The Perturbative Structure of Schwarzschild Black Hole Ringdown

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Hole Ringdown

Dipartimento di Fisica dell’Università di Pisa Corso di Laurea Magistrale in Fisica

Candidate Jacopo Mazza ID number 506177 Thesis Advisor Dr. Giancarlo Cella Academic Year 2018/2019

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The Perturbative Structure of Schwarzschild Black Hole Ringdown

Master thesis. Università di Pisa

This thesis has been typeset by LA_{TEX and the Pisathesis class.} Author’s email: [email protected]

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Abstract

Black hole (BH) perturbation theory represents a strategy for determining the gravitational field under the assumption of ‘small’ deviations from a BH geometry. In this context, one splits the full metric g_µν, as well as matter fields, into a non-dynamical background and a set of perturbations that propagate on top of it. Einstein’s equations are therefore expanded into a series of terms that fall naturally into a hierarchy of increasing perturbative order—thus allowing for an approximate description.

When dropping all but the leading term, the resulting equations are linear and much about their solutions can be understood analytically. One key result of such analysis is that the gravi-tational perturbation is described by a superposition of quasinormal modes (QNM): fundamental solutions characterised by a set of (infinitely many) frequencies, determined solely by the back-ground geometry. Since these frequencies are complex, the corresponding oscillating modes are damped (hence the modifier ‘quasi-’).

BH QNMs have been studied extensively since the first pioneering work on Schwarzschild perturbation theory—by Regge and Wheeler, in 1957—, and have now entered the canon of BH dynamics. They are object of interest in many areas of gravitational physics, including quantum gravity and BH thermodynamics. In particular, they play a central role in modelling gravitational wave (GW) signals generated by astrophysical sources, such as binary BH mergers: namely, they constitute the theoretical basis for understanding ringdown.

The first-order approximation is believed to break down in many dynamical regimes of phys-ical interest; the linearised theory alone, however, does not allow its limit of validity to be determined accurately. In the context of BH binaries, for instance, the merger appears to be in-trinsically non-linear and one expects the QNMs picture to be reliable only some time thereafter: what remains unclear—and in fact cannot be decided within the linear approximation—is when exactly the linear regime begins. In this regard, Giesler et al. recently claimed that including enough QNMs allows for fitting the GW time series (obtained numerically, in their case) up until the signal’s peak, i.e. very close to the time of merger.

With this thesis, we aim at contributing to this debate by considering second-order gravita-tional perturbations. We do so in the relatively simple context of a Schwarzschild background; this allowed our treatment to remain mostly analytic, with no need to resort to numerical tech-niques.

It is known that higher-order perturbations satisfy the same equations as the first, with the addition of a source that is completely determined by lower orders. For the second order, in particular, such source is quadratic in the first-order modes: we characterise it in terms of radial, temporal and asymptotic radial dependence—thus reproducing published results.

One might expect, as has indeed been claimed, that second-order modes exhibit new frequen-cies in addition to those of the first order, equal to sums and differences of the QN frequenfrequen-cies. This conclusion has however been disputed and the matter remains unsettled. In this regard, we put forward an improved way of matching initial conditions, which we believe yields a better understanding of the problem. In passing, we note that particular coincidences in the values of QN frequencies could lead to quasi-secular terms in the second-order perturbation. We propose an approach, based on the renormalisation group, to resum such terms and obtain a uniformly valid solution.

These conclusions required a careful analysis of the space of solutions to the relevant partial differential equation and a thorough understanding of the ringdown’s perturbative structure.

We expect our analysis to have impact on GW modelling, and on all areas that make use of BH QNMs. Our results could be of interest also in contexts in which gravitational perturbation

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theory is performed on backgrounds other than BHs, such as in cosmology.

Key words Schwarzschild black hole, Quasi-normal Modes, Ringdown, Regge-Wheeler equa-tion.

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Notations and Terminology

Geometry We always consider a four dimensional Lorentzian manifold whose metric has ‘mostly plus’ signature (−, +, +, +).

The metric compatible covariant derivative will be denoted with ∇. The Lie derivative with respect to a vector field ξ will be written as £ξ. The Riemann tensors will have the following expression in components:

Rα_µνρ = Γα_νσΓσ_µρ− Γα

ρσΓσµν+ ∂νΓαµρ− ∂ρΓαµν.

Fourier transform Our definition for the Fourier transform will be: f (t) = Z +∞ −∞ dω 2πe −iωt_{f (ω).}

Functions and their Fourier transforms will be distinguished by their arguments but no ty-pographical sign—the context and the text will make clear which of the two is considered. The same specifications apply for the Laplace transform, whose definition is given in the text.

Units We will use GN = ~ = c = 1 unless otherwise specified.

Schwarszschild metric For reference, we will write Schwarzschild metric in spherical coordinates (t, r, θ, φ) as ds2= −f (r)dt2+ 1 f (r)dr 2_{+ r}2_dΩ2_, f (r) = 1 −2M r . Glossary of abbreviations

The following initialisms are used:

ADM Arnowitt-Deser-Misner;

(A)dS (anti) de Sitter; AF asymptotically flat; BH black hole;

BBH black hole binary; CFT conformal field theory; CG Clebsch-Gordan (coeffi-cients);

ECO exotic compact object; GR general relativity; GW gravitational wave; NR numerical relativity; PN post-Newtonian; RG renormalisation group; RW Regge-Wheeler; RWZ Regge-Wheeler-Zerilli; SNR signal-to-noise ratio; TT transverse-traceless; QN quasi normal;

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1

Introduction

The first direct detection of gravitational waves (GWs) by the LIGO and Virgo collaborations [1] has been a tremendous achievement. Not only did it crown the century-long collective effort of hundreds of scientists; most importantly, it marked the beginning of a new era for gravitational physics and astrophysics, opening to unprecedented observational opportunities.

After the first event, GW150914, many more followed. In August 2017 two black holes (BHs) were seen colliding by three detectors together for the first time [4]; three days later a neutron star binary merger was recorder and its electromagnetic counterpart was followed up by the astrophysical community [5,6,3]. By December 2018, as the second LIGO-Virgo observing run (O2) came to an end, the first catalog of gravitational wave transients was released [81]: it listed a total of eleven events, four of which had never been reported before. Further results from the third observing run are expected in the near future.

Mergers of stellar-mass black hole binaries (BBHs) are a prominent source for the GWs that can be detected from Earth. The signal they produce has frequencies that fall well within Advanced LIGO’s and Advanced Virgo’s sensitive bandwidth—which ranges between ∼ 10 Hz and ∼ 10 000 Hz—. Indeed, they make up for all but one of the events recorder to date.

Their dynamics is usually understood as a sequence of three successive stages, each corre-sponding to a different regime (Figure1.1). During the inspiral, the two BHs are well separated and they orbit around their center of mass. As the system loses energy and angular momentum to gravitational radiation, the BHs come closer together and become increasingly relativistic. The gravitational field is still rather weak and well described in a post-Newtonian (PN) ap-proximation [39]. At the merger, the two BHs lose their individuality and coalesce into each other: a single horizon forms, surrounding both singularities. This is a regime of strong gravity, characterized by highly non-linear dynamics. Einstein’s equation cannot be solved analytically and their solution must be evaluated numerically [58].

After merger, the system keeps radiating energy and eventually settles down to a stationary, typically spinning, BH. This ringdown phase is what this thesis focuses on: The signal is well fitted by a superposition of exponentially damped sinusoids, whose frequencies and damping times are completely determined by the remnant’s final mass and spin. As could be guessed on dimensional grounds, the scale with the final mass of the BH: The least damped mode dies off within milliseconds for stellar-mass remnants, or within minutes in the case of supermassive BHs with M ∼ 106M.

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Figure 1.1. The three phases of a coalescence [10].

The ringdown can be described analytically within perturbation theory around a Kerr ge-ometry. Regge and Wheeler pioneered this approach by publishing [76], in 1957, a version of Einstein’s vacuum equation linearised around a Schwarzschild background; they managed to decouple the angular dependency from the radial-temporal one, and reduced the complicated system of equations to a simpler, Schrödinger-like master equation that has been named after them ever since. Their results were then extended by many authors in the following decades: most notably Zerilli [93]—who introduced another master equation, later named after him, necessary to fully account for all the components of the metric perturbation—and Teukolsky [80]—who wrote a similar master equation for a Kerr BH.

Nowadays, it is well understood that solutions to the linearised Einstein’s equations, supple-mented by suitable boundary conditions at spatial infinity and at the BH horizon, can be written as a sum over a discrete set of quasinormal modes (QNMs). Each of these modes has a unique, complex ‘quasinormal’ (QN) frequency: Its real part determines the oscillating component of the solution while the imaginary part has always the correct sign to produce damping—hence the adverb ‘quasi’. There exist and infinity of quasi-normal frequencies. They can be organised in multiplets (`, m) depending on the angular behaviour of the corresponding mode; each multiplet contains infinitely many frequencies, labeled by an index n and further orderable according to their damping time, which decreases indefinitely starting from a maximum value. The least damped mode (n = 0) in each multiplet is usually referred to as the fundamental mode, while all the others are called overtones in analogy with acoustic theory. It should be noted, however, that the overtone frequencies are not integer multiples of the fundamental, as often happens in acoustic theory; in particular, overtones are not necessarily subdominant at all times.

Relevant information concerning QNMs will be exposed in later chapters. Excellent reviews on the topic are Kokkotas [54], Nollert [72], several chapters in book by Chandrasekhar [30], Berti et al. [14] (thorough and more technical), chapter 12 in Maggiore [65] (very recent). They mostly come with a rich bibliography that includes historical as well as recent, more technical papers.

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which has not yet be proven—the theoretical understanding of QNMs is now sound. Further work is needed, however, to blend this knowledge into the context of GW phenomenology for astrophysically relevant processes, such as BBH coalescences. An example of a piece of informa-tion that one would like to access is the following: When does the transiinforma-tion from the non-linear to linear regime take place? When can one start to trust the QNM picture? Linear perturbation theory alone seems not the right setting to address these questions, since—this comes as no surprise—it provides no clue as to when the perturbative hypothesis is satisfied.

The question of the start time is hardly merely academical: since it enters the models used to fit recorded GW data, its correct determination is key to extract best-fit parameters and to assess their reliability [17]. One wants to pick it sufficiently late that all non-linear behaviour has damped away, but not too late, as the fitted time interval should last long enough before the signal is submerged by noise. The problem affects both real GW signal and numerical relativity (NR) simulations, but in the latter case one has complete control over the remnant’s final parameters and NR can therefore be used as a testing ground for data analysis techniques [59,60].

In a recent work, Giesler, Isi, Scheel and Teukolsky [40] investigated the subtle interplay between ringdown starting time and number of overtones included in the fitting model. Not surprisingly, they found that the more overtones are included, the earlier one has to start fitting. In other words, if one wants to extract the least damped mode, one has to fit it to the signal when all other contributions have died away; if instead one wishes to extract several QNMs, they can start earlier—in fact, they should, since each overtone introduces at least three new parameters to be fitted and more statistics, i.e. a longer time interval, is required. As a result of their analysis, they formulate the surprising (if not unsettling) claim that including enough overtones (seven) allows to describe the GW signal at all times after the GW strain peak. This would entail that the linear regime starts at very early times.

This thesis aims at contributing to this debate by analising the dynamics of second-order gravitational perturbations, in the simplified case of a non-rotating Schwarzschild BH.

The interest in QNMs is all but unmotivated. Observationally, one could use them to improve the analysis of a detected GW waveform and infer information on the compact binary that generated it. Indeed, QNMs carry plenty of information on the progenitors: With them, one can confidently tell BHs from other compact sources, and estimate the binary’s parameters (progenitors’ masses, spins, etc.). See for instance [51].

Boosting current observational capabilities, however, is just one (minor) piece of a much more ambitious puzzle that has become known as BH spectroscopy. This program was laid down in [35], and further discussed in [9] and references therein; for a discussion on the feasibility and on techniques to extract QNMs from the ringdown signal, see [59, 60]; see also [18] for an approach that makes full use of state-of-the-art GW modeling. At its very heart lies an evocative parallelism with atomic spectroscopy at the beginning of the last century. The absorption and emission of electromagnetic radiation by atoms—as the analogy goes—provided insights into their inner structure, with the discovery of electrons and atomic nuclei; soon afterwards, scattering experiments yielded further discoveries into the nature of subatomic particles and of electromagnetism as a fundamental interaction, as well as into their quantum behaviour. Today, the QNM spectrum could serve as a probe into the very concept of BH—the ‘atom’ for the gravitational interaction—as predicted by general relativity (GR); also, it may come as Nature’s deus ex machina to bring order to the overcrowded landscape of theoretical alternatives, improvements and modifications of Einstein’s theory. Hopes are that BH QNMs will be our scope into the realm of quantum gravity (QG).

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highest signal-to-noise ratio (SNR) so far, came with [29], by Carullo, Del Pozzo and Veitch (but see also [28]). In that reference, the authors conduct two separate searches for QNMs in the ringdown signal: an ‘agnostic’ one, whereby one or more otherwise unconstrained damped sinusoids where fitted for; and one rooted in the theoretical knowledge of Kerr’s QN ringing. The first search found evidence for one mode with f = 234+11₋₁₂ Hz and τ = 4.0+1,5_−1.0 ms, most likely having (` = 2, m = 2, n = 0) or (` = 3, m = −3, n = 0) or a combination thereof, and starting time 3.9+0.3_−0.3 ms after the waveform’s peak; they found no evidence for more than one mode. For the second search, they fixed the values of f, τ to those predicted by linear perturbation theory around a Kerr BH with mass and spin equal to those of GW150914’s remnant; the analysis including one and more modes yielded results compatible with the first search. They also note that the inability to resolve the degeneracy in the ` = 2 and ` = 3 subspace is likely to persist in future detections, unless the SNR is particularly high or in systems in which the final spin is not dominated by the orbital angular momentum.

The merit of work [29] was to show that currently operating detectors are ready for BH spectroscopy. It frustrated the most enthusiastic expectations, however, since it concluded that louder events are to be waited for before any answer can be given. Indeed, one QNM is sufficient to determine the remnant’s final parameter, but it is generally believed that at least one more needs to be measured in order to test for deviations from the standard picture.

Nonetheless, such deviations are, in principle, detectable. Previous studies by the LIGO and Virgo Collaborations [2,7,82] tested several aspects of GR—among others: the propagation of GWs from the sources to the detectors, in search for modifications to the graviton’s dispersion relation; consistency between the low-frequency (inspiral) and high-frequency (merger-ringdown) parts of the signal; and, most importantly for the purpose of this thesis, accordance of the ringdown frequencies and damping times with their GR predictions. They found no evidence for deviations.

One way to perform the last of these tests is to parametrise deviations from a theoretical GW waveform and check whether these deviations are consistent with zero. When restricting to the ringdown phase, this is a test of the no-hair theorem for the remnant BH. Specifically, one would write for the frequency and damping time

ω`mn = ω`mnGR(M, J )(1 + δ ˆω`mn) τ`mn = τ`mnGR(M, J )(1 + δ ˆτ`mn)

and use Bayesian methods to constrain them. See for instance [92,68,50] and references therein (but note also [84]). To give a taste of the motivation for parametrisations of this sort, we make reference to scalar-tensor theories, whereby a scalar field is coupled non-minimally to GR—e.g. Brans-Dicke’s S = Z d4x√−gM 2 P l 2 φR −Ω φ∇µφ∇ µ_{φ − m}2_φ2_,

Ω being a constant parameter. Typically, GR BH solutions have straightforward generalisations to configurations in which the scalar takes up a constant value. The presence of the scalar, however, manifests itself in the dynamics of linear perturbations: The resulting equations are very similar in form to those of GR, but the scalar is coupled to the tensor. Hence, the ringdown of a scalar-tensor BH contains three degrees of freedom—one more than GR—and differs from that of standard GR because of additional QNMs. See [79] for a general discussion of pertur-bations in modified gravity theories and [38] for an effective field theory approach to QNMs in scalar-tensor theories (also cf. [37]).

A different yet equally appealing possibility amounts to testing, instead of GR, the concept of BH. Countless alternatives to BHs have been proposed: ultra-compact, horizonless and/or

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regular (meaning, non-singular) objects, that resemble general-relativistic BHs at sufficiently large distances but differ from them in the close limit and in their interior. Such objects are collectively called ECOs (for Exotic Compact Objects)—see [27] for a review. They are however very diverse and find their motivations in different physical scenarios: for instance, boson stars are condensates of some bosonic field; gravastar (with a de Sitter interior) and black stars (sustained by vacuum polarization pressure) arise from quantum effects in curved spacetimes; fuzzballs are inspired by string theory, etc.

Figure 1.2. Echoes in the GW signal from the

ring-down of a Clear Photosphere Object (ClePhO), a very compact ECO; from [26].

Despite not having an horizon, these ob-jects would still exhibit a ringdown, which in-deed hallmarks the presence of a light ring, rather than an horizon [24]. At fairly early times this would be indistinguishable from that of a BH; later on, though, a pulse falling towards the surface of the object would inter-act with it and get reflected. Thus, part of the gravitational signal would be semitrapped between the object and the photosphere, and portions of it would be leaked to infinity in regular intervals. As a result, ‘echoes’ of grav-itational radiation would appear in the wave-form detected by a distant observer, some time after the ‘prompt’ (exponentially damped si-nusoids) ringdown signal—see [25,26] and ref-erences therein.

ECOs can be characterised by a small pa-rameter, , than quantifies the amount by which they deviate from their general-relativistic BH counterpart. In the case of spherical sym-metry, for instance, one may think of the surface of the ECO as being located at r0 = rS(1 + ), rS standing for the Schwarzschild radius. Thus the timescale of repeated reflections would be

τecho ∼ 2rS c |log |;

this is roughly the interval between two successive echoes in the GW late ringdown signal. Note that the amplitude of the signal decreases at each repetition, see Fig. (1.2).

In spite of its simplicity, the analogy with atomic spectroscopy proved surprisingly power-ful in the hands of researches whose interested leaned further towards the hic sunt leones of gravitational theory: quantum gravity. In 1998, Hod [46] noted a possible connection between the QN spectrum, in particular its highly-damped sector, and BHs quantisation as proposed by Bekenstein [11]. Semiclassical arguments by the latter lead to the following conjecture for the discrete eigenvalues for the area of a BH’s surface:

An= γL2_{P l}n, n = 1, 2, 3, . . .

with the bound, deriving from statistical physics, γ = 4 log k, k ∈ N, . Since according to Bohr’s correspondence principle “transition frequencies at large quantum numbers should equal classical oscillation frequencies”, Hod swiftly took up a then recent discovery by Nollert: that the real part of Schwarzschild’s QN frequencies tends to a finite value. Ergo, highly damped QNMs must correspond to the states with large quantum number of the quantised BH! Nollert’s numerical value, 0.0437132, is close to log 3/8π, hence Hod deduced γ = 4 log 3.

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Thereupon, publications on the matter flourished. A noticeable contribution came from Maggiore in 2008 [64]: He proposed to look at the asymptotic behaviour of a ‘physical’ frequency, given by the quadrature sum of the real and imaginary parts of the QN frequency. Indeed, if one regards QNMs in analogy to a damped harmonic oscillator, such quadrature sum is the actual frequency and looking solely at the real part is justified only if the imaginary part is correspondingly negligible. Since on the Schwarzschild background the imaginary parts grow unbounded, this is clearly not the case. With this premises, Maggiore derived γ = 8π. It is not clear, however, if these arguments can be extended to more general geometries. Whether Hod’s idea is correct or not, the temptation to picture QNMs as a bridgehead to conquer the fortified realm of QG remains, nonetheless, strong.

In an attempt to motivate our interest in QNMs, we revisited some topics and left out many more. To mention only two: connections with the AdS/CFT correspondence (QNMs of asymptotically anti de Sitter BHs correspond to poles of thermal Green’s functions in a conformal quantum field theory living on its boundary); and the possibility of using QNMs of compact starts to constrain their equations of state. The rest of the thesis will be devoted to a formal treatment of gravitational perturbation theory: The concrete case of the ringdown following a BBH merger will remain in the back of our heads, but our exposition will stay largely general and be applicable to contexts different from GW phenomenology. By including higher-order metric perturbations, we aim at improving the linear approximation to Einstein’s equations; we will put investigate the possibility of resumming part of the perturbative series, thus improving the acces to the non-linear regime of GR.

A plan of the following chapters follows. In Chapter 2 we will describe the framework for dealing with gravitational perturbations over a background in the context of GR, including the meaning of gauge freedom; we will then specify our treatment to the fairly simple case of a spherically symmetric vacuum background, quickly derive Schwarzschild solution—in order to fix notations—and discuss at large its linear perturbations. Chapter3will be devoted entirely to second order perturbations. Chapter4 will contain our original contribution to the subject, as we will propose our resummation method. Finally, Chapter 5will close the thesis by reviewing our main results and speculating on future developments.

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2

Perturbations over a

background

Perturbation theory is the master tool in physics to gain information about a system when the equations that describe it cannot be solved exactly or when the solution, though known, is too intricate to be intelligible. The latter might at times be the case for GR, but the former most definitely is: Einstein’s equations, though only second order in derivatives, involve both the spacetime metric and its inverse and are therefore highly nonlinear.

Some solutions, derived imposing a high degree of symmetry, are known but do not suffice to describe the physics of systems in which the symmetry is only approximate. One can start from these ‘background’ solutions and study the dynamics of small departures therefrom. If only aiming for an approximate description of the physical phenomena, one can consistently drop terms whose contribution is negligible at the required level of precision, thus obtaining significantly simpler equations.

The definition of perturbations in the presence of gravity, however, is in itself ambiguous. Indeed, the perturbed spacetime and the background are effectively two different manifolds and the invariance of the theory under diffeomorphisms entails that a prescription to identify points between the two manifolds must be given: such assignment, which is in principle arbitrary, is called a gauge choice. Since the physics must not dependent on any arbitrary choice, one must be able to relate results derived in different gauges so as to render said ambiguity harmless.

The following section (2.1) will be devoted to putting these ideas into the rigorous language of differential geometry. Later on we will restrict to the particular case we are interested in: perturbations of a Schwarzschild BH (2.2). We will write down the dynamical equations satisfied by linear perturbation and describe the space of their solutions. For completeness, we will also provide few examples of other interesting special cases (2.3).

2.1 Generalities of gravitational perturbation theory

At the heart of the perturbative expansion lies the identification of a parameter, in the fol-lowing, whose powers order the terms in the series, and therefore keep track of the degree of approximation. Such parameter might arise directly from the physical problem at hand as a dimensionless combination of the relevant dimensionfull quantities; more often, and always for what concerns this thesis, it will be merely formal and can be safely removed at the end of the calculations.

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All quantities of interest, such as matter fields and the metric, will be Taylor-expanded in powers of . Since we always assume that the differentiable structure on the spacetime manifold is that compatible with the metric, however, such expansion should somehow affect the geometry. First of all, hence, one must address the following question: What does it mean to Taylor-expand a tensor field? Recall that tensors defined at different points belong to different tangent spaces, and their difference is not defined. For basic definitions we refer to Hawking and Ellis [45].

In what follows, we will follow closely to [23]. We will always assume sufficient regularity to make our statements well defined, without bothering spelling out our hypotheses in detail.

Let M be a differentiable manifold and ξ a vector field on M. The vector field generates a one-parameter group of diffeomorphisms (diffeo’s)—i.e. a flow—

φ : R × M → M

(, p) 7→ φ(p) (2.1)

with φ₀(p) = p ∀p ∈ M.

Consider now a tensor field T on M, e.g. the metric or any matter field. We can compare the values that the field takes at different points using the pullback under φ. The new tensor field φ∗T , evaluated at a point, can be Taylor-expanded in powers of and in terms of the original tensor field T evaluated at the same point:

φ∗T = +∞ X n=0 k n!£ n ξT, (2.2) so formally φ∗_λ = exp(λ£_ξ).

Note that when the tensor field is a set of scalar functions coordinatising the manifold we get x0µ(p) := φ∗xµ(p) = xµ(φ(p)) = xµ+ ξµ+ 2 2∂νξ µ_ξν_{+ . . .} ! (p) (2.3)

i.e. a standard infinitesimal change of coordinates, pushed to second order (“infinitesimal point transformation” in the parlance of the cited reference).

This procedure can be extended by composing several flows φ(1), φ(2), . . . , φ(k), . . . —generated by ξ₍₁₎, ξ₍₂₎, . . . , ξ_(k), . . . —in the following, peculiar way:

Φ= · · · ◦ φ(k)k k!

◦ · · · ◦ φ(2)₂

2

◦ φ(1) ; (2.4)

the authors of [23] call this a knight diffeomorphism, with a terminology inspired by the game of chess. The resulting Taylor expansion of T is

Φ∗T = ∞ X l1=0 ∞ X l2=0 · · · ∞ X lk=0 . . . l1+2l2+···+klk... 2l2. . . (k!)lk. . . l₁!l₂! . . . l_k! . . .£ l1 ξ(1)£ l2 ξ(2). . . £ lk ξ(k). . . T. (2.5)

It is a fact that any one-parameter family (not group) of diffeo’s can be written as a family of knight diffeo’s, in general of infinite order. If one wishes to truncate the expansion at O(n_), then knight diffeo’s approximate any diffeo up to that order. The formalism outlined above is thus very general.

To recap: Given a family of diffeo’s, we have a natural definition of Taylor expansion. The reason why we need to introduce such a family will be elucidated in the following subsection.

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2.1.1 The perturbative expansion

Consider a ‘background’ manifold M₀ populated by a metric g₀ and some matter fields collec-tively denoted as τ0. To represent deviations from the background, we introduce a whole family of manifolds {(M, g, τ)}, all diffeomorphic to one another. The metric and the matter fields will satisfy, on each manifold, a set of equations of the form

E[g, τ] = 0. (2.6)

The situation is most easily described introducing a new manifold N , one dimension larger than M and foliated by manifolds diffeomorphic to it, so that N = M × R and each copy of M is labeled by the corresponding value of ∈ R. Tensor fields on the M’s are naturally lifted to tensor fields in N .

Now it is natural to think of the foils as connected through a one-parameter family of diffeo’s ϕ : N → N such that ϕ|M0 : M0 → M. This is a flow on N , so we might as well give the

vector field on N that generates it—let it be X.

We pause to make a remark: The choice of ϕ, the diffeo that connects the background to one of its perturbed copies, is completely arbitrary. This diffeo is indeed a gauge, as hinted to in the opening of this Chapter, and a change in the choice of ϕ is a gauge transformation.

Being ϕ a flow, one can apply the machinery developed above to a generic tensor field T defined on M: One can pull it back onto the background and Taylor-expand it. The difference ∆T:= ϕ∗T |_M 0 − T0, in particular, is: ∆T = ∞ X n=1 n n!δ n_T, _{where δ}n_{T :=} dnϕ∗T dn _=0,M 0 . (2.7)

Note that ∆Tas well as δnT are defined on M0, hence the perturbations are fields propagating on the background.

For perturbations of the metric we will use a notation inspired by tradition and consistent with extant literature on higher-order gravitational perturbation theory, e.g. [19, 20, 22]; in components: g_µνfull= g_µνbkg+ ∞ X n=1 n n! {n} hµν; (2.8)

henceforth we will drop the suffix ‘-bkg’ and denote the linear perturbation simply by hµν. We will mostly consider vacuum spacetimes, but whenever matter fields will be present their perturbations will be written in an analogous way.

The advantage of the perturbative expansion is that eq. (2.6) generates on the background a hierarchy of equations for the perturbations. Such equations are much simpler than the original. In particular, at first order they are of the form

L[h,{1}δτ ] = 0 (2.9)

with L linear; at second order, one finds

L[{2}h,{2}δτ ] = S[h,{1}δτ ], (2.10) where L is the same linear differential operator as in eq. (2.9) while S is built out of first-order perturbations only and thus acts as a source driving the second-order perturbations.

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Component expressions in pure GR In the absence of matter, eq. (2.6) written in com-ponents is the system of Einstein’s equations

Rµν− 1

2gµνR = 0. (2.11)

They can be derived by varying Einstein-Hilbert’s action with vanishing cosmological con-stant, what we will call with a slight abuse of terminology ‘standard GR’: More general possibil-ities are of course possible, and we will consider them later on as examples; but since the focus of this thesis is on GR we will make some relevant considerations here. As is well known, eq. (2.11) is equivalent to

Rµν = 0; (2.12)

the explicit expression of the linear operator L in eq. (2.9) in components can thus be found by linearising the Ricci tensor. After a fair amount of algebra (see for instance Chapter 1 in [63]), one finds δRµν = (Lh)µν = − 1 2[∇ α_∇ αhµν− ∇α∇µhαν − ∇α∇νhµα+ ∇µ∇νhαα] . (2.13) At second order, the nonlinear term in the expansion of the Ricci—i.e. minus the ‘source’ in eq. (2.10)—is rather given by

−{2}Sµν := 1 2 " − ∇αhβ_β∇αhµν+ 2∇αhµν∇βhβα+ 2hαβ∇α∇βhµν− 2hαβ∇µ∇βhαν − 2hαβ∇_ν∇_βhµα− 2∇αhνβ∇βhαµ+ 2∇βhνα∇βhαµ+ ∇αhββ∇µhαν − 2∇_βhβ_α∇µhα_ν + 2hαβ∇ν∇µhαβ+ ∇µhαβ∇νhαβ+ ∇αhββ∇νh α µ − 2∇_βhβ_α∇νhαµ # = 0. (2.14)

Note that, here and above, ∇ is the background covariant derivative. Expression (2.14) in itself is not particularly informative: It is worth noticing, however, that it only contains terms of the form h∇∇h and (∇h)(∇h); in particular, it is quadratic in the firs-order perturbation.

Expressions (2.13) and (2.14) have been (re)derived using the xAct bundle of Wolfram Math-ematica packages [66,67,21], a powerful computer algebra framework for tensor calculations. 2.1.2 Gauge transformations in perturbation theory

Back to eq. (2.7), it is clear that the definition of the perturbation ∆T is dependent on the particular diffeo ϕ or, equivalently, on the vector field X that generates it—i.e. it depends on the gauge. Choosing a different gauge, generated by a different vector field Y , would lead, in general, to a different perturbation: Call it ∆YT and rename the previous one as ∆XT. The two are physically equivalent, hence it must be possible to relate one to the other.

Call TX,Y := T₀+ ∆X,YT. It may well be that TX = TY

for all X, Y , then we will say that T is completely gauge invariant. This is a very stringent con-dition though, since it requires that T be a combination of Kronecker’s deltas with -dependent coefficients. This condition can be weakened by demanding that

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for all X, Y and k ≤ n for some n, in which case we will say that T is gauge invariant up to order n.

In general, if T is not gauge invariant, we can relate its power expansions in different gauges as follows. We define a diffeo Ψ: M₀→ M₀ such that Ψ:= φX₋◦ φY

. Then

TY = Ψ∗TX; (2.15)

this last expression can be Taylor-expanded as shown before. Explicitly, to second order: TY = TX + £_ξ (1)T X + 2 2 £2_ξ (1)+ £ξ(2) TX + . . . ; (2.16) here ξ₍₁₎ and ξ₍₂₎ are the first two generators of Ψ, i.e. of the gauge transformation.

Further note that

δTY − δTX _{= £} ξ(1)T0 δ2TY − δ2TX = (£_ξ₍₂₎+ £2_ξ (1))T0+ 2£ξ(1)δT X .. . (2.17)

Gauge invariance of the metric For perturbations of the metric, the theory at any given order n possesses an invariance with respect to a group of local—in the lingo of field theory, i.e. point-dependent—transformations; such group is smaller than the group of diffeo’s on the spacetime: In this sense, perturbation theory causes the breaking of general covariance, the cornerstone of GR, to a subgroup thereof.

At linear order, such transformations are of the form

h0_µν = hµν+ £ξgµν = hµν+ ∇µξν+ ∇νξµ, (2.18) where ξ is a ‘small’ vector field in the sense outlined above (recall that in our notation g is the background metric!). To this order, h_µν and h0_µν describe the same physics.

At O 2

, the perturbation transforms as {2}

h0_µν = {2}hµν+ £_ξ0gµν+ £2_ξgµν+ 2£ξhµν; (2.19)

here ξ is the vector field that fixes the gauge at first order and ξ0 is a new, independent vector field. In other words, the gauge at O 2

can be fixed in any way one chooses, but the explicit transformation will depend on the choice made at first order. Essentially the same pattern repeats itself at higher orders.

To avoid possible confusion, it is convenient to think of perturbation theory as a sequence of steps: First deal with the linear order, fix the gauge and—if that is the case—solve the equations; then study the second order, fix the gauge anew and solve the same equations as at O() but with a source that is completely determined by the background and the first-order perturbation; and so on for higher orders.

2.2 Linear perturbations of a Schwarzschild black hole

This thesis focuses on the simplest system that could have been considered: a non-spinning BH. Any more realistic (spinning) or, from a theoretical point of view, more interesting (hairy BHs, BH mimickers, etc.) situation would introduce considerable complications that would hinder the effectiveness of our exposition. We will therefore devote this section to explore the well-known physics of a Schwarzschild BH ringdown: We will largely build on top of it in subsequent chapters.

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2.2.1 The Schwarzschild solution

Karl Schwarzschild was the first, in 1915, to investigate spherically symmetric solutions to Ein-stein’s equations1. In modern terms, spherical symmetry means that the spacetime possesses three independent Killing vectors Vi that satisfy the algebra of rotations so(3)

[Vi, Vj] = ε_ijkVk; (2.20) equivalently, the spacetime M can be foliated into two-dimensional spheres: M = M₂× S₂.

This turns out to be a very stringent requirement, so much so that—in vacuum—it suffices to single out an essentially unique (up to an integration constant) solution to Einstein’s equations. Indeed, if one chooses coordinates XI with I = (0, 1) on M₂ and Yi with i = (0, 1) on S₂, spherical symmetry ensures that the metric has the following form:

ds2 = gIJ(X)dXIdXJ+ α(X)γij(Y )dYidYj (2.21) where γ_ij can be taken to be the round metric on the sphere and α is a function of the coordinates on M2. One can further choose coordinates so as to make the cross term g12 vanish.

Plugging this ansatz into Einstein’s equations in the absence of matter leads to the following solution:

ds2 = −f (r)dt2+ 1 f (r)dr

2_{+ r}2_(dθ2_{+ sin}2_θdφ2_). _(2.22) Here θ, φ are angular coordinates on the sphere, so that rotations on M act as translations in θ and φ and leave t, r unchanged; t is the coordinate time and r the coordinate radius, meaning that the spherical surface at constant r has area 4πr2. (Component indeces will take values in both the sets {0, 1, 2, 3} and {t, r, θ, φ} indistinctly.) The function f is

f (r) = 1 − 2M

r , (2.23)

where M is the mass of the spacetime—which enters the large-r Newtonian potential. As mentioned before, this solution is unique and parametrised by M only. Moreover, it is static: This non-trivial result (along with the property of asymptotic flatness) is the content of Birkhoff’s theorem.

The singularity at r = 2M is only due to the choice of coordinates. For instance in the so-called tortoise coordinate

r∗ = r + 2M ln(r/2M − 1), (2.24)

defined by dr∗ = dr/f (r), the metric reads

ds2 = f (r(r∗))[−dt2+ dr2_∗] + r2(r∗)[dθ2+ sin2θdφ2];

now r = 2M corresponds to r∗ = −∞ but none of the metric components diverges in that limit. The lightlike surface r = 2M is the horizon. Kruskal-Szekeres coordinates cover both regions, outside and inside the horizon, at the same time, thus proving the regularity of the (maximally-extended) spacetime all the way down to r = 0. The origin r = 0, on the contrary, is a physical singularity as can be seen by computing for instance the Kretschmann’s scalar:

RµνρσRµνρσ = 48M2

r6 .

1

To be fair, the solution that Schwarzschild published is almost, but not exactly, the one that nowadays carries his name; indeed, he required continuity of the metric coefficients everywhere, thus deriving a solution that differs from eq. (2.22) by a change of coordinates. Hence, in his ‘radius’ R there is no hint to an horizon.

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A peculiar feature of BHs’ spacetimes is the presence of a photon sphere, i.e. a surface on which light travels on closed orbits; for Schwarzschild, it is located at r = 3M .

There are only six independent connection coefficients: Γt_rt= −Γr_rr= −f 0 2f = M r(r − 2M ) Γ r tt= − 1 2f f 0 = M (r − 2M ) r3 Γr_φφ= sin2θΓr_θθ= −r sin2θf = − sin2θ(r − 2M ) Γθ_rθ = Γφ_rφ= 1/r Γθ_φφ= − sin θ cos θ Γφ_θφ= cos θ

sin θ,

(2.25)

f0 standing for the derivative of f with respect to its argument. 2.2.2 Regge-Wheeler and Zerilli equations

Studying linear perturbations of the Schwarzschild background means specifying eq. (2.13) using the connection coefficients (2.25). As anticipated above, this approach was pioneered by Regge and Wheeler in a truly remarkable publication [76] aimed at assessing the stability of the Schwarzschild solution; although the question turned out to be extremely subtle and was settled only years later, their work’s legacy has been profound and far-reaching.

An important leap forward was the realisation that the angular and spatio-temporal depen-dencies decouple if the perturbation is written in an appropriate basis. This is no surprise, since (2.13) is akin to a wave equation, though on a curved background—in fact, with appropriate gauge choices it can be put in the canonical ‘_{h = 0’ form. Experience with e.g. electrostatics} of spherically symmetric sources suggests a spherical harmonics-like decomposition.

2.2.2.1 Tensor spherical harmonics

The metric perturbation is a symmetric, rank-two, Lorentz tensor. Hence, it carries ten components— all independent prior to the gauge choice—having different transformation properties with re-spect to spatial rotations. Schematically:

h =      S S V1 V2 S S V1 V2 V1 V1 T11 T12 V2 V2 T21 T22      , (2.26)

where S stands for scalar, Vi are the components of a two-vector and Tij make up a symmetric two-tensor. In group-theoretic terms, h belongs to a reducible representation of SO(3), whose irreducible components are the scalar, vector and tensor representations.

To decompose h, therefore, one needs first to construct the spin-one and spin-two counter-parts of the usual Y_`m(θ, φ). A complete set for vectors and symmetric rank-two tensors on the sphere can be written by simply combining the scalar spherical harmonics with the spin-one and spin-two wave functions and the appropriate Clebsch-Gordon coefficient. In group-theoretic terms, this amounts to writing a representation with total angular momentum j, of dimension 2j + 1, in terms of the tensor product of irreducible representations of the orbital angular mo-mentum and the spin. The resulting tensors are called pure orbital rank-two tensor harmonics. Many alternatives exist, though—see Thorne [83] for a review. Indeed, pure orbital harmonics are neither transverse nor longitudinal with respect to the radial direction, and are therefore not well suited for describing radiation. We choose a different set, named after Zerilli and akin to those employed by Regge and Wheeler, in the notation of Maggiore [63,65].

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This ten independent tensors will be labeled by an index a taking values in {L0, T 0, tt, Rt, Et, E1, E2, Bt, B1, B21}.

In spherical coordinates they are written as

(Ta_`m)_µν = ca(r) (ta_`m)_µν, (2.27) where cL0= ctt= 1, cT 0= r2/√2, cRt = 1/√2, cEt= cE1= −cBt= −cB1= r/ q 2`(` + 1), cE2 = cB2= r2 s 1 2 (` − 2)! (` + 2)!; (2.28)

and (we suppress µν)

ttt_`m=      1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      Y`m(θ, φ), tRt_`m =      0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0      Y`m(θ, φ), tL0_`m=      0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0      Y`m(θ, φ), tT 0_`m =      0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 sin2(θ)      Y`m(θ, φ), tEt_`m =      0 0 ∂θ ∂φ 0 0 0 0 ? 0 0 0 ? 0 0 0      Y`m(θ, φ), tE1`m =      0 0 0 0 0 0 ∂θ ∂φ 0 ? 0 0 0 ? 0 0      Y`m(θ, φ), tBt_`m =      0 0 _{sin θ}1 ∂φ − sin θ∂θ 0 0 0 0 ? 0 0 0 ? 0 0 0      Y`m(θ, φ), tB1`m =      0 0 0 0 0 0 _{sin θ}1 ∂φ − sin θ∂θ 0 ? 0 0 0 ? 0 0      Y`m(θ, φ), tE2_`m =      0 0 0 0 0 0 0 0 0 0 W X 0 0 ? − sin2_θW      Y`m(θ, φ), tB2_`m =      0 0 0 0 0 0 0 0 0 0 −_{sin θ}1 X sin θW 0 0 ? sin θX      Y`m(θ, φ), (2.29) having called X = 2∂θ∂φ − 2 cot θ∂φ, W = ∂_θ2 − cot θ∂θ_sin12_θ∂_φ2; ? stands for the components

determined by symmetry.

Beware that the labels `, m really refer to the total angular momentum: we chose this notation in stead of the clearer but heavier j, jz. Such harmonics are indeed not eigenfunctions of the orbital angular momentum, but only of the spin and of the total angular momentum. Only the harmonics E2 and B2 are transverse to the radial direction and therefore contribute to radiation at infinity—which is typically described in a transverse traceless (TT) gauge; the others are longitudinal.

These harmonics are orthonormal in the sense that

Z dΩηµρηνσ(Ta_`0_m0)∗_µν Tb_`m ρσ = ε a_δab_δ ``0δ_mm0, (2.30)

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where εa = −1 for a = Rt, Et, Bt and +1 otherwise. Moreover, they are complex but inherit the conjugation properties of the scalar harmonics:

(Ta_`m)∗_µν = (−1)m Ta_{` −m}

µν; (2.31)

therefore, if hµν is real, h∗_`m = (−1)mh` −m.

Therefore, the metric perturbation can be decomposed as hµν(t, r) =X

a

X

lm

ha_`m(t, r)(ta_`m)µν(θ, φ); (2.32)

for other rank-two symmetric tensors that might appear (e.g. for sources) we intend to use the same decomposition.

This harmonics possess an additional interesting property under a parity transformation, whereby θ → π − θ, φ → φ + π: The harmonics of type B take up a sign (−1)`+1 while all the others take up (−1)`. Thus, we witness the separation into two sectors—which we will call axial, the former, and polar, the latter2—that at linear level never mix. The treatment of axial perturbations is significantly simpler and is the one given by Regge and Wheeler in [76]; the polar sector was not dealt with until the work of Zerilli [93] in 1970, thirteen years after Regge and Wheeler’s! We will consequently write

hµν = haxµν+ hpolµν. (2.33)

2.2.2.2 Dynamical equations

Zerilli’s harmonics are not eigenfunctions of the operator (2.13), but this behaves as a scalar in the sense that it does not change the `, m content of the tensor it acts on. In other words, its effect, as far as the angular dependency is concerned, is that of mixing components with different a but the same `, m. Hence, the angular variables are effectively separated.

For future reference, we consider the non-homogeneous version of (2.13), bearing in mind that in the vacuum, linear case the source vanishes:

(Lh)_µν = S_µν. (2.34)

Regge-Wheeler gauge A second important leap in Regge and Wheeler’s work was the de-termination of a fortunate gauge choice that greatly simplifies the expression of h_µν in (2.32) and hence eq. (2.34).

Such gauge choice has been known as Regge-Wheeler (RW) gauge ever since. For the axial sector, it amounts to eliminating the component hB2_`m, so that no second derivatives of Y_`mappear in the expansion of hax_µν; for the polar sector, it consists in eliminating hE2_`m, hE1_`m and hEt_`m, so that the expansion of hpol_µν is free from derivatives of Y_`m.

Details, such as the explicit expression of the gauge-fixing vector, can be found in Maggiore

2

Several alternatives to the axial/polar terminology exist, for instance: magnetic/electric and odd/even—we particularly dislike this last one, because strictly speaking axial/polar harmonics can be both odd (for ` even/odd) and even (for ` odd/even) functions of the coordinates.

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[65], Chapter 12. The result of gauge fixing is reported here for clarity: hax_µν =      0 0 hBt_`m(t, r)_{sin θ}1 ∂Y`m_∂φ −hBt `m(t, r) sin θ ∂Y`m ∂θ 0 0 hB1_`m(t, r)_{sin θ}1 ∂Y`m ∂φ −h B1 `m(t, r) sin θ ∂Y`m ∂θ ? ? 0 0 ? ? 0 0      µν hpol_µν =      htt_`m(t, r)Y_`m hRt_`m(t, r)Y_`m 0 0 hRt_`m(t, r)Y`m hL0`m(t, r)Y`m 0 0 0 0 hT 0_`m(t, r)Y_`m 0 0 0 0 sin2θhT 0_`m(t, r)Y_`m      µν. (2.35)

Axial sector For the axial sector in RW gauge, eq. (2.34) gives rise to the following system (we drop the multipolar indeces):

1 2     −f ∂2 r− h 2 rf 0₋`(`+1) r2 i f∂r+2_r ∂t 0 −1 f ∂r−2 r ∂t _f1∂_t2+(`−1)(`+2)_r2 0 1 f∂t −f 0_{− f ∂} r 0        hBt hB1 0   =    sBt sB1 sB2   . (2.36)

Note that it is second-order in time and radial derivatives, and that the coefficients are functions of r only. The angular dependency enters through the multipole index ` only; in particular, each ` is completely degenerate in m—a direct consequence of spherical symmetry.

The three equations are clearly redundant for the determination of the two unknown functions hBt and hB1; indeed the matrix above becomes invertible only if appropriately restricted. In particular, one can show that in the homogeneous case the time derivative of the first equation is automatically satisfied upon use of the second and third.

The third equation gives

∂thBt= f (r)[+2sB2+ ∂r(f (r)hB1)], (2.37) which can be used in the second:

∂2_thB1− ∂r[f ∂r(f hB1)] + 2f r ∂r(f h B1₎ + f (r)`(` + 1) − 2 r2 h B1_{= 2}_{f (r)s}B1₊_∂ r− 2 r (f sB2) . (2.38) This form of the equation does not yet render manifest its simplicity. Let us introduce a RW function Z_`A(t, r) := 1 rf (r) " −hB1_`m−1 2 ∂rh B2 `m− 2hB2_`m r !# RW = −1 rf (r)h B1 `m (2.39)

(the combination in square brackets is gauge invariant and reduces to the last expression in RW gauge) and resort to the tortoise coordinate defined in (2.24). We get:

(∂2_r_∗− ∂2 t)Z`mA (t, r) − V`A(r)Z`mA (t, r) = S`mA (2.40) where V_`A(r) := f (r) _{`(` + 1)} r2 − 6M r2 (2.41) is the Regge-Wheeler potential; and

S_`mA = 2f r f (r)sB1_`m+ ∂r−2 r (f sB2_`m) . (2.42)

Eq. (2.40) is the Regge-Wheeler equation. (These equations, derived by the author, agree with those of Maggiore [65].)

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Polar sector For the polar sector, the system analogous to eq. (2.36) involves seven equations. It is possible to introduce a function similar to (2.39). This time, however, it is first necessary to take the Fourier transform with respect to time; the Zerilli function can than be defined as

Z_`P(ω, r) := 1 λr + 3Mh T 0 `m(ω, r) + rf (R) iω(λr + 3M )h Rt `m(ω, r), (2.43) where λ := (` − 1)(` + 2)/2. This function satisfies

∂_r2_∗Z_`mP (ω, r) + [ω2− VP ` (r)]Z`mP (ω, r∗) = S`mP (ω, r∗) (2.44) with V_`P(r) := f (r)2λ 2_{(λ + 1)r}3_{+ 6λ}2_{M r}2_{+ 18λM}2_{r + 18M}3 r3_{(λr + 3M )}2 (2.45) and SP_`m= f (r)∂r _{rf (r)} iω(λr + 3M )(rs Rt `m+ 2sEt`m) − 2s Et `m iω f (r)[λ(λ + 1)r2+ 3M rλ + 6M2] r(λr + 3M )2 −s Rt `m iω λr2_f2_(r) (λr + 3M )2 + (rs L0 `m+ 2sE1`m) rf2_(r) λr + 3M − 4f (r) r s E2 `m (2.46) The potential is referred to as Zerilli potential while the equation is the Zerilli equation.

Some properties of the RWZ equation The RW (2.40) and Zerilli (2.44) (RWZ) equations, when written in Fourier space with respect to t and when all sources vanish, have the form of a Schrödinger’s equation and describe, essentially, a scattering problem off a potential: It is fairly remarkable that the complicated dynamics of gravitational perturbations on a curved background could be simplified to this extent.

The two ‘master functions’ Z_`A,P account for the two physical degrees of freedom of grav-itational radiation and are sufficient to reconstruct all the components of the perturbation— although the 1/ω factor in Z_`P gives rise to non-local contributions when Fourier-transformed back to t-space. This means that the inverse of the operator L in eq. (2.13) is entirely determined in terms of the Green’s functions of the RW and Zerilli equations.

As noted before, the only reminiscence of the angular dependency is the index ` that appears in the potentials and labels the RWZ functions. In particular, there is no dependency on m whatsoever: In the space of solutions, each subspace with given ` is (2` + 1) times degenerate. Thus, for the sake of computational simplicity, one could have treated the simpler m = 0 case only.

The two equations, in fact, differ only for the potentials. Despite their different functional form, however, these are numerically very similar—see fig. (2.1)—and, most importantly, they have the same qualitative behaviour. Both are always positive and go to zero at infinity and at the horizon; specifically:

V_`A,P ∼ 1/r2 as r → ∞ (r∗→ +∞) V_`A,P ∼ 1/r as r → 2M (r∗ → −∞).

(2.47)

As a consequence, there can be no bound state. They both have a single maximum, located in the vicinity of r = 3M (Schwarzschild’s photon sphere).

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2

3

4

5

6

7

0.0

0.2

0.4

0.6 V

A, P photon sphere

= 2

= 3

= 4

RW

Zerilli

2

3

4

5

6

7 r/M

0.01

0.00

0.01 (V

A

V

P

)

= 2

= 3

= 4

Figure 2.1. RW and Zerilli potentials for a few values of ` (top panel), as functions of r/M ; difference

between RW and Zerilli potential for the same few values of ` (bottom panel). The vertical axis is in units of 1/M2.

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In passing, we note that a similar treatment for perturbations of different spin—scalars and vectors—leads to master equations of the form of RW’s, with essentially the same potential. One can write

V`(r) = f (r) _{`(` + 1)} r2 + σ 2M r3 (2.48) with σ = 1−s2and s = 0, 1, 2 for scalar, vector (electromagnetic waves) and tensor perturbations respectively.

2.2.3 Schwarschild quasinormal modes

Eq.s (2.40) and (2.44) are hyperbolic second-order partial differential equations; along with suit-able boundary and/or initial conditions, they form a well-posed Cauchy problem—equivalently, in Fourier space they constitute a Sturm-Liouville problem.

In this context, however, the domain of interest is non-compact and one therefore chooses more physically motivated ‘asymptotic’ conditions, namely that the solutions behave in some particular way at infinity and at the horizon. Indeed, let

φ00(ω, r∗) + [ω2− V (r∗)]φ(ω, r∗) = 0 (2.49) be any of the Fourier-transformed RW and Zerilli eq.s (we drop the multipole indes `). Since V → 0 as r∗ → ±∞, in these limits (2.49) reduces to a free equation, solved by e±iωr∗_—

recall that, in our convention, the inverse Fourier transform brings about a factor e−iωt, hence solutions with the plus (minus) sign correspond to out-going (in-going) plane waves. Thus, a general solution should be asymptotic to a superposition of such waves in the large |r∗| regions.

One can safely assume, however, that no waves come out from the BH. Hence we take φ(ω, r∗) ∼ e−iωr∗ _{as r}

∗ → −∞. (2.50)

Typically, one further requires no incoming radiation from infinity. This is a well motivated assumption when studying the ringdown subsequent to a BBH merger, or more generally the intrinsic properties of the BH spacetime; but may not be valid in different circumstances such as when dealing with the scattering of GWs by a BH. Anyhow, we will require

φ(ω, r∗) ∼ e+iωr∗ _{as r}

∗ → +∞. (2.51)

Thus, we demand solutions to be out-going at both boundaries, meaning that radiation always leaves the domain of interest.

We will show momentarily that boundary conditions (2.50) and (2.51) are satisfied when ω takes up some discrete values, the QN frequencies. As we anticipated in the Introduction, such frequencies are complex and their imaginary part has always the sign to produce damping— negative, with our convention on the Fourier transform.

This fact has an unexpected consequence, though. In the large |r∗| limit, the solution φ(ω, r∗) ∼ ei Re ω|r∗|_{· e}+|Im ω||r∗| _(2.52)

diverges exponentially. The physical interpretation of QNMs should therefore be considered carefully: They are not stationary states describing the system over all of space at any given time; rather, they represent how the system responds to a perturbation that is switched on at a specific instant of time, when observed at a sufficiently late time and at a specific point in space. Moreover, one would like to represent any initial perturbation as a superposition of QNMs, in analogy to acoustic theory, but this does not seem to be possible: It is hard to think how a bounded initial data could be written combining divergent basis functions. Indeed, as we will show in due time, the QNMs do not form a complete set.

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QNMs in Laplace picture A better understanding of QNMs is achieved by recasting RWZ equation in the language of Laplace transform (we largely follow the clear explanation in Mag-giore [65], Chapter 12)

φ(s, r∗) :=

Z +∞

0

dte−stφ(t, r∗) (2.53)

In the definition s ∈ R, although for bounded functions the transform admits an analytic con-tinuation in Re(s) ≥ 0; under this condition, the transfrom can be inverted (for t ≥ 0):

φ(t, r∗) = Z +i∞ −i∞ ds 2πie +st_{φ(s, r∗}₎ _(2.54)

(the integration path runs parallel to the Im(s)-axis in the right half plane). The equivalent of eq. (2.49) is

φ00(s, r∗) − [s2+ V (r∗)] = J (s, r∗) (2.55) where J (s, r∗) is determined by initial conditions (it is not an external source):

J (s, r∗) = − [sφ(t, r∗) + ∂tφ(t, r∗)] _t=0. (2.56) The solution φ(s, r∗) can be determined by means of a Green’s function. Given any two independent solutions φ±(s, r∗) of the homogeneous equation, a Green’s function can be built as follows

G(s; r∗, r_∗0) = 1

W (s)φ−(s, r <

∗)φ+(s, r>∗) (2.57) where r<_∗ (r<_∗) is the smallest (greatest) between r∗ and r∗0, while W is the Wronskian of the two solutions

W (s) := φ−(s, r∗)φ0₊(s, r∗) − φ0−(s, r∗)φ+(s, r∗) (2.58) (which is independent on r∗).

Note that there are as many Green’s functions as possible choices of the two homogeneous solutions. As we will show momentarily, however, demanding that the solution be bounded in r∗ selects univocally φ±. Thus

φ(s, r∗) = 1 W (s)

Z +∞

−∞

dr0_∗φ−(s, r_∗<)φ+(s, r>∗)J (s, r0∗). (2.59) The antitransform to real space can be computed by closing the contour of integration with a semicircle in the left half plane. If φ± were analytic in s, the integral would be given by two contributions: one due to the semicircle at infinity, the other coming from the residues of the integrand’s poles enclosed by the contour. Such poles would come from zeros of the Wronskian, which form a discrete set—this is the mathematical origin of QNMs.

One of the two solutions is indeed analytic, but the other contains an essential singularity at s = 0 and a branch cut corresponding to the negative real s-axis. Consequently, a third contribution must be taken into account. Thus, the solution to the RWZ equation is given by

φ(t, r∗) = −

Z +∞

−∞

dr0_∗[∂tG(t; r∗, r0∗)φ(t = 0, r∗) + G(t; r∗, r∗0)∂tφ(t = 0, r∗)], (2.60) where G(t; r∗, r0∗), the antitransformed version of G(s; r∗, r∗0), can be schematically decomposed as:

The Perturbative Structure of Schwarzschild Black Hole Ringdown

Hole Ringdown

Abstract

Notations and Terminology

Contents

1

Introduction

2

Perturbations over a

background

2.1

Generalities of gravitational perturbation theory

2.2

Linear perturbations of a Schwarzschild black hole

2

3

4

5

6

7

0.0

0.2

0.4

0.6

V

= 2

= 3

= 4

RW

Zerilli

2

3

4

5

6

7

r/M

0.01

0.00

0.01

(V

V

)

= 2

= 3

= 4