Black hole information recovery from gravitational waves

Naively, black holes at rest are featureless and seem to destroy information as they evaporate. In the original paradox, Hawking showed the density matrix of the Hawking radiation was diagonal [13], which he noted to be information-theoretically inconsistent with an initial 'pure' quantum state. Therefore before discussing gravitational waves, we begin by an introduction to quantum information conservation and motivate the study of black hole perturbations.

Compared to wavefunctions, density matrices are generalized representations of states. States with well-defined wavefunctions, called 'pure states', have N × N density matrices $\rho = \vert \psi\rangle\langle \psi\vert$, where $\vert \psi\rangle = \sum_i a_i\vert n_i\rangle$ is the wavefunction in some basis $\vert n_i\rangle_{i\in[1,\ldots,N]}$ and N is the dimensionality of the Hilbert space. Off-diagonal elements $\rho_{i\neq j}$ can then appear as the phase coherence of the state. These phases are important as they guarantee the existence of a basis in which we can write $\rho = \unicode{x1D7D9}_{N\times N}\delta_{i,1}$. Pure states, however, are more often characterised using $\text{Tr}(\rho^2) = 1$. More generally, a state with well-defined probabilities but partial or absent phase coherence (e.g. classical statistical ensembles) will exhibit a partial or total basis independent absence of off-diagonal matrix elements and is called a partially or totally 'mixed' state. Mixedness can be quantified as $\text{Tr}(\rho^2)\lt1$. Unitary evolution of the system, such as the time evolution of gravitationally collapsing radiation and/or matter, implies mixedness and vice versa purity remain constant since:

$$\begin{align} \textrm{Tr}(\rho^2)=\textrm{Tr}(U\rho U^\dagger U\rho U^\dagger). \end{align} \tag{ 1 }$$

This implies the mixedness/purity is a measure of information since information is assumed preserved in unitarily evolving systems. Further the Von Neuman entropy of a state, defined as $S_\textrm{vN} = -\textrm{Tr}(\rho \ln(\rho))$, is also a measure of mixedness of the system and remains constant. It is identically zero for a pure state only. But what of the classical entropy of a system described by a pure state which has thermalized? It is non-zero as expected, and in fact can be found as the upper bound in the space of subsystems, which will be mixed, such that $S_\textrm{BH}(\Omega) = \max_{\Omega_0\subset\Omega}S_\textrm{vN}(\Omega_0)$. Indeed, [4, 21] showed that random subsystems from a thermal ensemble (such as blackbody radiation escaping a black hole) are mixed because they are entangled subsystems of a globally pure state. This is why Von Neuman entropy is sometimes called 'entanglement entropy'. This can be illustrated with a simple two particle-two spin state example with states $\vert \psi_\textrm{AB}\rangle = \frac{1}{\sqrt{2}}(\vert +-\rangle-\vert -+\rangle)$, taking the partial trace of the density matrix over the A states such that $\textrm{Tr}_A(AB) = B~\textrm{Tr}(A)$ gives:

$$\begin{align} \rho_\textrm{AB}=\frac{1}{2}\left(\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{matrix}\right)\enspace\longrightarrow\enspace \rho_\textrm{B}=\frac{1}{2}\left(\begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix}\right) \end{align} \tag{ 2 }$$

$$\begin{align} S_\textrm{vN}(\rho_\textrm{AB})=0\enspace\longrightarrow \enspace S_\textrm{vN}(\rho_\textrm{B})=\ln 2, \end{align} \tag{ 3 }$$

where we expressed ρ_AB in the basis $\{\vert -\rangle,\vert -+\rangle,\vert +-\rangle,\vert ++\rangle\}$, and summed over values (+ and −) of $\textrm{A}$ for each value of $\textrm{B}$ in the basis $\{\vert +\rangle,\vert -\rangle\}$ to obtain ρ_B. Note this matches the classical entropy $S_\textrm{cl}(\textrm{AB}) = \ln 2 = S_\textrm{vN}(\rho_\textrm{B})$ since $\textrm{AB}$ has two possible microstates. Since the particles A and B are entangled, the density matrix of a subsystem can only contain partial information. Let us now assume a black hole is a thermalized body made of entangled quanta, which escape during evaporation. Since the global state is pure, as more particles escape the black hole the mixedness of the subsystem of evaporated quanta will eventually start to decrease after the 'Page time' until purity is restored. In the above example, assuming AB evaporates one particle at a time, the detector will initially see $S_\textrm{vN}(\rho_\textrm{vacuum}) = 0$, then $S_\textrm{vN}(\rho_\textrm{A or B}) = \ln 2$ and finally $S_\textrm{vN}(\rho_\textrm{AB}) = 0$. This yield a so-called Page curve. However this is in contrast to Hawking's result which states the evaporation particles are maximally mixed, such that mixedness keeps increasing until the end of the evaporation, and emitting an entropy $S_\textrm{BH}$ (illustrated in figure 1).

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Nevertheless recovering entanglement at late times has led the community to believe the information could be recovered if the black hole could be shown to evolve unitarily from a pure state and was referred to as the 'central dogma' of black hole information in [11]. However, this requires finding correlations between patches of the black hole metric, which could be transferred to the radiation. These correlations would naturally appear in a microscopic model of gravitational degrees of freedom that have become highly entangled through scatterings and eventually thermalizing. This also introduces along with it the idea of thermal metric fluctuations at the horizon, albeit small, as seen in figure 2. This also revisits the classical assumptions of a featureless background behind Hawking's initial calculation. Nevertheless, since the original formulation of the paradox, many have revisited the classical assumption of a purely classical the background metric (e.g. [24–38]).

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These various works have described black hole horizons as a thermal microscopic ensemble in order to recover correlations. Assuming a pure state, the corresponding off-diagonal elements can be seen as quantum fluctuations [16] and we will assume them to be present in rest of this work, although they have been subject of debate [23, 40]. Indeed, similarly to a classical state, a quantum many-body state also exhibits quantum fluctuations in addition to classical fluctuations at late times, when the system has 'equilibrated'. This is defined as the moment the average local entanglement entropy has reached its maximum, or scrambling time, as illustrated in figure 2. The classical fluctuations can be seen as arising from the distribution of diagonal elements (representing probabilities) whereas quantum fluctuations can be seen as arising from the off-diagonal ones. These fluctuations appear due to interactions of the smaller microscopic degrees of freedom of the system. They can be used to recover the initial state in a closed system and therefore contain information. In the context of metrics, fluctuations are called 'hair', and have been studied extensively [41–43]. An important classical result in 'vanilla' general relativity is that asymptotically in time 'black holes have no hair' [44–46]. Black hole hair appears therefore purely in models with quantum features.

As we have established quantum and classical fluctuations in an equilibrated system carry classical and quantum information (often referred to as 'coarse grained' and 'fine grained' respectively), we may study how to recover this information. The more entangled the final state, the more measurements and operations are necessary to recover quantum information (see e.g. [47, 48], although the algorithm may depend on the microscopic physics). After the scrambling time any algorithm to recover the information from fluctuations requires a larger number of steps, contrary to the initial state where the information is more readily available. Furthermore the size of fluctuations in the system reach their minimum after this time, and their size can be well modelled using the Eigenstate Thermalization Hypothesis (ETH), which has experimental and numerical success with a variety of generic systems [16]. The ETH describes local operators as:

$$\begin{align} \hat{O}=f(\bar{E})\delta_{ij} + e^{-S_\textrm{cl}(\bar{E})/2}R_{ij}g(\bar{E},\omega), \end{align} \tag{ 4 }$$

where $(f(\bar{E}),g(\bar{E},\omega))$ are smooth functions of $(\bar{E},\omega) = ((E_i-E_j)/2,E_i-E_j)$ and R_ij is a matrix of random variables with zero mean and order one variance. Quantum fluctuations are therefore highly suppressed, and the classical fluctuations will dominate.

In our case, we are interested in fluctuations before the scrambling time, i.e. at times near when a particle fell in, since they are larger and depend more heavily on the disentangled initial state $\vert \text{infalling particle}\rangle\otimes \vert \text{black hole}\rangle$. In the absence of a theory of quantum gravity, these can be calculated as gravitational waves using classical perturbation theory of a black hole metric with a source, which is exempt of the assumptions of the no-hair theorem ² . The procedure is heuristically illustrated in figure 3. Corrections due to fluctuations of the metric from the scrambled past history of the black hole will still be present, but we may reasonably expect them to be smaller in a variety of scenarios. Indeed, assuming the thermodynamic limit of many microscopic gravitational degrees of freedom (quanta) in thermal equilibrium with temperature $T_\mathrm{BH} = T_\mathrm{Hawking}$, the energy fluctuations of the system can simply be estimated as the standard deviation of a Bose–Einstein condensate, which is $\mathcal{O}(kT_\mathrm{BH} = M_p^2/(8\pi M_\textrm{BH}))$. Meanwhile, the signal energy of early fluctuations of a radially infalling particle has been found to be $\int\vert u(\omega,r)\vert^2d\omega dr \approx 10^{-2} m_0^2/M_\textrm{BH}\ll m_0$ [49, 50] where $u(\omega,r)$ is the GW signal while m₀ and $M_\textrm{BH}$ are the particle and black hole masses respectively (similar dependence on m₀ and $M_\textrm{BH}$ appear for more general trajectories). The recovered information will therefore have an SNR$^2(\propto\chi^2)\approx m_0^2/M_p^2$ where M_p is the Planck mass. Assuming the black hole and gravitational wave system is thermal such that information is spread evenly throughout the system, the quantum information recovered in this (sub)system can be quantified as the change in the Von Neuman entropy when taking the partial trace of the black hole density matrix over the subsystem such that:

$$\begin{align} \rho_{\textrm{BH} \otimes \textrm{GW}}=\left(\begin{matrix} & \vdots & \\ \cdots & \vert \textrm{BH}_i\rangle\vert \textrm{GW}_j\rangle\langle \textrm{GW}_k\vert\langle \textrm{BH}_l\vert & \cdots \\ & \vdots & \end{matrix}\right) \enspace \nonumber\\ \longrightarrow\enspace \rho_\textrm{BH}=\left(\begin{matrix} & \vdots & \\ \cdots & \vert \textrm{BH}_n\rangle\langle \textrm{BH}_n\vert & \cdots \\ & \vdots & \end{matrix}\right), \end{align} \tag{ 5 }$$

$$\begin{align} S_\textrm{vN}(\rho_{\textrm{BH} \otimes \textrm{GW}})=0 \enspace\longrightarrow\enspace S_\textrm{vN}(\rho_\textrm{BH})\gt0, \end{align} \tag{ 6 }$$

where (i, j) and (k, l) run over the Hilbert space of the black hole and the associated gravitational waves, in the columns and rows respectively, and whose states include all possible histories of the black hole given its initial conserved charges of mass, angular momentum, and possible gauge charges. Note this makes the simplest possible assumptions about $\rho_{\textrm{BH}\otimes\textrm{GW}}$. A more accurate estimate would require study of the thermalization and information loss of open quantum systems.

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Further, although the information recovered by the gravitational wave spectrum will be recoverable up to small quantum fluctuations of the metric, the information in these fluctuations can also be retrieved in principle by measuring them. Assuming Hawking radiation arises from the decay of microscopic gravitational degrees of freedom at the horizon, one expects it contains information that is approximately spread out through the horizon and proportional to the size of the fluctuations which is given by the temperature. These fluctuations represent the late time tail of the black hole history [26–28, 30, 33–35, 37, 38, 51, 52]. We may therefore think of information retrieval in two parts: the first is reading large fluctuations in the form of gravitational waves from which information is easily retrievable, and the second is reading the late time tail fluctuations occurring after the scrambling time which encode the black hole history. Again note this tail of quantum and classical fluctuations appears as a consequence of the quantum features attributed to the horizon, since classical metric equations predict exponentially small signal with time in line with the no hair theorem. Further, the slow escape of the information from the horizon during the late time tail is in sharp contrast with early phase gravitational waves, where the infalling particle is still mostly unentangled and coherently scatters with the microscopic degrees of freedom of the black hole, allowing for easy information recovery away from the horizon.

We may also venture to estimate the share of the total information contained in the gravitational wave signal, setting aside for now the difficulty of recovering the information. Assuming the Von Neumann entropy of a small subsystem grows linearly with the energy (which is an extensive quantity), we may estimate the information content of gravitational waves as their share of the total energy of the BH and GW system. This corresponds to taking the entanglement entropy of the black hole after formation but without the gravitational waves. It can be expressed as the average share of energy m₀ emitted as GWs, which we call $E_\textrm{GW}$, during infall:

$$\begin{align} \int E_\textrm{GW}/m_0dt\approx \int 10^{-2}m_0/M_\textrm{BH}dt, \end{align} \tag{ 7 }$$

obtained over the black hole history, assuming most of the signal is recoverable (SNR $\gt1$). However, this fails to account for the potentially large share of black hole history information present in the gravitational waves, which reduces/collapses the Hilbert space further. As we will see in section 4 this is $\mathcal{O}(1)$, such that the share of information of an observable can be estimated as $\mathcal{O}(1)\times\max(\textrm{SNR},1)$. The total entropy in the system $S_\textrm{cl}$ is therefore reduced by a factor of $E_\textrm{fluct}/(E_\textrm{GW}+E_\textrm{fluct}) = 1/(1+\textrm{SNR}^2)$, and the fraction of the information in GWs becomes $1-1/(1+\textrm{SNR}^2) = \textrm{SNR}^2/(1+\textrm{SNR}^2)$, which is large for information regarding any constituent of the black hole larger that the Planck mass. Thus gravitational waves contain the black hole information down to a Planck mass granularity, and the share of black hole information may be found as:

$$\begin{align} \int_0^{M_\textrm{BH}}\textrm{SNR}^2(m_0)/(1+\textrm{SNR}^2(m_0))\text{d}m_0\approx 1-M_p/M_\textrm{BH}, \end{align} \tag{ 8 }$$

which is quite large for astrophysical black holes. Note we assume an $\mathcal{O}(1)$ share of the information is obtained for SNR$\gt1$, although a more conservative estimate would use SNR$\gt10$, and depends on the experimental setup [53]. However the main purpose of this work is to motivate information retrieval using gravitational waves and we leave this for future work.

The scope of black hole information retrieval in gravitational waves in different regimes is summarized in table 1. In the classical picture ($\hbar\rightarrow 0$ such that $M_p\rightarrow 0$), black holes are eternal and information is lost behind the horizon. However the information of the initial state is also copied and reflected near the horizon in the form of gravitational waves, preserving the initial state information. In the semiclassical picture, where fields are quantized on a smooth metric background, black holes evaporate into maximally mixed states, containing no information besides the conservation of total energy [13]. In this context, gravitational waves provide the only source of information. Finally, in a purely quantum regime, one expects fluctuations of the black hole metric to contain information along with Hawking radiation. However by using only the GW signals (early fluctuations) to extract information we obtain the same share of information as in the semi-classical case ('quantum' picture in table 1), i.e. a sizeable portion of the initial state information, which converges to the classical result in the limit $M_p\rightarrow 0$.

Table 1. Summary of share of information estimates of the initial collapsing state in GWs in both the fully classical picture (eternal black hole) and quantum case (including Hawking radiation and backreactions).

	Information in GWs	Information in BH	Equations
Classical picture	1	0	(36), (40) and (46)
Quantum picture	$\mathcal{O}(1-M_p/M_\textrm{BH})$	$\mathcal{O}(M_p/M_\textrm{BH})$	(8)

With this estimate we are left to quantify the other advantage of gravitational waves: the complexity, or relative simplicity, of recovering information from gravitational wave signals, which we do for black hole-particle mergers in the following sections. First however we comment on their role within the current body of literature on gravitational waves.

Gravitational wave signals from black hole perturbations have been studied extensively (e.g. [17, 18, 49]), and obtaining them to good numerical/computational accuracy is non-trivial. However as we will see, once a few key quantities which can be computed in advance are known the dependence on the various quantum/classical numbers of the infalling particles and initial black hole is trivial, such that with full knowledge of the signal (i.e. in the absence of noise) one can determine them exactly. This is in fact a common implicit assumption behind many gravitational wave detection works [53–56]. However remarkably it has not been explicitly shown before, although mentioned in passing in [57], partly due to its irrelevance in the face of traditional experimental constraints such as intrinsically noisy detectors, irreducible background signals and numerical accuracy. The information is therefore usually determined with a Bayesian approach using catalogues of waveforms (e.g. [58–60]) such that the widths of the significance contours go linearly with these uncertainties:

$$\begin{align} P(\theta_S|d,\theta_N)=\frac{P(d|\theta_S,\theta_N)P(\theta_S|\theta_N)}{P(d|\theta_N)}, \end{align} \tag{ 9 }$$

where $(\theta_S,\theta_N)$ represent vectors of parameters describing the signal and noise respectively and d is the detector data. Nevertheless in an experimentally finite SNR regime, we may recast our previous statement regarding the accuracy of an observable by instead claiming that Bayesian contours do not give disjoint regions of high statistical significance when in the limit of highly sensitive/low noise experiments.

We will consider here a Schwarzschild black hole formed from particles radially falling in, perturbing the metric and emitting gravitational radiation. Other types of perturbation have been considered, and many are solvable to first order [61]. Schwarzschild black hole perturbations are separable into ten scalar degrees of freedom using a tensorial harmonic basis, which can be reduced to four polar (P-even) and two axial (P-odd) degrees of freedom by an appropriate choice of gauge [62]. We may then solve the Einstein field equations of each set separately, resulting in the Zerilli and Regge–Wheeler equations [62]. Furthermore Chandrasekhar proved these equations were equivalent (isospectral) [63] through a supersymmetry transformation. Chandrasekhar also proved [64] that the limit of the Teukolsky equations at zero spin, where Kerr spacetimes become Schwarzschild, is equivalent to the Regge–Wheeler and Zerilli equations, as expected. Thus we expect our method to be generalizable to Kerr perturbations. We also expect the method to generalize to Reissner-Nortstrom black holes as it reduces to a Schwarzschild metric in the limit of zero charge, and since perturbations on this background follow at Zerilli-like equation.

The Zerilli equation is obtained by perturbing a background Schwarzschild metric $g_{\mu\nu} = \textrm{diag}(-f(r),1/f(r),r^2,r^2\sin^2\theta)$, where $f(r) = (1-2M/r)$ and use $M = M_\textrm{BH}$ for notational brevity, with a small parity-even perturbation $h_{\mu\nu}^{even}$ such that $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}^{even}$ and solving the field equations. Writing

$$\begin{align} h_{\mu\nu}^{even}= \begin{pmatrix} -f(r)H_0Y_{lm} & -H_1Y_{lm} & 0 & 0 \\[3pt] -H_1Y_{lm} & -\frac{1}{f(r)}H_2Y_{lm} & 0 & 0 \\[3pt] 0 & 0 & -r^2KY_{lm} & 0 \\[3pt] 0 & 0 & 0 & r^2\sin\theta K Y_{lm}, \\ \end{pmatrix} \end{align} \tag{ 10 }$$

where Y_lm are spherical harmonics and $r_* = r+2M\;\ln(r/2M-1)$ is the tortoise coordinate ³ , one can express the resulting equations in terms of just two out of the initial four scalar degrees of freedom, which are chosen to be H₁ and K such that we discard H₀ & H₂. With the following choice of variable

$$\begin{align} \hat{u}=\alpha K+i\beta H_1\nonumber\\ \alpha=\frac{r^2}{\eta r+3M} \,\,\,\,\,,\,\,\,\,\, \beta=\frac{2M-r}{\omega(\eta r+3M)}, \end{align} \tag{ 11 }$$

one can write the radial Zerilli equation [65]:

$$\begin{align} \frac{d^2\hat{u}(\omega,r_*)}{dr_*^2}+(\omega^2-V(r_*))\hat{u}(\omega,r_*)=I(\omega,r_*), \end{align} \tag{ 12 }$$

where

$$\begin{align} V(r)=\left(1-\frac{2M}{r}\right)\frac{2\eta^2(\eta+1)r^3+6\eta^2Mr^2+18\eta M^2r+18M^3}{r^3(\eta r+3M)^2}, \end{align} \tag{ 13 }$$

with $\eta = (l-1)(l+2)/2$ and as usual $(M,\omega,l)$ are the black hole mass, frequency (eigenvalue of the Zerilli equation) and spherical harmonic l-number. Therefore as expected from the e.g. de Donder gauge, we see perturbations of a Schwarzschild metric (gravitational waves) are solutions of a wave equation with admit two real degrees of freedom: polarizations K and H₁. In what follows we will assume that there is a one-to-one mapping between experimental GW data, H₁ and K, and $\hat{u}$. We will further assume a gedanken experiment which allows us to obtain GW signals and $\hat{u}$ from any point x^µ and with arbitrary accuracy.

The right hand side of (12) describes the source of the perturbations:

$$\begin{align} I(\omega,r)& = \frac{4m_0e^{i\omega T(r)}}{\eta r+3M}\left(l+\frac{1}{2}\right)^{1/2}\left(1-\frac{2M}{r}\right)\left[\left(\frac{r}{2M}\right)^{1/2}-\frac{2i\eta}{\omega(\eta r+3M)}\right]\nonumber\\[3pt] & \quad + e^{i\omega t_0}\left[i\omega u(t,r)-\frac{\partial u(t,r)}{\partial t}\right]_{t=t_0}, \end{align} \tag{ 14 }$$

where m₀ is the infalling particle mass $m_0\ll M$. The first sum term of I is called the 'source term' denoted $\mathcal{S}$ whereas the second term represents perturbations present at t₀ when we start observing GWs, and appears in the frequency domain as an artifact of the Laplace transformation of the time domain Zerilli equation:

$$\begin{align} \frac{\partial^2u}{\partial r_*^2}-\frac{\partial^2u}{\partial t^2}+V(r_*)u=\mathcal{S}, \end{align} \tag{ 15 }$$

where the Laplace transform is:

$$\begin{align} \mathcal{L}u(t,r_*)=\hat{u}(\omega,r_*)=\int_{t_0}^\infty u(t,r_*)e^{i\omega t}dt. \end{align} \tag{ 16 }$$

For now we may ignore the second term in $I(\omega,r)$ if the signal is detected such that u ≈ 0 for $t_0\rightarrow -\infty$. T(r) describes the coordinate time at which the particle can be observed at radius r, which for radial infall from rest is [49]

$$\begin{align} \frac{T(r)}{M}=-\frac{4}{3}\left(\frac{r}{2M}\right)^{3/2}-4\left(\frac{r}{2M}\right)^{1/2}+2\log\left[\left(\left(\frac{r}{2M}\right)^{1/2}+1\right)\left(\left(\frac{r}{2M}\right)^{1/2}-1\right)^{-1}\right]. \end{align} \tag{ 17 }$$

While the expression for T(r) and hence $I(\omega,r)$ appear cumbersome, remarkably m₀ only appears as an overall factor in $I(\omega,r)$, which allows us to extract information about the falling mass relatively easily, as we will see in the next section.

One may solve the Zerilli equation (12) using a Green's function such that:

$$\begin{align} G(\omega,r_*,\zeta)=\frac{1}{W(\zeta)} \begin{cases} \hat{u}_1(\omega,r_*)\hat{u}_2(\omega,\zeta) \,\,\text{if }r_*\leqslant \zeta\\ \hat{u}_1(\omega,\zeta)\hat{u}_2(\omega,r_*) \,\,\text{if }r_*\geqslant \zeta, \end{cases} \end{align} \tag{ 18 }$$

where $\hat{u}_1$ (respectively $\hat{u}_2$) is a solution to the homogeneous Zerilli equation, i.e. I = 0, that respect the desired boundary condition at the horizon (respectively infinity) and $W = \frac{\partial\hat{u}_1}{\partial r_*}\hat{u}_2-\frac{\partial \hat{u}_2}{\partial r_*}\hat{u}_1$. The solution to (12) is then:

$$\begin{align} \hat{u}(\omega,r_*)=\int_{-\infty}^\infty I(\omega,\zeta)G(\omega,r_*,\zeta)d\zeta. \end{align} \tag{ 19 }$$

The Green's function is a solution to the Zerilli equation with a point-like source function I located at ζ. Since solutions to the Zerilli equation are linear in the perturbations, one may sum over a distribution of point-like potentials. The solution to (19) is then the weighting of the Green's function against a distribution I, which is a plane wave with time dependent phase for radially infalling point particles.

More specifically, the $r_*\rightarrow\pm\infty$ boundary conditions on the $\hat{u}_i(\omega,r_*)$ solutions allow us to write in a convenient plane wave basis:

$$\begin{align} \lim_{r_*\rightarrow -\infty}\hat{u}_1(\omega,r_*)=e^{-i\omega r_*} \,\,\,\,\,\,\text{, }\,\,\,\,\,\, \lim_{r_*\rightarrow +\infty}\hat{u}_1(\omega,r_*)=A_{out}(\omega)e^{i\omega r_*}+A_{in}(\omega)e^{-i\omega r_*}, \end{align} \tag{ 20 }$$

and

$$\begin{align} \lim_{r_*\rightarrow +\infty}\hat{u}_{2\pm}(\omega,r_*)=e^{\pm i\omega r_*} \,\,\,\,\,\,\text{, }\,\,\,\,\,\, \lim_{r_*\rightarrow -\infty}\hat{u}_{2\pm}(\omega,r_*)=B_{out}(\pm\omega)e^{\pm i\omega r_*}+B_{in}(\pm \omega)e^{\mp i\omega r_*}, \end{align} \tag{ 21 }$$

We define quasinormal modes (QNMs) ω_nl as having $A_{in}(\omega_{nl}) = 0$. Note that we define two $\hat{u}_2$ functions for later use but will only use $\hat{u}_2 = \hat{u}_{2+}$ in the Green's function since we assume no incoming radiation from infinity ⁴ . From (20) and (21) we find that $W = 2i\omega A_{in}(\omega)$. We can now write the solution in ω-space:

$$\begin{align} \hat{u}(\omega,r_*)=\frac{\hat{u}_2(\omega,r_*)}{2i\omega A_{in}(\omega)}\int_{-\infty}^{r_*}I(\omega,\zeta)\hat{u}_1(\omega,\zeta)d\zeta + \frac{\hat{u}_1(\omega,r_*)}{2i\omega A_{in}(\omega)}\int_{r_*}^{+\infty}I(\omega,\zeta)\hat{u}_2(\omega,\zeta)d\zeta. \end{align} \tag{ 22 }$$

If the source has compact support near the horizon and the experiment, located at $r_*$, is far from the horizon and the source, this simplifies to:

$$\begin{align} \hat{u}(\omega,r_*)= \frac{e^{i\omega r_*}}{2i\omega A_{in}(\omega)}\int_{-\infty}^{+\infty}I(\omega,\zeta)\hat{u}_1(\omega,\zeta)d\zeta. \end{align} \tag{ 23 }$$

This may further simplify if the source is sufficiently close to the horizon to replace $\hat{u}_1\sim e^{-i\omega\zeta}$. Although this may not be our case, in what follows we will focus on this integral for simplicity, since the result and techniques can be straightforwardly generalized to the case of (22). At first glance the integral in (23) is typically divergent at the horizon. Nevertheless we may regularize it by imposing appropriate boundary conditions (see appendix).

We now want to understand the relation with the time series which is measured at our gedanken experiment. We must therefore return to the time domain via an inverse Laplace transform. This integral is non-trivial and requires defining a contour in the complex ω-plane (see figure 4). The contour contains poles at the quasinormal frequencies ω_nl where $A_{in}(\omega_{nl}) = 0$, which following the residue theorem contribute to the result:

$$\begin{align} \nonumber u(t,r_*)&=\sum_{n,l}\frac{A_{out}(\omega_{nl})}{2\omega}\left(\frac{dA_{in}(\omega_{nl})}{d\omega}\right)^{-1}e^{-i\omega_{nl}(t-r_*)}\int_{-\infty}^{+\infty}\frac{I(\omega_{nl},\zeta)\hat{u}_1(\omega_{nl},\zeta)}{A_{out}(\omega_{nl})}d\zeta \nonumber\\ &\quad - \frac{1}{2\pi}\int_{HC+BC}\left( e^{-i\omega(t-r_*)}\int_{-\infty}^{+\infty}I(\omega,\zeta)\hat{u}_1(\omega,\zeta)d\zeta\right) d\omega. \end{align} \tag{ 24 }$$

Note the inverse derivative of A_in is an intrinsic property of the black hole metric, and combined with the source integral is often called 'quasinormal excitation coefficient' (QNEC). The observed radiation time series is thus a sum over QNMs ω_nl and a continuous spectrum of half circle modes and the branch cut. We discuss in the following sections the effect of the contour on information recovery.

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Although the A_out 's cancel, the residue coefficient requires knowledge of A_in in the neighborhood of ω_nl . We follow [61, 66] in determining these using the following series ansatz in ω-space and Schwarzschild coordinates with $2M = 1$: ⁵

$$\begin{align} \hat{u}_1(\omega,r) & =(r-1)^{-i\omega}r^{2i\omega}e^{i\omega(r-1)}\sum_{j=0}^\infty a_j\left(\frac{r-1}{r}\right)^j \end{align} \tag{ 25 }$$

$$\begin{align} \hat{u}_{2\pm}(\omega,r) & =(2\omega)^{\mp i\omega}e^{i\phi_\pm}\left(1-\frac{1}{r}\right)^{-i\omega}\sum_{j=-\infty}^{+\infty}b_j(G_{j+\nu}(-\omega,\omega r)\pm iF_{j+\nu}(-\omega,\omega r)), \end{align} \tag{ 26 }$$

where $G_{j+\nu}$ and $F_{j+\nu}$ are the regular and irregular Coulomb wave functions with ν chosen to yield a minimal solution for the series b_j (i.e. with a fixed initial value a₀, which (20) fixes to $b_0 = 1$) and

$$\begin{align} \phi_\pm=\pm i\ln\left[\sum_{j=-\infty}^{+\infty}b_j\left(\frac{\Gamma(j+\nu+1-i\omega)}{\Gamma(j+\nu+1+i\omega)}\right)^{\pm 1/2}e^{\mp i(j+\nu)\pi/2}\right]. \end{align} \tag{ 27 }$$

These expansions respect the correct boundary conditions e.g.

$$\begin{align} \hat{u}_1(\omega,r)\sim (r-1)^{-i\omega}e^{i\omega(r-1)} \sim e^{-i\omega r_*} \,\,\,\,\,\,\,\text{as }\,\,\, r\rightarrow 1. \end{align} \tag{ 28 }$$

Although the above expansion holds for all $r_*$, the series a_j diverges as $r\rightarrow\infty$, and is therefore impractical for direct evaluation of $A_{out}(\omega_{nl})$. In (20) and (21), we have defined three solution to the homogeneous Zerilli equation, namely u₁, $u_{2-}$ and $u_{2+}$, each respecting a different boundary condition. Any pair of these three solutions will form a basis of the space of solutions to the homogeneous Zerilli equation, which is a second order linear ODE. Hence, any one of these solutions can be expressed as a linear superposition of the other two (the combination is oftentimes referred to as a Bogoliubov transformation). The coefficient of the Bogoliubov transformation can be found from the boundary conditions that defines the various solutions, in particular we can write

$$\begin{align} \begin{cases} \hat{u}_1(\omega,r_*)=A_{in}(\omega)\hat{u}_{2-}(\omega,r_*)+A_{out}(\omega)\hat{u}_{2+}(\omega,r_*) \\ \hat{u}_1^{^{\prime}}(\omega,r_*)=A_{in}(\omega)\hat{u}_{2-}^{^{\prime}}(\omega,r_*)+A_{out}(\omega)\hat{u}_{2+}^{^{\prime}}(\omega,r_*). \end{cases} \end{align} \tag{ 29 }$$

Thus we can solve for A_out and A_in for all ω and some chosen finite $r_*$ ⁶ :

$$\begin{align} \begin{cases} A_{in}(\omega)= \dfrac{u_1(\omega,r_*)u_{2+}^{^{\prime}}(\omega,r_*)-u_1^{^{\prime}}(\omega,r_*)u_{2+}(\omega,r_*)}{u_{2+}^{^{\prime}}(\omega,r_*)u_{2-}(\omega,r_*)-u_{2+}(\omega,r_*)u_{2-}^{^{\prime}}(\omega,r_*)} \\[10pt] A_{out}(\omega)= \dfrac{u_1^{^{\prime}}(\omega,r_*)u_{2-}(\omega,r_*)-u_1(\omega,r_*)u_{2-}^{^{\prime}}(\omega,r_*)}{u_{2+}^{^{\prime}}(\omega,r_*)u_{2-}(\omega,r_*)-u_{2+}(\omega,r_*)u_{2-}^{^{\prime}}(\omega,r_*)}, \end{cases} \end{align} \tag{ 30 }$$

from which we can calculate $dA_{in}/d\omega_{nl}$ once we obtain $(\hat{u}_1,\hat{u}_{2\pm})$. To do this we plug the series into the Zerilli equation. The recursion relation obeyed by the coefficient of the series is given by (re-establishing mass units where $2M\neq 1$):

$$\begin{align} \alpha_0 a_1+\beta_0 a_0=0 \nonumber\\ \alpha_{nl}a_{n+1}+\beta_{nl}a_n+\gamma_{nl}a_{n-1}=0 \end{align} \tag{ 31 }$$

$$\begin{align} \begin{split} \alpha_{nl}= & \, n^2+(2-4i\omega M)n+1-4i\omega M \\ \beta_{nl}= & -(2n^2+(2-16i\omega M)n-32\omega^2 M^2-8i\omega M+l(l+1)-3) \\ \gamma_{nl}= & \, n^2-8i\omega M n-16\omega^2 M^2-4, \end{split} \end{align} \tag{ 32 }$$

where n corresponds to the series index. Building on the three term recurrence relation, Leaver showed that the solution to the Zerilli equation will converge [66, 68, 69] if the $(a_n)_{n\in\mathbb{N}}$ is a minimal solution to the above recurrence relation. The existence of a minimal solution implies that the following continued fraction holds:

$$\begin{align} \frac{a_1}{a_0} = \frac{-\gamma_{1l}}{\beta_{1l}-}\frac{\gamma_{2l}\alpha_{1l}}{\beta_{2l}-}\frac{\gamma_{3l}\alpha_{2l}}{\beta_{3l}-}\ldots, \end{align} \tag{ 33 }$$

where we have used the standard notation for continued fraction in the above equation. Equation (33) will be valid for a discrete set of complex frequencies which are the QNM frequencies. The index labeling the element of this set is called the 'overtone' number. In practice one numerically solves (33) or its $n{\text{th}}$ inversion to find the $n{\text{th}}$ QNM frequencies. This method of finding QNM frequencies, commonly called Leaver's method has been improved and generalized to a wide range of space-time beyond the Schwarzschild metric considered here. QNM solutions ($A_{in}(\omega) = 0$) occur at fixed discrete solutions of $u_i(\omega,r_*)$ and $\{\alpha_{nl},\beta_{nl},\gamma_{nl}\}_{n\in\mathbb{N}}$. By inspection of (32), the coefficients remain constant if ω_nl is simply related to the mass:

$$\begin{align} \omega_R\propto M_\textrm{BH}^{-1} \,\,\, , \,\,\,\,\,\, \omega_I\propto M_\textrm{BH}^{-1}. \end{align} \tag{ 34 }$$

The proportionality coefficients are determined solely by the structure of the black hole perturbation equations and (n, l) and can be determined up to arbitrary accuracy (i.e. given sufficient computational resources) by inverting the continued fraction. For the 'fundamental' frequency $(n,l) = (0,2)$ they are often quoted as $(12.07,18.06)$ kHz $M_\odot$. Having explained how to calculate each of the quantities of (24) our gedanken GW experiment may find the mass of the black hole and the rest of the parameters with arbitrary accuracy, as we will see in the following section.