Exotic Compact Objects Interacting with Fundamental Fields Engineering

Exotic compact objects interacting 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. Emgeral, 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 inwhat 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 > M 2...... 13

2.2 Carter-Penrose of the maximal analytical extension of Kerr spacetime along the axis of symmetry (θ = 0) for a2 < M 2 and a2 = M 2...... 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 2 objects 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, 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

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 2 a superspinar with a perfectly-reflecting|R| ( = 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 2 (|R| = 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 2 (|R| = 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 2 (|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, 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 2 compact objects with a perfectly-reflecting (|R| = 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 electromagnetic 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 2 frequencies of a superspinar with a perfectly-reflecting|R| ( = 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- 2 reflecting (|R| = 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-reflectingR ( = −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 BH7 [ ,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 event5 [ ,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 onthe 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], 1 8πG Rµν − R gµν = Tµν , (1.1) 2 c4 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]

−1 2 2M 2 2M 2 2 2 2 2 ds = − 1 − dt + 1 − dr + r dθ + sin θ dϕ . (1.2) r r

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],

EM 1 ρ 1 ρσ T = FµρFν − gµν FρσF , (1.3) µν 4π 4 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 µ [15,24] in free space, compactly written in the form ∇ Fµν = 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 EM [25]. This electro-vacuum solution Tµν = Tµν 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]

2 2 ∆ 2 2 sin θ 2 2 2 Σ 2 2 ds = − dt − a sin θ dϕ + (r + a ) dϕ − a dt + dr + Σ dθ , (1.4) Σ Σ ∆

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]

Qr 2 Aµ = −1, 0, 0, a sin θ . (1.5) Σ

The Kerr-Newman metric (1.4) actually contains the classic set of solutions of the EFE in GR: the Schwarzschild, the Kerr, the Reissner-Nordstr¨om and the Kerr-Newman metrics [16]. While the Schwarzschild and the Reissner-Nordstr¨om 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.

JQ BH solution Event horizon Uniqueness theorem = 0 ≠ 0 = 0 ≠ 0

Schwarzschild ✓ ✓ r = 2M Israel theorem √ Kerr ✓ ✓ r = M + M 2 − a2 Carter-Robinson theorem p Reissner-Nordstrom¨ ✓ ✓ r = M + M 2 − Q2 Generalization of Israel theorem p Kerr-Newman ✓ ✓ r = M + M 2 − 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 √ 2 2 (Q = 0). It has coordinate singularities at ∆ = 0, which solves for r± ≡ M ± M − a . These roots

define the outerr ( +) 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 √ + 2 2 2 the event horizon and the ergosphere, the hypersurface rE = M + M − a cos 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 solution28 [ ], 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 ismore 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 ≳ 1.4 M⊙, 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 ≳ 3 M⊙, 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 − 70 M⊙); intermediate BHs 3 5 10 (M ∼ 10 M⊙); and supermassive BHs (M ∼ 10 − 10 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 besent 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 electromagnetic 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, thenthe 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 madeupof 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

+4.0 +0.05 GW150914 62.0−4.0 0.67−0.07 +6.1 +0.06 GW151226 20.8−1.7 0.74−0.06 +5.7 +0.09 GW170104 48.7−4.6 0.64−0.20 +3.2 +0.07 GW170814 53.2−2.5 0.70−0.05 +4.8 +0.04 GW170608 18.0−0.9 0.69−0.05

Table 1.2 Inferred mass and angular momentum of the final BH candidates for all GW observations ever reported.

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 ECOsinto 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 photosphere 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 ata −35 microscopic or Planck distance (lP = 1.616229 × 10 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)

√ 2 2 is the location of the surface and rH = M + M − a 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 scalar particles; Heisenberg uncertainty principle prevents gravitational collapse. Anisotropic 1974 [50] Relativistic sphere with locally anisotropic equations of state. star Wormhole 1988 [51] Regular geometry which in general is characterized by deviations from the Kerr metric. Gravastar 2004 [52] Gravitational Bose-Einstein condensate star; the presence of a thin layer of perfect fluid between an interior de Sitter condensate phase and a Schwarzschild exterior prevents the formation of an event horizon. Fuzzball 2005 [53] Made up of a ball of strings which ends outside the would-be event horizon; avoids the information loss paradox. Black star 2008 [54] Slowly free-falling matter whose collapse is delayed ad infinitum by vacuum polarization. Superspinar 2009 [55] BH-like object whose rotation parameter exceeds its mass (Kerr bound violation). Proca star 2016 [56] Macroscopic Bose-Einstein condensate of massive, complex vector particles; Heisenberg uncertainty principle prevents gravitational collapse. Collapsed 2017 [57] Bound state of highly excited strings. polymer 2 − 2 hole 2017 [58] Horizonless matter-sourced solution of classical quadratic gravity which closely matches the exterior Schwarzschild solution down to about 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 Kerrmetric, 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 onand 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]

2 2 ∆ 2 2 sin θ 2 2 2 Σ 2 2 ds = − dt − a sin θ dϕ + (r + a ) dϕ − a dt + dr + Σ dθ , (2.1) Σ Σ ∆ 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)

(a) with the 1-forms e , 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

 q  (0) ∆ 2 √1 2 2  e = Σ dt − a sin θdϕ  e(0) = r + a ∂t + a ∂ϕ  q  qΣ∆  (1) Σ  ∆  e = ∆ dr  e(1) = Σ ∂r √ (2.5) (2) 1  e = Σdθ  e = √ ∂θ   (2) Σ    e(3) sin√ θ r2 a2 ϕ − a t  e √ 1 a 2 θ ∂ ∂ = Σ + d d (3) = Σ sin θ sin t + ϕ

In the dual basis, the inverse line element is written as

2 2 2 2 2 ∂s = −[e(0)] + [e(1)] + [e(2)] + [e(3)] , (2.6) which yields

2 1 2 2 2 ∆ 2 1 2 1 2 2 ∂ = − r + a ∂t + a ∂ϕ + ∂ + ∂ + ∂ϕ + a sin θ ∂t (2.7) s Σ∆ Σ r Σ θ Σ sin2 θ 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],

2 2 g ≡ det(gµν ) = −Σ sin 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 cos ϕ, y = r2 + a2 sin θ sin ϕ 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]

2 2 2M −3 2 4aM sin θ −3 ds = − 1 − + O r dt − + O r dϕ dt r r 2M + 1 + + O r−2 dr2 + r2 dθ2 + sin2 θdϕ2 . (2.10) r

This result can be derived solving EFE in vacuum in the weak-field approximation60 [ ]. 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]

2 µνσλ 48M 2 2 2 2 2 2 2 RµνσλR = r − a cos θ Σ − 16r a cos θ . (2.11) Σ6 2.2 Symmetries 11

The scalar blows up on the hypersurface Σ = 0, but remains finite on ∆ = 0. Indeed,

π r = 0, θ = (2.12) 2 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

p 2 2 r± = M ± M − a , (2.13)

2 2 which are only real when a ≤ M . The hypersurfaces r = r+ and r = r− are the outer and inner event 2 2 2 2 horizons, respectively. Note that they coincide when a = M . In the case when a > M , 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  T : t → −t or, equivalently, .  Pϕ : ϕ → −ϕ  Pa : a → −a

2.2.2 Continuous symmetries

The continuous symmetries of a manifold manifest themselves in isometries. An isometry is a coordinate µ ′µ ′ transformation x → x which leaves the metric tensor gµν form-invariant, i.e. gµν (y) = gµν (y) for µ ′ all coordinates y , where gµν 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 µ satisfies the Killing’s equations ξ(µ;ν) = 0, in which case ξ is called a Killing vector field [40]. Computing Killing vector fields is often a laborious task. However, some continuous symmetries ofa manifold can be easily identified, and their corresponding Killing vectors readily guessed, just by looking at its line element.

1 PX 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 > M 2, 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)

µ ν where ηµν is the metric of flat spacetime and l is a null vector with respect to ηµν , i.e. ηµν l l = 0. To do so, the Boyer-Lindquist coordinates t and ϕ are replaced by the new variables [7]

r2 + a2 a dt˜= dt + 1 − , dϕ˜ = dϕ − dr, (2.18) ∆ ∆ which bring the Kerr line element (2.1) to the form [7,28,60]

ds2 = −dt˜2 + dx2 + dy2 + dz2 2 2Mr3 r a z + dt + (x dx + y dy) + (y dx − x dy) + dz , (2.19) r4 + a2z2 r2 + a2 r2 + a2 r 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

r 2Mr3 rx + ay ry − ax z lµ = 1, , , . (2.21) r4 + a2z2 r2 + a2 r2 + a2 r 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 inthe (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 > M 2. Source: [7].

2 2 The case when a < M 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

r2 + a2 a du+ = dt + dr, dϕ+ = dϕ + dr, (2.25) ∆ ∆ yielding [28,27]

2 2 2 2 2 2 2Mr 2 2 (r + a ) − ∆a sin θ 2 2 ds = − 1 − du + 2du+dr + Σdθ + sin θdϕ Σ + Σ + 2 2 4aMr sin θ − 2a sin θdrdϕ+ − du+dϕ+, (2.26) Σ 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

r2 + a2 a du− = dt − dr, dϕ− = dϕ − 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 < M 2. The Carter-Penrose diagram of Kerr spacetime along the axis of symmetry (θ = 0) for a2 < M 2 is depicted in the left panel of Figure 2.2, where three different types of regions are identified: regionsr I( + < 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 2 2 (−∞ < r < r−) feature the ring singularity (2.12). In the extreme case (a = M ), 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 of symmetry (θ = 0) for a2 < M 2 (left) and a2 = M 2 (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 = x˙ µ, 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]

ϕ˙ gϕt 2Jr ≡ − ΩZAMO = = 2 2 (2.28) t˙ gϕϕ (r2 + a2) − a2∆ sin θ

As one would expected, the angular velocity vanishes at spatial infinityr ( → +∞). 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

µ µ A timelike curve C• with parametric equation x = x (t), where t is the Boyer-Lindquist time µ coordinate, defines a static observer if its tangent vector T• is proportional to ξt, i.e.

dxµ T µ = = (1, 0, 0, 0) (say). (2.30) • dt

µ Since the curve C• is timelike, the norm of T• satisfies the condition

µ ν 2Mr gµν T T = gtt = − 1 − < 0, (2.31) • • Σ

− + which solves for r < rE (θ) ∨ r > rE (θ), where

± p 2 2 2 rE (θ) ≡ M ± M − a cos θ. (2.32) 16 The Kerr metric

± µ The hypersurfaces r = rE are known as infinite redshift surfaces. Therefore, the tangent vector T• is null ± on the hypersurfaces r = rE and becomes spacelike in the spacetime region

− + rE (θ) < r < rE (θ), (2.33)

− + 2 2 2 2 where observers are never static. Note that rE ≤ r− < r+ ≤ rE . When a < M (a > M ), the + − + + spacetime region r+ < r < rE (rE < r < rE ) is known as ergoregion and its outer boundary, r = rE , 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]

2π π r+ Z Z Z 0 √ VE = dϕ dθ dr grrgθθgϕϕ, (2.34) − − 0 θ0 r0

− − + − − where the limits of integration θ0 , r0 and r0 are defined according to the value of a. θ0 = 0, r0 = r+ + + 2 2 − − − + + 2 2 and r0 = rE for a < M , whereas θ0 = arcos(a/M), r0 = rE and r0 = rE for a > M . 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±.

200

150 3

/M 100 E V

0 0 1 2 3 4 5 |a/M|

Figure 2.3 Proper volume of the ergoregion of Kerr spacetime as a function of |a/M|.

µ µ On the other hand, a timelike curve C◦ with the parametric equation x = x (t), where t is the Boyer- µ Lindquist time coordinate, defines a stationary observer if its tangent vector T◦ is a linear combination of

ξt and ξϕ, i.e.

µ µ dx T = = ξt + Ωξϕ = (1, 0, 0, Ω) (say). (2.35) ◦ dt 2.6 Penrose process 17

µ Since the curve C◦ is timelike, the norm of T◦ satisfies the condition

µ ν 2 gµν T◦ T◦ = gtt + 2gtϕΩ + gϕϕΩ < 0, (2.36)

µ ν whose solution depends on the value of Ω. The condition gµν T◦ T◦ = 0 solves for

q 2 −gtϕ ± gtϕ − gttgϕϕ Ω± = . (2.37) gϕϕ

2 2 Since gtϕ − gttgϕϕ = ∆ sin 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.

µ The tangent vector T◦ 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 Penrose29 [ ] 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 < rE ), the time translation Killing vector field ξt is spacelike, meaning that the particle can have negative energy.

(0) (0) Consider that a massive particle (0) with energy E = −p · 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)

(1) (1) (2) (2) with E = −p · ξt and E = −p · 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 initsmass M to first order is

κ δM = δAH + ΩH δJ, (2.43) 8π where dAH and dJ are the changes in the area of the event horizon and in the angular momentum of the BH, respectively, and

r+ − M κ = (2.44) 2Mr+ 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

−i(ωt−mϕ) ψ• ∝ e , (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 m = . (2.47) δM ω

It follows immediately from (2.43) and (2.45) that

δM − ΩH δJ ≥ 0. (2.48)

Plugging (2.47) into (2.48), one obtais

m 1 − ΩH δM ≥ 0, (2.49) ω 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 thanits 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 ∂2 − 2 + 2 − V (r) Ψ = 0, (3.1) ∂t ∂r∗ 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, Z +∞ 1 −iωt Ψ(r∗, t) = dω e Ψ(˜ r∗, ω), (3.2) 2π −∞ 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,

2 d 2 ˜ 2 + ω − V (r) Ψ = 0. (3.3) dr∗

1Although not strictly necessary, this condition is common in the literature and will be adopted hereafter for the sake of simplicity. 22 Quasinormal modes

In asymptotically-flat spacetimes, the effective potential satisfies the relations

 2  µ , r∗ → +∞ V (r) ∼ , (3.4) 2 2  ω − ϖ , r∗ → −∞ 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]

 −iωr∗ +iωr∗  e + A+e , r∗ → +∞ Ψ˜ +(r) ∼ (3.5) −iϖr∗  B+e r∗ → −∞  +iωr∗  B−e r∗ → +∞ Ψ˜ −(r) ∼ (3.6) +iϖr∗ −iϖr∗  e + A−e , r∗ → −∞

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

1 Z +∞ φ(x, t) = dω C(ω) e−i(ωt−kx), (3.7) 2π −∞ 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

∂k −1 k vg = , vp = . (3.8) ∂ω ω

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˜,

C(˜ω) ˜ φ(x, t) ≃ e−i(˜ωt−kx), (3.9) 2π where k˜ ≡ k(˜ω), with group velocity

∂k −1 vg = . (3.10) ∂ω ω=˜ω

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 3 (incident) waves (Table 3.2). As expected, sign(vg) = sign(vp) at infinityr ( ∗ → +∞). On the other hand,

−iϖr∗ near the would-be event horizon (r∗ → −∞), the ingoing wave e 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 Wave ωϖ > 0 ωϖ < 0 e−iωr∗ −1 −1 e−iωr∗ Incident Incident e+iωr∗ +1 +1 e+iωr∗ Reflected Reflected e−iϖr∗ −1 −ϖ/ω e−iϖr∗ Incident Reflected e+iϖr∗ +1 +ϖ/ω e+iϖr∗ Reflected Incident

Table 3.1 Group and phase velocities of the Table 3.2 Nature of the quasi-monochromatic waves quasi-monochromatic waves (3.5) and (3.6). (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 4 Wronskians of equation (3.3), one can write the following relations between the coefficients A± and B±:

˜ ˜ ∗ ˜ ˜ ∗ 2 2 W[Ψ+, Ψ+]|r∗=+∞ = W[Ψ+, Ψ+]|r∗=−∞ → ω(1 − |A+| ) = ϖ|B+| (3.11) ˜ ˜ ∗ ˜ ˜ ∗ 2 2 W[Ψ−, Ψ−]|r∗=+∞ = W[Ψ−, Ψ−]|r∗=−∞ → ϖ(1 − |A−| ) = ω|B−| (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

2 |A±| < 1, if sign(ωϖ) = +1, (3.15)

2 |A±| > 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∗ → +∞,

+iωr∗ −iωr∗ 2 A+e , carries more energy than the ingoing (incident) wave, e , whenever |A+| > 1, i.e. for ω > 0 and ϖ < 0 or ω < 0 and ϖ > 0. Thus, the field modes satisfying the foregoing conditions are

+iϖr∗ superradiant modes. On the other hand, the outgoing wave at r∗ → −∞, e , carries more energy

3 It 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

−iϖr∗ 2 than the ingoing wave, A−e , as seen by an observer at infinity, whenever |A−| < 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, ifone requires purely outgoing waves at infinity,

+iωr∗ Ψ(˜ ω, r∗) ∼ e 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]

−iϖr∗ Ψ(˜ ω, r∗) ∼ e 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]

∗ ˜ ∗ −iϖr0 Ψ−(ω, r0) = (1 + R)A−e , (3.19)

∗ where r0 ≡ r∗(r0) < 0 and R is a complex parameter encoding the reflective features of the surface at 5 r = r0 (hereafter referred to as reflectivity) . 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]

−i2ϖr∗ RA−e 0 = 1. (3.20)

5For the sake of simplicity, the reflectivity R will be considered model-independent, i.e. the underlying physical mechanisms giving rise to the surface’s reflective properties (e.g. frictional dissipation) shall be ignored and a heuristic approach 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, r0) = 0, Ψ−(ω, r0) = 0, (3.21)

respectively, where the prime denotes differentiation with respect to the tortoise coordinate r∗. Explicitly,

+iϖr∗ −iϖr∗ e 0 ± A−e 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, r0) + sin(ξ)Ψ−(ω, r0) = 0, (3.23)

∗ ∗ ˜ ∗ −iϖr0 ˜ ′ ∗ −iϖr0 where ξ ∈ [0, π). Plugging relations Ψ−(ω, r0) = (1+R)A−e and Ψ−(ω, r0) = −iϖ(1−R)A−e into (3.23) and solving for R, one gets

cos(ξ) − iϖ sin(ξ) R = − , (3.24) cos(ξ) + iϖ sin(ξ) 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).

iΦ If the coefficient A− is represented in the form A− = |A−|e [68], where Φ = arg A−, equation (3.22) solves for6  1  2n + 1, for DBCs ϖ = − ∗ [pπ − Φ + i ln |A−|] , p = , (3.25) 2r0  2n, for NBCs 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

1 1 ωR = mΩH − ∗ [pπ − Φ] , ωI = − ∗ ln |A−|. (3.26) 2r0 2r0

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

2 Z = |A+| − 1. (3.27)

6 Recall that the principal value of the logarithm of z ∈ C is defined as ln z = ln |z| + i arg z. 26 Quasinormal modes

Starobinski˘i and Churilov showed that the amplification factors of the superradiant scattering ofa low-frequency neutral bosonic wave of spin s off a Kerr BH have the form [75]

l 2 Y 4Q 2l+1 Z = 4Qβsl 1 + [ωR(r+ − r−)] , (3.28) n2 n=1 where

2 (l − s)!(l + s)! 2Mr+ βsl = ,Q = (mΩ − ωR). (3.29) (2l)!(2l + 1)!! r+ − r−

This analytic expression, valid when MωR ≪ 1, offers an expression for |A−|. Assuming ω ∼ mΩH , then Q2 ≪ 1 and one obtains7

βsl 2Mr+ 2l+1 ωI = ∗ [ωR(r+ − r−)] (ωR − mΩ). (3.30) r0 r+ − r−

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

N c(n) +iωr∗ X ∞ ψ˜(r) = e , r∗ = r∗(r) (3.31) rn n=0

7 ln |A−| = ln(1 + Z) ≃ Z, since Z ≪ 1. 3.3 Methods for computing quasinormal modes 27

(n) where the coefficients c∞ 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 + ω2 − V (r) Ψ˜ = 0, (3.32) dr2

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 (1) (N) (0) write c∞ , . . . , c∞ in terms of c∞ . The latter is usually set to 1. The choice of N should be a trade-off between computational time and accuracy.

(n) 2. Once the coefficients c∞ 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

dΨ˜ dψ˜ Ψ(˜ r∞) = ψ˜(r∞) and = , (3.33) r r d r=r∞ d r=r∞

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 foundin 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]

Z 1 4 √ µ S[Φ] = − d x −g [(∇ Φ) (∇µΦ)] , (4.1) 2 which yields the equation of motion □Φ = 0, (4.2)

µ where □ ≡ ∇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

−iωt +imϕ Φ = e F0(r, θ)e . (4.3)

Plugging the field mode (4.3) into the equation of motion (4.2), one gets

2 tt tϕ 2 ϕϕ 1 rr 1 θθ −ω g + 2mωg − m g Σ + ∂r (Σg ∂rF0) + ∂θ Σ sin θ g ∂θF0 = 0. (4.4) F0 F0 sin θ

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 1 r2 + a2 − a2 sin2 θ ω2 + − m2 + 2 1 − amω ∆ ∆ sin2 θ ∆ 1 d dR0 1 1 d dS0 + ∆ + sin θ = 0. (4.6) R0 dr dr S0 sin θ dθ dθ

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 m side, one can equate each side of the equation to the separation constant −0E l and get two second-order linear ODEs [24,69]: the radial equation,

2 d dR0 T T K ∆ − V R0 = 0,V = λ0 − , (4.7) dr dr 0 0 ∆

2 2 m 2 2 where K ≡ ω(r + a ) − am and λ0 ≡ 0E l + a ω − 2amω, and the angular equation,

2 1 d dS0 m 2 2 2 m sin θ − − a ω cos θ − E S0 = 0. (4.8) sin θ dθ dθ sin2 θ 0 l

When |aω| ≪ 1, the separation constant admits the power series expansion [77–79]

+∞ m X (n) n (0) (1) (2) 0E l = f0lm(aω) with f0lm = l(l + 1), f0lm = 0, f0lm = h0(l + 1) − h0(l) − 1,... (4.9) n=0

2 2 2 where h0(l) = 2l(l −m )/(4l −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 of −iωt +imϕ e F0(r, θ)e . 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

2 2 dr∗ r + a = (4.10) dr ∆ together with the new radial function

2 2 Y0(r) = (r + a )R0, (4.11) the radial equation (4.7) becomes [24,69]

2Y G K2 − λ d 0 ¯ T ¯ T 2 d 0 ∆ s 2 − V0 Y0 = 0, V0 = G0 + − 2 2 2 , (4.12) dr∗ dr∗ (r + a ) where

r∆ G0(r) = . (4.13) (r2 + a2)2

2 T The effective potentialω ( + V¯0 ) 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 2 m constant 0E l 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). T Note that, when a = 0, the potential V¯0 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.

2 m The power series expansion (4.9) is a good approximation of 0E l when |aω| ≲ 1. This inequality is satisfied by the fundamental 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 great accuracy. However, it is worth pointing out that the numerical results are more sensitive to the number of terms of the expansion when a/M approaches 1±. 3n is the overtone number. 32 Scalar perturbations of exotic compact objects

0.30 10-2 Dirichletl=1 l=2 0.25 Dirichlet Neumannl=1 10-4 Neumannl=2 0.20

I -6 R 10

Mω 0.15 -Mω 10-8 0.10

-10 0.05 10

-7 -6 -5 -4 -3 10-7 10-6 10-5 10-4 10-3 10 10 10 10 10 δ/M δ/M

Figure 4.1 Real (left) and imaginary (right) parts of the fundamental (n = 0) |l| = 1, 2 scalar QNM 2 frequencies of a Schwarzschild-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 the corresponding Schwarzschild BH, as a function of 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]

(0) (1) 2 ωR,I ∼ ωR,I + mωR,I χ + O(χ ), (4.14)

(0) (0) where ωR and ωI are the real and imaginary parts of the frequency in the absence of rotation (a = 0), (1) respectively, and ωR,I are first-order corrections which depend on l and δ/M. The coefficient of the (1) dimensionless parameter χ in (4.14) is proportional to m, meaning that the first-order correction ωI (0) (1) controls the stability of the QNM as δ/M approaches 0. Since ωI → 0 in that limit (Figure 4.1), ωI 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.

4 (ωR, 10 ωI ) a/M 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- reflectingR ( = −1) surface at r = r0 = rH + δ, 0 < δ ≪ M, where rH is the would-be event horizon of the corresponding Kerr BH. The shaded rows represent the experimental band for the rotation parameter of the 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 imaginary part yields the same numerical value. 34 Scalar perturbations of exotic compact objects

0.4 δ/M=10-7 0.4 δ/M=10-7 δ/M=10-6 δ/M=10-6 0.3 -5 -5 δ/M=10 0.3 δ/M=10 δ/M=10-4 δ/M=10-4 0.2 δ/M=10-3 0.2 δ/M=10-3 R R 0.1 Mω Mω 0.1 0.0 0.0 -0.1

-0.2 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a/M a/M

-5 10-4 10 | -6 | -7 I 10 I 10 M|ω M|ω 10-8 10-9

-10 10 10-11

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a/M a/M

Figure 4.2 Real (top) and imaginary (bottom) part of the fundamental (n = 0) l = m = 1 scalar QNM 2 frequencies of a Kerr-like ECO with a perfectly-reflecting (|R| = 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 refer to negative (positive) frequencies. The shaded regions represent the experimental band for the rotation parameter of the end-product of compact-binary coalescences, based on the currently available data from GW 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 polynomialsm ( = 1)

(0) (1) 2 ωR = ωR + ωR χ + O(χ ) (4.15) (0) (1) (2) 2 3 ωI = ωI + ωI χ + ωI χ + O(χ ) (4.16)

(0) (1) (2) for values of χ over the range [0, 0.25]. The coefficients ωR,I , ωR,I and ωI 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

DBCs NBCs δ/M (0) (1) 3 (0) 3 (1) 3 (2) (0) (1) 4 (0) 4 (1) 4 (2) ωR ωR 10 ωI 10 ωI 10 ωI ωR ωR 10 ωI 10 ωI 10 ωI 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 and imaginary 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 rotation parameter 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]

iν iν R0(x) = cP Pl (2x + 1) + cQQl (2x + 1), (4.17)

iν iν where x ≡ (r −r+)/(r+ −r−) and ν ≡ 2mac/(r+ −r−), Pl and Ql are the associated Legendre functions of the first and second kinds, respectively, and cP , cQ ∈ R. The asymptotically-flat nature of Kerr iν spacetime requires that the field modes remain finite at infinity. Given that Pl (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

 iν  Ql (2x + 1) = 0 for DBCs x=x0 , (4.18) d  Qiν (2x + 1) = 0 for NBCs dx l x=x0 where x0 ≡ δ/(r+ − r−). Since δ ≪ M, one can write the associated Legendre function of the second kind in the approximate form [9]

−πν iν e −i ν Γ(−iν)Γ(l + 1 + iν) +i ν Q ∼ x 2 + x 2 (4.19) l 2Γ(iν) Γ(iν)Γ(l + 1 − iν) and, therefore, obtain the compact analytical expression

δ π(p + 1) i Γ(1 − iν)Γ(l + 1 + iν) ln ∼ − + ln , (4.20) r+ − r− ν ν Γ(1 + iν)Γ(l + 1 − iν) 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

0.8 Dirichletl=m=1 l=m=1 0.7 Neumann

0.6 /M

c 0.5 a

0.4

0.3

0.2 -7 -6 -5 -4 -3

log10(δ/M)

Figure 4.3 Critical value of the rotation parameter above which the fundamental (n = 0) scalar l = m = 1 QNM 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.

0.35 0.25 δ/M=10-7 δ/M=10-6 0.30 δ/M=10-5 0.20 -4 δ/M=10 0.25 I δ/M=10-3 I Mω 0.15 Mω 0.20 4 4 10 10 0.15 0.10 0.10

0.05 0.05 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 a/M a/M

Figure 4.4 Detailed view of the imaginary part of the fundamental (n = 0) scalar l = m = 1 QNM 2 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 the corresponding Kerr BH, as a function of the rotation parameter 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),

2 M X (m) m τins ∼ τ [log (δ/M)] [s], (4.21) M ins 10 10 ⊙ m=0

(m) whose coefficients τins 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.

5 Dirichletl=m=1 Neumannl=m=1 4

/M) [ s] (m) ⊙ 3 τins DBC NBC (0) (10M τins 0.03340 0.05623 ins

τ (1) 2 τins −0.10331 −0.11122 (2) τins 1.44978 0.56792

1 -7 -6 -5 -4 -3

log10(δ/M) Figure 4.5 Timescale of the scalar ergoregion instability of rapidly-rotating Kerr-like ECOs with a Table 4.3 Numerical value of the coefficients 2 perfectly-reflecting (|R| = 1) surface at r = r0 ≡ of the second-order polynomial (4.21) which fits rH + δ, 0 < δ ≪ M, where rH is the would-be event the timescale of the scalar ergoregion instability horizon of the corresponding Kerr BH, as a function of a perfectly-reflecting|R| ( 2 = 1) Kerr-like of δ/M, for l = m = 1 and both DBCs and NBCs. ECO, 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 2 (R ≳ −1) and quasi-NBCs (R ≲ 1). Note that the greater the absorption coefficient1 ( − |R| ), 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 2 instability for any spin value. When |R| < 0.996, the setup is stable (ωI > 0) whatever the value of a. 38 Scalar perturbations of exotic compact objects

0.2 0.2 ℛ=-1.0000 ℛ=1.0000 ℛ=-0.9995 ℛ=0.9995 0.1 0.1 ℛ=-0.9990 ℛ=0.9990 ℛ=-0.9985 ℛ=0.9985

I 0.0 ℛ=-0.9980 I 0.0 ℛ=0.9980 Mω Mω

4 -0.1 4 -0.1 10 10

-0.2 -0.2

-0.3 -0.3

0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 a/M a/M

Figure 4.6 Imaginary part of the fundamental (n = 0) l = m = 1 scalar QNM frequencies of a Kerr-like 2 ECO with a partially-reflecting (|R| < 1) surface at r = r0 ≡ rH + δ, where rH is the would-be event 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.

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),

2 X (k) k Max(MωI ) ∼ a R , (4.22) k=0 whose coefficients a(k) are listed in Table 4.4 for values of R in the range [−0.9980, −1] (quasi-DBCs).

2.5 δ/M=10-3 δ/M=10-4 2.0 (0) (1) (2) δ/M=10-5 δ/M a a a -6

) δ/M=10 I 1.5 −7 . . . δ/M=10-7 10 0 69571 1 39987 0 70417 −6 . . . 1.0 10 0 78566 1 58100 0 79535 −5 Max(Mω 10 0.91735 1.84597 0.92864 5 0.5 10 10−4 1.11362 2.24057 1.12696 0.0 10−3 1.40146 2.81878 1.41734

-0.5 0.9980 0.9985 0.9990 0.9995 1.0000 Table 4.4 Numerical value of the coefficients |ℛ| of the second-order polynomial (4.22) which fits Figure 4.7 Fit of the maximum value of the imag- the maximum value of the imaginary part of the inary part of the fundamental (n = 0) l = m = 1 fundamental (n = 0) l = m = 1 scalar QNM scalar QNM frequency of a Kerr-like ECO with re- frequency of a Kerr-like ECO with reflectivity flectivity R in the range [−0.9980, −1] (quasi-DBCs) R in the range [−0.9980, −1] (quasi-DBCs), for to the polynomial (4.22), for different values of δ/M. different 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 > M 2 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 2 2 when a > M , 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 ≥ rE , 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

0.5 0.5 r 0/M=0.1 r0/M=0.1 r /M=0.3 r 0.4 0 0.4 0/M=0.3 r 0/M=0.5 r0/M=0.5 r 0.3 0/M=0.7 0.3 r0/M=0.7 r 0/M=0.9 r0/M=0.9 R 0.2 R 0.2 Mω Mω 0.1 0.1

0.0 0.0

-0.1 -0.1 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.00 1.05 1.10 1.15 1.20 1.25 1.30 a/M a/M

-3 10 10-3

10-5 10-5 | | I I 10-7 10-7 M|ω M|ω

10-9 10-9

-11 10-11 10 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.00 1.05 1.10 1.15 1.20 1.25 1.30 a/M a/M

Figure 4.8 Real (top) and imaginary (bottom) parts of the fundamental (n = 0) l = m = 1 scalar QNM 2 frequencies of a superspinar with a perfectly-reflecting|R| ( = 1) surface at r = r0 > 0, as a function of a/M, for DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions of the imaginary part refer to positive (negative) frequencies.

2 Partial- and over-reflection (|R| ≶ 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 2 of some absorption (|R| < 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 2 slightly exceeds 1 (|R| ≳ 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 thestability 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

1.6 1.25 1.4 1.20

/M) [s] 1.2 ⊙ /M

c 1.15 a (10M 1.0 ins

1.10 τ

Dirichletl=m=1 0.8 Dirichletl=m=1 1.05 Neumannl=m=1 Neumannl=m=1

0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8

r0/M r0/M

Figure 4.9 Critical value of the rotation parame- Figure 4.10 Timescale of the scalar ergoregion in- ter under which the fundamental (n = 0) l = m = 1 stability of superspinars with a perfectly-reflecting 2 scalar QNM frequency of a perfectly-reflecting (|R| = 1) surface at r = r0 > 0, as a function of 2 (|R| = 1) superspinar is unstable, for both DBCs r0, for l = m = 1. and NBCs. The shaded regions refer to the domain of the ergoregion instability.

0.5 1.0 ℛ=-0.9995 ℛ=0.9995 0.4 0.8 ℛ=-1.0000 ℛ=1.0000 0.3 ℛ=-1.0005 0.6 ℛ=1.0005 ℛ=-1.0010 ℛ=1.0010 I I 0.2 ℛ=-1.0020 0.4 ℛ=1.0020 Mω Mω

4 0.1 4 0.2 10 10 0.0 0.0

-0.1 -0.2

-0.2 -0.4 1.00 1.01 1.02 1.03 1.04 1.05 1.00 1.02 1.04 1.06 1.08 1.10 a/M a/M

Figure 4.11 Imaginary part of the fundamental (n = 0) l = m = 1 scalar QNM frequencies of a superspinar featuring a surface with reflectivity R at r = r0 => 0, as a function of a/M. The introduction of absorption (|R|2 < 1) enhances the ergoregion instability, whereas over-reflecting|R| ( 2 > 1) BCs mitigate 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.

1 R ωR ωI 0.100 0.03239 −0.01894 −0.25 0.12043 −0.01997 0.010 (%) 0.21400 −0.02142

011 . − .

Z 0 03246 0 00947 0.001 ℛ=0.00 −0.50 0.12053 −0.00998 ℛ=-0.25 - 0.21414 −0.01069 10 4 ℛ=-0.50 ℛ=-0.75 0.03248 −0.00393 10-5 −0.75 0.12056 −0.00414 0.01 0.02 0.05 0.10 0.20 0.21417 −0.00443 MωR

Figure 4.12 Amplification factors for superradiant (0 < ω < mΩH ) l = m = 1 scalar field perturbations scattered Table 4.5 Scalar l = m = 1 QNM off Kerr-like ECOs with a/M = 0.9 and featuring a surface frequencies corresponding to the reso- with reflectivity R at r = r0 ≡ rH + δ, where rH is the nance peaks displayed in Figure 4.12. would-be event horizon of the corresponding Kerr BH and δ/M = 10−5.

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,

2 ˆ d R0 2 l(l + 1) + ω − Rˆ0 = 0, (4.23) dr2 r2 where Rˆ0(r) ≡ rR0. The general solution of this ODE is a linear combination of Bessel functions of the first kind [81],

1 R0(r) = √ αJ (ωr) + βJ (ωr) , (4.24) r l+1/2 −l−1/2 where, in general, α, β ∈ C. The large-r behavior of the asymptotic solution (4.24) is

r 2 1 R0(r) ∼ [α sin(ωr − lπ/2) + β cos(ωr + lπ/2)] , (4.25) πω r which can be written as a superposition of ingoing and outgoing waves,

ilπ/2 e 1 h i(l+1/2)π +iωr iπ/2 ilπ −iωri R0(r) ∼ √ β − αe e + αe + e β e . (4.26) 2πω r

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 fluxesat 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, 2 2 a, l, m, δ and R to compute the amplification factors (3.27), where A+ = A− = |O| /|I| . The small-r behavior of the asymptotic solution (4.24) is

l+1/2 −l−1/2 (ω/2) l (ω/2) −l−1 R0(r) ∼ α r + β r . (4.29) Γ(l + 3/2) Γ[−l + 1/2]

Near the would-be event horizon, the radial equation (4.7) reduces to

d dR r4 (ω − mΩ)2 ∆ + + − l(l + 1) R = 0. (4.30) dr dr ∆

iωˆ l+1 Introducing the radial coordinate z = (r − r+)/(r − r−) and the definition R0(z) = z (1 − z) Q0(z) for 2 the radial function, where ωˆ = (ω − mΩ)r+/(r+ − r−), one can bring the radial equation into the form

2 d Q0 dQ0 z(z − 1) + [c − (2b + c)z] − abQ0 = 0, (4.31) dz2 dz 44 Scalar perturbations of exotic compact objects with a = l + 1 + 2iωˆ, b = l + 1 and c = 1 + 2iωˆ. 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

−2iωˆ R0(z) = A z F (a − c + 1, b − c + 1, 2 − c; z) + BF (a, b, c; z), (4.32)

with A, B ∈ C. The small-z (r ∼ rH ) behavior of the asymptotic solution (4.32) is

r+ r+ −iωˆ +iωˆ −i(ω−mΩ) r∗ +i(ω−mΩ) r∗ R0(r) ∼ Az + Bz ∼ Ae 2M + Be 2M , (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

−i(ω−mΩ)r∗ +i(ω−mΩ)r∗ R0(r) ∼ Ae + Be . (4.34)

Therefore, the BC (3.20) to be imposed at the reflecting surface r = r0 is

B −2i(ω−mΩ)r∗ = Rˆ ≡ R e 0 , (4.35) A

∗ where, as defined in Chapter 3, r0 ≡ r∗(r0) < 0. On the other hand, the large-z (large-r) behavior of the asymptotic solution (4.32) is

l Γ(2l + 1) Γ(1 − 2iωˆ) Γ(1 + 2iωˆ) r R0(r) ∼ A + B Γ(l + 1) l + 1 − 2iωˆ Γ[l + 1 + 2iωˆ] r+ − r− −l−1 Γ(−2l − 1) Γ(1 − 2iωˆ) Γ(1 + 2iωˆ) r + A + B , (4.36) Γ(−l) Γ(−l − 2iωˆ) Γ(−l + 2iωˆ) r+ − r− 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

Γ(l + 3/2) Γ(2l + 1) Γ(1 − 2iωˆ) Γ(1 + 2iωˆ) α Rˆ = l+1/2 l + (4.37) (ω/2) (r+ − r−) Γ(l + 1) Γ(l + 1 − 2iωˆ) Γ(l + 1 + 2iωˆ) Γ(−l + 1/2) Γ(−2l − 1) Γ(1 − 2iωˆ) Γ(1 + 2iωˆ) β Rˆ . = −l−1/2 −l + (4.38) (ω/2) (r+ − r−) Γ(−l) Γ(−l − 2iωˆ) Γ(−l + 2iωˆ)

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

1 ℛ=0.00 1 ℛ=-0.25

Numerical 0.100 Analytical 0.100

0.010 0.010 (%) (%) 011 011 Z 0.001 Z 0.001

10-4 10-4

10-5 10-5 0.01 0.02 0.05 0.10 0.20 0.01 0.02 0.05 0.10 0.20 MωR MωR

1 ℛ=-0.50 1 ℛ=-0.75

0.100 0.100

0.010 0.010 (%) (%) 011 011 Z 0.001 Z 0.001

10-4 10-4

10-5 10-5 0.01 0.02 0.05 0.10 0.20 0.01 0.02 0.05 0.10 0.20 MωR MωR

Figure 4.13 Numerical and analytical values for the amplification factors of superradiant0 ( < ω < mΩH ) l = m = 1 scalar field perturbations scattered off Kerr-like ECOs 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 BH 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 forwardin[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 donein 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

  0 1 0 0     (µ)(ν) 1 0 0 0  η(µ)(ν) = η =   . (5.3)   0 0 0 −1   0 0 −1 0

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

µ ν µ ν 1 µ ν µ ν − κ = lµ;ν m l − ρ = lµ;ν m m¯ − ε = (lµ;ν n l − mµ;ν m¯ l ) 2 µ ν µ ν 1 µ ν µ ν − σ = lµ;ν m m µ = nµ;ν m¯ m − γ = (lµ;ν n n − mµ;ν m¯ n ) 2 µ ν µ ν 1 µ ν µ ν λ = nµ;ν m¯ m¯ − τ = lµ;ν m n − α = (lµ;ν n m¯ − mµ;ν m¯ m¯ ) 2 µ ν µ ν 1 µ ν µ ν ν = nµ;ν m¯ n π = nµ;ν m¯ l − β = (lµ;ν n m − mµ;ν m¯ m ) . (5.4) 2

It is useful to introduce the following notation for the directional derivatives

µ µ µ µ D ≡ l = l ∂µ, ∆ ≡ n = n ∂µ, δ ≡ m = m ∂µ, δ¯ ≡ m¯ =m ¯ ∂µ. (5.5)

5.2 Maxwell’s equations

The action for classical electrodynamics in curved spacetime is [24]

Z Z 1 √ 4 µν √ 4 µ S[A] = − −g d x Fµν F + −g dx AµJ , (5.6) 16π V V 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 ;ν Fµν = 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

µ ν 1 µ ν µ ν µ ν ϕ0 = Fµν l m , ϕ1 = Fµν (l n +m ¯ m ), ϕ2 = Fµν m¯ n , (5.8) 2 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 − δϕ¯ 0 = (π − 2α)ϕ0 + 2ρϕ1 − κϕ2

δϕ1 − ∆ϕ0 = (µ − 2γ)ϕ0 + 2τϕ1 − σϕ2 (5.9) Dϕ2 − δϕ¯ 1 = −λϕ0 + 2πϕ1 + (ρ − 2ε)ϕ2

δϕ2 − ∆ϕ1 = −νϕ0 + 2µϕ1 + (τ − 2β)ϕ2.

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 − δϕ¯ 0 = (π − 2α)ϕ0 + 2ρϕ1, (5.10)

δϕ1 − ∆ϕ0 = (µ − 2γ)ϕ0 + 2τϕ1, (5.11)

Dϕ2 − δϕ¯ 1 = 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∗) 3 and subtracting one equation from the other, the terms in ϕ1 vanish by the identity [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]

r2 + a2 a D = ∂t + ∂r + ∂ϕ ∆ ∆ 1 2 2 ∆ = r + a ∂t − ∆∂r + a ∂ϕ (5.17) 2Σ 1 i δ = √ ia sin θ ∂t + ∂θ + ∂ϕ , ρ¯ 2 sin θ 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]

1 h 2i ρ = −(r − ia cos θ)−1 γ = Mρ ρ − iaρ∗ cos θ + (a|ρ| sin θ) 2 1 h ∗ 2i 1 1 µ = ρ 1 + M(ρ + ρ ) + (a|ρ| sin θ) α = ρ iaρ sin θ − cot θ 2 2 2 ia τ = −√ |ρ|2 sin θ ρ∗ cot θ 2 β = − √ . 2 2 ia π = √ ρ2 sin θ 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

−i(ωt−mϕ) ϕi ∝ e , (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]

∆ † 1 † ∗ 1 D = D0, ∆ = − D0, δ = √ L0, δ = √ L0, (5.19) 2ρ2 ρ¯ 2 ρ¯∗ 2

† † where the derivative operators Dn, Dn, Ln, Ln (n ∈ Z) are given by

iK r − M Dn = ∂r − + 2n , Ln = ∂θ − Q + n cot θ, ∆ ∆ (5.20) † iK r − M † D = ∂r + + 2n , Ln = ∂θ + Q + n cot θ, n ∆ ∆ 5.2 Maxwell’s equations 51

2 2 † and K = (r + a )ω − am and Q = aω sin θ − m cosec θ. While Dn and Dn are purely radial operators, Ln † † † and Ln are purely angular operators. Note that Dn(ω, m) = Dn(−ω, −m) and Ln(ω, m) = Ln(−ω, −m). † ∗ † ∗ Additionally, although Dn = (Dn) , Ln ≠ (Ln) . Finally, using the tetrad (5.19) together with the foregoing notation, one can rewrite equations (5.15) and (5.16) in the form

1 † 1 † ia sin θ ia sin θ ∆ D1 + D − + L + L1 − Φ0 = 0, (5.21) ρ¯∗ 1 ρ¯∗ 0 ρ¯∗ ρ¯∗ † 1 † 1 ia sin θ † ia sin θ ∆ D + D − + L0 + L − Φ2 = 0, (5.22) 0 ρ¯∗ 0 ρ¯∗ ρ¯∗ 1 ρ¯∗

−2 where Φ0 = ϕ0 and Φ2 = 2ρ ϕ2. 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]

1 † 1 † 2iK ∆ D1 + D − = ∆D1D + , ρ¯∗ 1 ρ¯∗ 1 ρ¯∗ (5.23) † ia sin θ ia sin θ † 2ia sin θ L + L1 − = L L1 − Q, 0 ρ¯∗ ρ¯∗ 0 ρ¯∗ equation (5.21) takes the simpler form

† † (∆D1D1 + L0L1 + 2iωρ¯)Φ0 = 0. (5.24)

In a similar way, using the identities [64]

† 1 1 † 2iK ∆ D + D0 − = ∆D0D − , 0 ρ¯∗ ρ¯∗ 0 ρ¯∗ (5.25) ia sin θ † ia sin θ † 2ia sin θ L0 + L − = L0L + Q, ρ¯∗ 1 ρ¯∗ 1 ρ¯∗ one can convert equation (5.22) into the form

† † (∆D0D0 + L0L1 − 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

−iωt +imϕ −iωt +imϕ Φ0 = e R+1(r)S+1(θ)e , Φ2 = e R−1(r)S−1(θ)e , (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,

† (∆D1D1 + 2iωr)R+1 =λ ¯+1R+1 (5.28) † (L0L1 − 2aω cos θ)S+1 = −λ¯+1S+1 (5.29) 52 Electromagnetic perturbations of exotic compact objects

and for R−1 and S−1,

† (∆D0D0 − 2iωr)R−1 =λ ¯−1R−1 (5.30) † (L0L1 + 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) = −Ln(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,

Z π 2 (S±1) sin θ dθ = 1, (5.32) 0 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 4 satisfy complex-conjugate equations . Indeed, using the readily verifiable relations ∆Dn+1 = Dn∆ and † † ∆Dn+1 = Dn∆, equation (5.28) can be written in the form

† (∆D0D0 + 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 ∆D0D0∆R+1 = BR−1 (5.34) † † L0L1S−1 = BS+1, L0L1S+1 = BS−1

q 2 2 2 where B = λ¯|±1| − 4a ω + 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 197369 [ ] 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

2 −s d s+1 dRs T T 2is(r − M)K − K ∆ ∆ − V Rs = 0,V = − 4isωr +λ ¯|s| − s(s + 1), (5.35) dr dr s s ∆

4 That 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

2 2 1 d dSs m + s + 2ms cos θ 2 2 2 m sin θ − − a ω cos θ + 2aωs cos θ − E Ss = 0, (5.36) sin θ dθ dθ sin2 θ s l

m 2 2 where sE l ≡ λ¯|s| + 2amω − a ω , 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).

m 6 It follows from the symmetries of the angular equation (5.36) that sE l satisfy the relations [77]

m m m −m sE l(aω) = −sE l(aω), sE l(−aω) = sE l(aω). (5.37)

In the case when aω ≪ 1, perturbation theory holds [77–79]

+∞ 2 m X (n) n (0) (1) 2ms (2) E = f (aω) with f = l(l + 1), f = − , f = hs(l + 1) − hs(l) − 1,..., s l slm slm slm l l slm n=0 ( + 1) (5.38) where

2 1 2 2 1 2 l − (α + β) l − (α − β) h l l2 − s2 4 4 , s( ) = ( ) 3 2 1 (5.39) 2l l − 4 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

2 2 s Ys(r) = (r + a )∆ 2 Rs, (5.40) the radial equation (5.35) becomes

2 2 d Ys 2 dGs 2is(r − M)K − K − ∆(4irωs − λ) 2 − Gs + + 2 2 2 Ys = 0, (5.41) dr∗ dr∗ (r + a ) where λs ≡ λ|s| − s(s + 1) and

s(r − M) r∆ Gs(r) = + . (5.42) r2 + a2 (r2 + a2)2

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, note that λ¯|0| = λ0 (Section 4.1). 6 m The first relation in (5.37) can be inferred from the very definition of sE l, 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]

r 1 ∂ ∂ ∆ ∂ e(t) = √ A + 2Mar e = ∂t ∂ϕ (r) ∂r AΣ∆ Σ√ 1 ∂ Σ ∂ e(θ) = √ e(ϕ) = √ Σ ∂θ A sin θ ∂ϕ

2 2 2 2 2 where A ≡ (r + a ) − a ∆ sin θ, E(α) and B(α) are defined by

µ ν 1 λτ µ ν E(α) = Fµν e(α) e(t) , B(α) = − ϵµνλτ F e(α) e(t) , 2 √ with ϵµνλτ ≡ −g [µνλτ], where [µνλτ] is the four-dimensional Kronecker delta. Explicitly [87],

2 2 ia sin θ 1 2 r + a E(r) = √ ϕ2 − ∆ρ ϕ0 + c.c. + 2 √ Re(ϕ1), 2Aρ 2 A " 2 2 r # r r + a ∆ 1 ϕ2 ∆ E(θ) = ϕ0 − + c.c − 2a sin θ Im(ϕ1), ρ∗Σ 2A 2 ∆ρ2 A r ∆ ϕ0 ϕ2 E(ϕ) = −i ρ + + c.c., 2 2 ∆ρ2 2 2 a sin θ 1 2 r + a B(r) = √ ϕ2 − ∆ρ ϕ0 + c.c. + 2 √ Im(ϕ1), 2Aρ 2 A " 2 2 r # r r + a ∆ 1 ϕ2 ∆ B(θ) = − i ϕ0 − + c.c + 2a sin θ Re(ϕ1), ρ∗Σ 2A 2 ∆ρ2 A r ∆ ϕ0 ϕ2 B(ϕ) = −ρ + + c.c.. 2 2 ∆ρ2

5.4 Perfectly-reflecting boundary conditions

EWs are totally reflected when their Poynting vector is perpendicular to the surface of a perfectcon- 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,

√ 2 2 ∗ ∗ a ∆ sin θ B(r) − (r + a ) E(θ) = 0 → Re(ρΦ0) = Re(ρ Φ2)/∆ √ 2 2 ∗ ∗ a ∆ sin θ B(r) + (r + a ) E(θ) = 0 → Im(ρΦ0) = Im(ρ Φ2)/∆ (5.44)

E(ϕ) = 0 → Im(ϕ1) = 0

−2 where, as before, Φ0 = ϕ0 and Φ2 = 2ρ ϕ2. The first two conditions in (5.44) can be reduced to the form

2 2 |Φ2| |Φ0| = at r = r0. (5.45) ∆2 5.5 Detweiler transformation 55

Given the field decompositions (5.27) and the normalization conditions (5.32), this can be further simplified to

2 2 2 iΥ ∆ |R+1| = |R−1| or ∆R+1 = e R−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,

 † ¯ iΥ  D0∆R+1 = Λ∆R+1 λ|s| + 2iωr − e B , Λ = , at r = r0. (5.47) iK  D0R−1 = ΛR−1 2

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]

   r−seiωr∗  r−2s−1eiωr∗ Ys(r) ∼ or Rs(r) ∼ as r → +∞ (5.48)  r+se−iωr∗  r−1e−iωr∗    ∆+s/2e+iϖr∗  eiϖr∗ Ys(r) ∼ or Rs(r) ∼ as r → r0, (5.49)  ∆−s/2e−iϖr∗  ∆−se−iϖr∗

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

s 2 2 1 s+1 dRs Xs = ∆ 2 (r + a ) 2 α(r)Rs + β(r)∆ , (5.50) dr which satisfies the second-order linear ODE

2 d Xs D D U∆ 2 dG 2 − Vs Xs = 0,Vs (r, ω) = 2 2 2 + G + , (5.51) dr∗ (r + a ) dr∗ where

dα dβ U(r) = V T + β−1∆−s 2 + ∆s+1 . (5.52) s dr dr 56 Electromagnetic perturbations of exotic compact objects

D The functions α(r) and β(r) are chosen so that the potential Vs 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˜∆ + 1 β˜∆ α(r) = p , β(r) = p , (5.53) 2B[Re(˜α∆) + 1] 2B[Re(˜α∆) + 1] where

2 2K − ∆(2iωr +λ ¯−1) 2iK α˜(r) = , β˜(r) = . (5.54) B∆2 B∆

D 7 2 D The explicit form of V−1 can be found in [88] (see Appendix A therein) . The effective potential (ω + V−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]

− −iϖr∗ +iϖr∗ R−1 ∼ A−∆e + B−e , as r → r0, (5.55)

(0) (1) (0) (1) where A− = A− + A− η + ... and B− = B− + B− 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

(0) (0) (1) A− = A− , B− = B− + B− η. (5.56)

(0) (0) − Following [89], A− and B− are related to the asymptotic coefficients of X−1 as r → r0 by

(0) 1 B (0) ϖ 2 2 2 − iB A = − 2(r + a ) , = − A−, (5.57) − B + (0) K K¯ A− 4 + 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

(1) ima 2ωr+ − iλs (0) B− = + B− . (5.58) M(r+ − r−) 4Mϖr+

7 D When finding numerical solutions to the ODE (5.51) using Mathematica the explicit form of V−1 should be used. An implicit definition, in terms of U, α and β, will not work properly. 8 − Recall 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,

+iϖr∗ −iϖr∗ e 0 ± A−e 0 = 0, (5.59)

∗ where r0 ≡ 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 m the separation constant ±1E l and 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 D both DBCs (R = −1) and NBCs (R = 1). When a = 0, the potential V−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.

0.30 10-2 Dirichletl=1 l=2 0.25 Dirichlet Neumannl=1 10-4 Neumannl=2 0.20

I -6 R 10

Mω 0.15 -Mω 10-8 0.10

-10 0.05 10

-7 -6 -5 -4 -3 10-7 10-6 10-5 10-4 10-3 10 10 10 10 10 δ/M δ/M

Figure 5.1 Real (left) and imaginary (right) parts of the fundamental (n = 0) |l| = 1, 2 electromagnetic 2 QNM frequencies of a Schwarzschild-type 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 the corresponding Schwarzschild BH, as a function 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 −4 −3 10 (10 ) 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

0.4 δ/M=10-7 0.4 δ/M=10-7 δ/M=10-6 δ/M=10-6 0.3 -5 -5 δ/M=10 0.3 δ/M=10 δ/M=10-4 δ/M=10-4 0.2 δ/M=10-3 0.2 δ/M=10-3 R R 0.1 Mω Mω 0.1 0.0 0.0 -0.1

-0.2 -0.1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a/M a/M

10-4 10-5

-6 | 10 | -7 I I 10 M|ω M|ω -8 10 10-9

-10 10 10-11

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 a/M a/M

Figure 5.2 Real (top) and imaginary (bottom) part of the fundamental (n = 0) l = m = 1 electromagnetic 2 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 the corresponding Kerr BH, as a function of a/M, for both DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions of the imaginary part refer to negative (positive) frequencies. The shaded regions represent the experimental band for the rotation parameter of the end product of compact-binary coalescences, based on the currently available 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 am iΥ l(l + 1) + i − (1 − e ) R−1 = 0 at r = r0. (5.60) dr ∆ 2am

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

am dR−1 iam R0 = −i + R−1 . (5.61) l(l + 1) dr ∆ 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

2 2 dR0 a m dR−1 am l(l + 1) ∆ = + i − R−1 . (5.62) dr l(l + 1) dr ∆ am

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 R0(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 R0(r0) = 0 and Y0 (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.

1.6 2.5 δ/M=10-7 1.4 δ/M=10-6 2.0 1.2 δ/M=10-5 -4 I 1.0 δ/M=10 I δ/M=10-3 1.5 Mω 0.8 Mω 4 4

10 0.6 10 1.0 0.4 0.2 0.5

0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 a/M a/M

Figure 5.3 Detailed view of the imaginary part of the fundamental (n = 0) l = m = 1 electromagnetic 2 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 the corresponding Kerr BH, as a function of a/M in 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 (m) δ/M, together with the fit to the general second-order polynomial (4.21) whose coefficients τins are listed −6 −6 in Table 5.1 for both BCs. For fixed δ/M, it is greater for objects with δ/M ≳ 10 (≲ 10 ) 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

0.60 Dirichletl=1 0.55 Neumannl=1 0.50

0.45 (m) τins DBC NBC /M) [s] ⊙ 0.40 (0) τins 0.21781 0.10261

(10M 0.35 (1) τins −0.02102 −0.02044 ins τ 0.30 (2) τins 0.00421 0.00690 0.25

0.20 -7 -6 -5 -4 -3

log10(δ/M) Figure 5.4 Timescale of the electromagnetic ergore- Table 5.1 Numerical value of the coefficients gion instability of rapidly-rotating Kerr-like ECOs of the second-order polynomial (4.21) which fits with a perfectly-reflecting|R| ( 2 = 1) surface at the timescale of the electromagnetic ergoregion 2 r = r0 ≡ rH + δ, δ ≪ M, where rH is the would-be instability of a perfectly-reflecting|R| ( = 1) event horizon of the corresponding Kerr BH, as a Kerr-like ECO, for l = m = 1 and both DBCs function of δ/M, for l = m = 1. and NBCs.

ℛ=-1.000 ℛ=1.000 1 1 ℛ=-0.995 ℛ=0.995 ℛ=-0.990 ℛ=0.990 ℛ=-0.985 ℛ=0.985 0 0 I I Mω Mω 4 4

10 -1 10 -1

-2 -2

0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 a/M a/M

Figure 5.5 Imaginary part of the fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of a 2 Kerr-like ECO with a partially-reflecting|R| ( < 1) surface at r = r0 ≡ rH + δ, where rH is the would-be event 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 −5 δ/M = 10 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).

1.5 δ/M=10-3 δ/M=10-4 -5 δ/M=10 (0) (1) (2) δ/M=10-6 δ/M a a a ) I 1.0 -7 δ/M=10 10−7 0.08824 0.18353 0.09537 10−6 0.10791 0.20759 0.10791 Max(Mω 4 0.5 −5 . . .

10 10 0 11586 0 24098 0 12524 10−4 0.14463 0.30017 0.15567 −3 0.0 10 0.12954 0.27145 0.14207

0.985 0.990 0.995 1.000 |ℛ| Table 5.2 Numerical value of the coefficients Figure 5.6 Fit of the maximum value of the imag- of the second-order polynomial (4.22) which fits inary part of the fundamental (n = 0) l = m = 1 the maximum value of the imaginary part of electromagnetic QNM frequency of a Kerr-like ECO the fundamental (n = 0) electromagnetic l = with reflectivity R in the range [−0.985, −1] (quasi- m = 1 QNM frequency of a Kerr-like ECO with DBCs) to the polynomial (4.22), for different values reflectivity R in the range [−0.985, −1] (quasi- of δ/M. 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.

2 Partial and over-reflection (|R| ≶ 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: 2 2 when |R| ≲ 1 (|R| ≳ 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

0.5 0.5 r0/M=0.1 r0/M=0.1 0.4 r0/M=0.3 0.4 r0/M=0.3 r0/M=0.5 r0/M=0.5 0.3 r0/M=0.7 0.3 r0/M=0.7 r0/M=0.9 r0/M=0.9 R 0.2 R 0.2 Mω Mω 0.1 0.1

0.0 0.0

-0.1 -0.1 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.00 1.05 1.10 1.15 1.20 1.25 1.30 a/M a/M

-3 10 10-3

10-5 10-5 | | I I 10-7 10-7 M|ω M|ω

10-9 10-9

-11 10-11 10 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.00 1.05 1.10 1.15 1.20 1.25 1.30 a/M a/M

Figure 5.7 Real (top) and imaginary (bottom) parts of the fundamental (n = 0) l = m = 1 electromagnetic 2 QNM frequencies of a superspinar with a perfectly-reflecting|R| ( = 1) surface at r = r0 > 0, as a function of a/M, for both DBCs (left) and NBCs (right). The left (right) arms of the interpolating functions of the imaginary part refer to positive (negative) frequencies.

0.25 Dirichletl=1 Neumannl=1

0.20 /M) [s] ⊙ 0.15 (10M ins τ

0.10

0.0 0.2 0.4 0.6 0.8 1.0

r0/M

Figure 5.8 Timescale of the electromagnetic ergoregion instability of superspinars with a perfectly- 2 reflecting (|R| = 1) surface at r = r0 > 0, as a function of r0, for l = m = 1. 64 Electromagnetic perturbations of exotic compact objects

4 ℛ=-0.995 8 ℛ=0.995 3 ℛ=-1.000 ℛ=1.000 ℛ=-1.005 6 ℛ=1.005 2 ℛ=-1.010 ℛ=1.010 I I ℛ=-1.015 4 ℛ=1.015 1 Mω Mω 4 4 2 10 10 0 0 -1 -2 -2 1.00 1.01 1.02 1.03 1.04 1.05 1.00 1.02 1.04 1.06 1.08 1.10 a/M a/M

Figure 5.9 Imaginary part of the fundamental (n = 0) l = m = 1 electromagnetic QNM frequencies of a superspinar featuring a surface with reflectivity R at r = r0 > 0, as a function of a/M. The introduction of absorption (|R|2 < 1) enhances the ergoregion instability, whereas over-reflecting|R| ( 2 > 1) BCs mitigate 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 Friedman10 [ ], 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-Nordstr¨om- 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 ofits 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. References

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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 ofthese 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

iK r − M Dn = ∂r − + 2n , Ln = ∂θ − Q + n cot θ, ∆ ∆ (A.1) † iK r − M † D = ∂r + + 2n , Ln = ∂θ + Q + n cot θ n ∆ ∆ 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:

† † † ∆(Dn − Dn) = 2iK, ∆Dn+1 = Dn∆, ∆Dn+1 = Dn∆, (A.2)

† † † Ln − Ln = 2Q, sin θ Ln+1 = Ln sin θ, sin θ Ln+1 = Ln sin θ, (A.3) Z π Z π † g(Lnf) sin θ dθ = − f(L−n+1g) sin θ dθ, (A.4) 0 0

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

 † (∆D1D1 + 2iωr)R+1 =λR ¯ +1 (A.6) † (L0L1 − 2aω cos θ)S+1 = −λS¯ +1 (A.7)

 † (∆D0D0 − 2iωr)R−1 =λR ¯ −1 (A.8) † (L0L1 + 2aω cos θ)S−1 = −λS¯ −1 (A.9)

Here, the angular functions S±1 are normalized to unity,

Z π 2 (S±1) sin θ dθ = 1. (A.10) 0

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(∆D0D0 − 2iωr)R−1 = (∆D1D1 + 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 ∆D0D0 and using the equation (A.6) and the elementary identities (A.2), one obtains

† † † † † † ∆D0D0(∆D0D0 + 2iωr)∆R+1 = (∆D0D0 − 2iωr)∆D0D0∆R+1 (A.13)

† † The operator acting on ∆D0D0∆R+1 is the same as the one acting on R−1 in equation (A.8), meaning

† † ∆D0D0∆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, ∆D0D0∆R+1 = BR−1, (A.15) where B is a complex constant to be determined. Combining the foregoing equations, one gets

2 † † † † |B| = ∆D0D0∆D0D0 = ∆D0D0∆D0D0 (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 =λ ¯2 − 4a2ω2 + 4aωm (A.17)

Acting on −λS¯ +1 with L0L1 and using equation (A.7) and the elementary identities (A.3), one gets

† † L0L1(L0L1 − 2aω cos θ)S+1 = (L0L1 + 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 L0L1 and using the equation (A.9) and the elementary identities (A.3), one obtains

† † † † † † L0L1(L0L1 + 2aω cos θ)S−1 = (L0L1 − 2aω cos θ)L0L1S−1 (A.20)

† † Note that the operator acting on L0L1S−1 is the same as the one acting on S+1 in equation (A.7), meaning that

† † L0L1S−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, L0L1S−1 = C2S+1, (A.22)

1 where C1 and C2 are real constants . Surprisingly, since the angular functions are normalized to unity, it turns out that C1 = C2. In fact,

Z π Z π 2 (A.10) 2 2 (A.22) C1 = C1 S−1 sin θ dθ = (L0L1S+1)(L0L1S+1) sin θ dθ 0 0 Z π Z π (A.4) † † (A.22) † † = (L0L1L0L1S+1)S+1 sin θ dθ = C1 (L0L1S+1)S+1 sin θ dθ 0 0 Z π (A.22) 2 (A.10) = C1C2 S+1 sin θ dθ = C1C2, 0 and, therefore, C1 = C2 ≡ C. Combining the identities (A.22), one gets

2 † † † † C = L0L1L0L1 = L0L1L0L1 (A.23)

1 † On the assumption that the angular functions S±1 are real and given that Im(Ln) = Im(Ln) = 0, the functions † † L0L1S+1 and L0L1S−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 =λ ¯2 − 4a2ω2 + 4aωm, (A.24) i.e. |B|2 = C2. Finally, the Teukolsky-Starobinsky identities for spin-1 fields are

† † ∆D0D0R−1 = B∆R+1 ∆D0D0∆R+1 = BR−1 , (A.25) † † L0L1S−1 = BS+1, L0L1S+1 = BS−1

p where B = λ¯2 − 4a2ω2 + 4amω is the Starobinsky constant2.

2The plus sign in the definition of the Starobinsky constant is not arbitrary. The careful reader should read thedetailed explanation in [64].