†
0∆R+1 ∝ R−1. (A.14)
Given that ∆R+1 and R−1 satisfy complex-conjugate equations, the relative normalization of the radial
functions can be chosen so that
∆D0D0R−1 = B∆R+1, ∆D†0D†
0∆R+1 = BR−1, (A.15)
where B is a complex constant to be determined. Combining the foregoing equations, one gets
|B|2 = ∆D†0D†
0∆D0D0 = ∆D0D0∆D†0D†
0 (A.16)
A.2 Teukolsky-Starobinsky identities for spin-1 fields 73
With the aid of the identities (A.2), the direct evaluation of the identity (A.16) holds
|B|2 = λ2 − 4a2ω2 + 4aωm (A.17)
Acting on −λS+1 with L0L1 and using equation (A.7) and the elementary identities (A.3), one gets
L0L1(L†0L1 − 2aω cos θ)S+1 = (L0L†
1 + 2aω cos θ)L0L1S+1 (A.18)
Note that the operator acting on L0L1S+1 is the same as the one acting on S−1 in equation (A.9), meaning
that
L0L1S+1 ∝ S−1. (A.19)
Similarly, acting on −λS−1 with L†0L†
1 and using the equation (A.9) and the elementary identities (A.3),
one obtains
L†0L†
1(L0L†1 + 2aω cos θ)S−1 = (L†
0L1 − 2aω cos θ)L†0L†
1S−1 (A.20)
Note that the operator acting on L†0L†
1S−1 is the same as the one acting on S+1 in equation (A.7), meaning
that
L†0L†
1S−1 ∝ S+1. (A.21)
Unlike the radial functions ∆R+1 and R−1, the angular functions S+1 and S−1 do not satisfy complex-
conjugate equations. As a result, the relations (A.19) and (A.21) can be expressed in the form
L0L1S+1 = C1S−1, L†0L†
1S−1 = C2S+1, (A.22)
where C1 and C2 are real constants1. Surprisingly, since the angular functions are normalized to unity, it
turns out that C1 = C2. In fact,
C21
(A.10)= C21
∫ π
0S2
−1 sin θ dθ (A.22)=∫ π
0(L0L1S+1)(L0L1S+1) sin θ dθ
(A.4)=∫ π
0(L†
0L†1L0L1S+1)S+1 sin θ dθ (A.22)= C1
∫ π
0(L†
0L†1S+1)S+1 sin θ dθ
(A.22)= C1C2
∫ π
0S2
+1 sin θ dθ (A.10)= C1C2,
and, therefore, C1 = C2 ≡ C. Combining the identities (A.22), one gets
C2 = L†0L†
1L0L1 = L0L1L†0L†
1 (A.23)
1On the assumption that the angular functions S±1 are real and given that Im(Ln) = Im(L†n) = 0, the functions
L0L1S+1 and L†0L†
1S−1 are also real. Therefore, one can take the constants of proportionality C1 and C2 as real.
74 Teukolsky-Starobinsky identities
With the aid of the identities (A.3) and (A.5), the direct evaluation of the identity (A.23) holds
C2 = λ2 − 4a2ω2 + 4aωm, (A.24)
i.e. |B|2 = C2.
Finally, the Teukolsky-Starobinsky identities for spin-1 fields are
∆D0D0R−1 = B∆R+1
L†0L†
1S−1 = BS+1,
∆D†0D†
0∆R+1 = BR−1
L0L1S+1 = BS−1
, (A.25)
where B =√λ2 − 4a2ω2 + 4amω is the Starobinsky constant2.
2The plus sign in the definition of the Starobinsky constant is not arbitrary. The careful reader should read the detailedexplanation in [64].