# Quadratic perturbations of the Schwarzschild black hole:
The algebraically special sector

Jibril Ben Achour  Arnold Sommerfeld Center for Theoretical Physics, Munich,
Germany  Munich Center for Quantum Science and Technology, Munich, Germany
Univ de Lyon, ENS de Lyon, Laboratoire de Physique, CNRS UMR 5672, Lyon 69007,
France  Hugo Roussille  Univ de Lyon, ENS de Lyon, Laboratoire de Physique,
CNRS UMR 5672, Lyon 69007, France

###### Abstract

We investigate quadratic algebraically special perturbations (ASPs) of the
Schwarzschild black hole. Their dynamics are derived from the expansion up to
second order in perturbation of the most general algebraically special
twisting vacuum solution of general relativity. Following this strategy, we
present analytical expressions for the axial-axial, polar-polar and polar-
axial source terms entering in the dynamical equations. We show that these
complicated inhomogeneous equations can be solved analytically and we present
explicit expressions for the profiles of the quadratic ASPs. As expected, they
exhibit exponential growth both at the past and future horizons even in the
non-linear regime. We further use this result to analyze the quadratic zero
modes and their interpretation in terms of quadratic corrections to mass and
spin of the Schwarzschild black hole. The present work provides a direct
extension beyond the linear regime of the original work by Couch and Newman.

###### Contents

  1. I Review of algebraically special modes
  2. II Twisting radiative spacetime and black hole perturbations
  3. III Linear algebraically special perturbations
    1. III.1 Dynamics
    2. III.2 Exact solutions
    3. III.3 Zero modes
  4. IV Quadratic algebraically special perturbations
    1. IV.1 Dynamics
    2. IV.2 Exact solutions
      1. IV.2.1 Axial perturbations
      2. IV.2.2 Polar perturbations
    3. IV.3 Quadratic zero modes
      1. IV.3.1 Monopole-monopole zero mode
      2. IV.3.2 Dipole-dipole zero mode
  5. V Discussion
  6. A Spherical harmonics in stereographic coordinates
    1. A.1 Definition
    2. A.2 Real spherical harmonics
    3. A.3 Decomposition coefficients
  7. B Reconstruction of the perturbed geometry
    1. B.1 Perturbative expansion of the twisting vacuum radiative metric
    2. B.2 Covariant perturbation formalism
    3. B.3 Reconstructing Regge-Wheeler perturbations
  8. C An illustrative concrete example for a quadratic ASP

## Introduction

The gravitational radiation emitted during the coalescence of a binary black
hole merger is well described by black hole perturbation theory at both early
and late time, corresponding respectively to the inspiral phase and the
relaxation to equilibrium. For spherically symmetric black holes, the
perturbation theory was set up in the late fifties by Regge and Wheeler in
their seminal paper Regge:1957td  where the dynamics of the linearized
perturbations around the Schwarzschild black hole were described. It is now
well understood that during the relaxation to equilibrium, the perturbed
Schwarzschild black hole radiates away all its higher multipoles Martel:2005ir
. While the early time signal does depend on the details of the merger, the
intermediate signal is dominated by damped waves dubbed quasi-normal modes
(QNMs) followed by power-law tails which drive the late-time signal. The QNMs
frequencies are universal and fully characterized by the black hole mass (and
spin in the case of a rotating black hole) which makes the QNM spectrum a
unique footprint of the black hole geometry. Therefore the potential
observation of long-lived QNMs in the near future provides a powerful test of
general relativity and its possible extensions Berti:2004md ; Berti:2009kk ;
Berti:2015itd ; Franchini:2023eda .

While the description of the relaxation process after the merger based on the
linear regime of the perturbation theory is well developed, several points are
still debated such as a characterization of the time domain during which QNMs
actually dominate the signal or their stability against small deformations
Cardoso:2024mrw ; Destounis:2021lum ; Berti:2022xfj ; Oshita:2024fzf . More
recently, characterizing non-linear corrections to the QNMs spectrum during
the relaxation process right after the merger has attracted much
attention111See DeLuca:2023mio ; Riva:2023rcm  for related investigations in
the static regime.. On the one hand, this requires pushing black hole
perturbation theory up to second order, a task which was undertaken by various
authors in the last decades Gleiser:1995gx ; Gleiser:1998rw ;
Campanelli:1998jv ; Garat:1999vr ; Nicasio:2000ge ; Thompson:2016fxe ;
Brizuela:2007zza ; Brizuela:2009qd . On the other hand, the non-linearities
can be investigated numerically, a target which has attracted considerable
attention recently London:2014cma ; Cheung:2022rbm ; Mitman:2022qdl ;
Ma:2022wpv ; Lagos:2022otp ; Bhagwat:2023fid ; Kehagias:2023ctr ;
Khera:2023oyf ; Redondo-Yuste:2023seq ; Perrone:2023jzq ; Bucciotti:2023ets ;
Cheung:2023vki ; Qiu:2023lwo ; Zhu:2024rej ; Ma:2024qcv ; Zhu:2024dyl ;
Rosato:2024arw  (see Yi:2024elj  for a recent prospect on the detectability of
these non-linearities). While the spectrum of quadratic QNMs (dubbed QQNMs)
has a straightforward relationship to the linear spectrum, the fate of the
amplitude of quadratic perturbations has been addressed only recently
Bucciotti:2024zyp ; Bourg:2024jme . Numerical investigations have shown that
the amplitude of the quadratic modes depends on the multipole structure,
dominating in some cases the one of their linear counterparts for large
multipoles London:2014cma ; Cheung:2022rbm ; Mitman:2022qdl ; Ma:2022wpv .
This growing body of results reveals that the black hole spectroscopy program
is more delicate than anticipated and underlines the necessity to further
investigate the response of black holes to perturbations beyond the linear
regime.

In this work, we shall focus on the characterization of a certain class of
quadratic perturbations which play a special role in black hole perturbation
theory. It corresponds to the perturbations which, while spoiling the Petrov
type D character of the background, preserve its algebraically special
character allowing for a perturbed black hole geometry of Petrov type
II222Generic perturbations usually break the algebraically special character
of the background geometry such that no symmetry survives Araneda:2015gsa .
For this reason, these perturbations are called algebraically special
perturbations (ASPs). So far, these modes have been discussed for the
Schwarzschild black hole only at the linearized level first by Chandrasekhar
in Chandrasekhar1984 , and later on by Couch and Newman in Couch:1973zc  (see
also Qi:1993ey  for a more recent discussion). The case of the linear ASPs of
the Kerr black hole was investigated by Wald in Wald:1973wwa . The first goal
of this work is to investigate the Schwarzschild ASPs at the quadratic level.
Studying these modes beyond the linear regime can be motivated from several
corners.

First, they stand aside from the other perturbations as they can be derived
from pure symmetry arguments, without invoking any boundary conditions. As we
shall see, they can be obtained by expanding to a given order in perturbation
theory the most general algebraically special twisting vacuum solutions of
general relativity (thus being of Petrov type II). Therefore, contrary to the
other perturbations, one has already access to a fully non-perturbative
description of these modes. This allows one to investigate how the non-
linearities manifest in the perturbative treatment, providing a rather unique
window to address this question.

Second, in dimension greater than 4, it has been shown that ASPs are purely
stationary, i.e. they stand as pure zero modes connecting different higher
dimensional black hole solutions Dias:2013hn . However, in the four
dimensional case, they correspond to propagating, albeit non-oscillating
modes. This rather peculiar property which shows up only in 4d comes with
another feature. At the linear level, these ASPs diverge either on the future
or on the past Schwarzschild horizon. This divergence prevent them from being
well defined excitations of the Schwarzschild black hole which allow for a
relaxation to equilibrium after a certain time. As such, they stand outside
the standard perturbations of compact support one usually considers in black
hole perturbation theory. For this reason, they are generally considered as
pathological. At best, one can consider these perturbations as valid only on a
finite portion of the null cone, as initially argued in Couch:1973zc .

Nevertheless, it turns out that among the well-defined perturbations of the
Schwarzschild black hole, one can find specific excitations with similar
spectral properties, namely a purely imaginary frequency which is numerically
very close from the theoretical damping time of the Schwarzschild ASP
MaassenvandenBrink:2000iwh . Whether the value of the damping time matches
exactly the ASP one is a matter of debate, but the main point is that one can
exhibit well-defined highly damped modes of the Schwarzschild black hole which
are spectrally close to the ASP Leung:1999fr ; Leung:2003eq ;
MaassenvandenBrink:2000iwh  (see Berti:2003jh ; Cook:2016ngj ; Cook:2016fge ;
Hod:2013fea  for the Kerr black hole). Therefore, in view of the growing
interest in the non-linear effects on the QNM spectrum, it is an interesting
task to describe the behavior of the quadratic ASPs of the Schwarzschild black
hole to later confront them to the QQNMs spectrum. This is one of the main
goals of this work. Let us further stress that these algebraically special
modes also play a key role in the understanding of the symmetries of black
hole perturbations and in particular in the proof of the isospectrality of the
QNM spectrum ISO ; Glampedakis:2017rar ; Yurov:2018ynn ; Lenzi:2021njy ;
Lenzi:2023inn . Hence, understanding their non-linear behavior could also shed
light on possible hidden structures of the QQNMs spectrum.

Finally, alongside the above motivations, the divergence of the ASPs naturally
led to the following question (already raised in Couch:1973zc ): can the
divergence of these linear modes at the horizon be cured by non-linear
corrections ? In particular, can the non-linear zero modes modify the
Schwarzschild background such that the divergences disappear ? The most
natural way to address this question is to push the original treatment of
Couch and Newman up to second order to analyze the resulting geometry and the
non-linear zero modes. As we shall see, the answer is negative and the
quadratic ASPs remain divergent.

The paper is organized as follows. We begin in Section I by a brief review of
the algebraically special modes as solutions of the RW and Zerilli wave
equations. Then, we present in Section II the most general algebraically
special twisted vacuum solutions of general relativity which will be our
starting point to derive the profile of the quadratic ASPs. Section III
presents the concrete dynamics of the linear ASPs, reviewing the results of
Couch and Newman Couch:1973zc . We use this opportunity to present in section
III.3 a concise and clear derivation of the action of the monopole and dipole
linear zero modes. Then, in Section IV we derive the dynamical equations for
the quadratic ASPs and present the explicit analytical solutions for their
profiles. We conclude this section by a careful treatment of the quadratic
zero modes and we use this result to discuss the non-linear behavior of the
dynamical quadratic ASPs. Section V contains a discussion of our results and
the perspectives they open. Appendix A is dedicated to a review of the main
properties of real spherical harmonics. Finally, the reconstruction of the
metric perturbations in the standard Regge-Wheeler parametrization
Regge:1957td  is presented for completeness in appendix B and the detailed
derivation of the explicit example is given in appendix C.

##  I Review of algebraically special modes

Linear perturbations of the Schwarzschild black hole geometry are described by
the well-known Regge-Wheeler and Zerilli equations. Using the spherical
symmetry of the background, these two equations reduce to simple 2d second
order differential equations dictating the time and radial profile of the
polar (even parity) perturbation, denoted $Y_{\text{RW}}$, and the axial (odd-
parity) perturbations denoted $Y_{\text{Z}}$. These decoupled equations read
respectively

$\displaystyle-\derivative[2]{Y_{\mathrm{RW}}}{t}+\derivative[2]{Y_{\mathrm{RW}}}{r_{*}}=V_{\mathrm{RW}}(r)Y_{\mathrm{RW}}\,,$
(1)
$\displaystyle-\derivative[2]{Y_{\mathrm{Z}}}{t}+\derivative[2]{Y_{\mathrm{Z}}}{r_{*}}=V_{\mathrm{Z}}(r)Y_{\mathrm{Z}}\,,$
(2)

Here, $r_{\ast}$ is the tortoise coordinate defined by

$\derivative{r_{*}}{r}=\Big{(}1-\frac{2M}{r}\Big{)}^{-1}\mbox{\quad
or\quad}r_{*}(r)=r+2M\log(r-2M)\,,$ (3)

such that $r_{\ast}\rightarrow-\infty$ at the horizon, i.e. when $r=2M$ and
the RW and Zerilli potentials are given explicitly by

$\displaystyle V_{\mathrm{RW}}(r)$
$\displaystyle=\Big{(}1-\frac{2M}{r}\Big{)}\frac{2({\lambda}_{\mathrm{a}}+1)r-6M}{r^{3}}\,,$
(4) $\displaystyle V_{\mathrm{Z}}(r)$
$\displaystyle=\Big{(}1-\frac{2M}{r}\Big{)}\frac{72M^{3}+72M^{2}r{\lambda}_{\mathrm{p}}+24Mr^{2}{\lambda}_{\mathrm{p}}^{2}+8r^{3}{\lambda}_{\mathrm{p}}^{2}({\lambda}_{\mathrm{p}}+1)}{r^{3}(6M+2r{\lambda}_{\mathrm{p}})^{2}}\,.$
(5)

We have written ${\lambda}_{\mathrm{a}}$ and ${\lambda}_{\mathrm{p}}$ the
values of $\lambda$ associated respectively to axial and polar perturbations.
In these expressions, we use the integer $\lambda$ defined from the angular
momentum $\ell$ of perturbations by

$2\lambda=\ell(\ell+1)-2\,.$ (6)

Solving these two wave equations requires choosing some boundary conditions
which lead to qualitatively different perturbations. The quasi-normal modes
(QNM) are obtained by demanding that the wave is purely in-going at the
horizon and purely out-going at ${\mathcal{I}}^{+}$. Another choice consists
in imposing that the wave is purely in-going at the horizon and at
${\mathcal{I}}^{+}$ which gives the so called totally transmitted from the
right modes (TTMR). Demanding instead that the wave is purely out-going at
both boundaries gives the so called totally transmitted from the left modes
(TTML).

Another way to solve for these equations is to introduce the ansätze

$Y_{\mathrm{RW}}=Y_{\mathrm{RW}}^{\mathrm{ret}}=e^{{\kappa}_{\mathrm{a}}(t-r_{*})}{f}_{\mathrm{a}}(r)\mbox{\quad
and\quad}Y_{\mathrm{Z}}=Y_{\mathrm{Z}}^{\mathrm{ret}}=e^{-{\kappa}_{\mathrm{p}}(t-r_{*})}{f}_{\mathrm{p}}(r)\,.$
(7)

where ${\kappa}_{\mathrm{a}}$ and ${\kappa}_{\mathrm{p}}$ are real constants.
Injecting these expressions into (1) and (2), one obtains the following exact
solutions

$\kappa=\frac{(\ell-1)\ell(\ell+1)(\ell+2)}{12M}\,,\quad{f}_{\mathrm{a}}=\frac{3M+{\lambda}_{\mathrm{a}}r}{r}\mbox{\quad
and\quad}{f}_{\mathrm{p}}=\frac{r}{3M+{\lambda}_{\mathrm{p}}r}\,.$ (8)

These solutions are dubbed _algebraically special modes_ for reasons that will
be clarified in the following. They correspond to non-oscillating modes which
violate the boundary conditions satisfied by the QNMs and by the TTMsL/R,
giving rise to a new family of modes. Indeed, introducing the retarded and
advanced time coordinates

$u=t-r_{\ast}\qquad v=t+r_{\ast}$ (9)

we see from (7) that while $Y_{\mathrm{RW}}$ diverges at ${\mathcal{I}}^{+}$,
i.e. when $u\rightarrow+\infty$, the polar perturbation $Y_{\mathrm{Z}}$
diverges at the horizon, i.e. when $u\rightarrow-\infty$. This couple of ASM
comes with another set of solutions obtained by reversing the time coordinate
$t$, leading to

$Y_{\mathrm{RW}}=Y_{\mathrm{RW}}^{\mathrm{adv}}=e^{-{\kappa}_{\mathrm{a}}(t+r_{*})}{f}_{\mathrm{a}}(r)\mbox{\quad
and\quad}Y_{\mathrm{Z}}=Y_{\mathrm{Z}}^{\mathrm{adv}}=e^{{\kappa}_{\mathrm{p}}(t+r_{*})}{f}_{\mathrm{p}}(r)\,.$
(10)

This second couple of solutions behaves exactly in the opposite way from the
former.

The status of these ASM, which are well defined exact solutions of the RW and
Zerilli equations (at least when considered in a finite region of spacetime
corresponding to a finite range of $u$ or $v$) has attracted considerable
attention over the years. Because they blow up on the horizon, these solutions
of the RW and Zerilli equations cannot be considered as perturbations per se,
as their evolution prevent the geometry to return back to the Schwarzschild
one. Nevertheless, it is known that among the well behaved perturbations of
the Schwarzschild black hole, one can find QNMs and TTMs exhibiting a
vanishing frequency and a decay time which is numerically close from the ASM
Leung:1999fr ; MaassenvandenBrink:2000iwh ; Leung:2003eq , giving rise to
well-behaved highly damped modes (see Cook:2016ngj ; Cook:2016fge ;
Hod:2013fea  for Kerr). Despite these results, the exact spectral equivalence
of these specific modes to the ASM remains an open question Berti:2009kk .

More importantly for the purpose of this work, this set of ASM turn out to
correspond to linearized algebraically special radiative solutions of general
relativity, hence their name. This result was shown by Couch and Newman in
Couch:1973zc  and their relation to the Chandrasekhar algebraically special
modes was stressed by Qi and Schultz in Qi:1993ey . This alternative
derivation of these ASM makes it explicit that contrary to QNM and TTMs, one
can derive their expressions without imposing boundary conditions, but simply
by imposing symmetry restriction (i.e. restricting the Petrov type of the
perturbed geometry). Remarkably, this strategy can be generalized to higher
order in perturbation theory, giving access to an analytic treatment of the
non-linear ASMs. As we shall see, studying these ASM beyond the linear order
allows one to contemplate the complicated mode-mixing in an analytical form as
well as to discuss the fate of non-linear zero modes in black hole
perturbation theory.

##  II Twisting radiative spacetime and black hole perturbations

In this section, we present the exact family of twisting vacuum radiative
solutions of general relativity which will be our starting point to derive
analytic expression for the quadratic algebraically special perturbations
(ASP). We follow closely Stephani:2003tm  where this family of solutions is
presented in detail. Let us introduce the tetrad $\omega$ given by

$\omega^{l}=\differential{u}+L\differential{z}+\bar{L}\differential{\bar{z}}\,,\quad\omega^{n}=\differential{r}+W\differential{z}+\bar{W}\differential{\bar{z}}+H\omega^{l}\,,\quad\omega^{m}=\frac{\differential{\bar{z}}}{P\rho}\,,$
(11)

where the functions $L$, $W$, $H$, $P$ and $\rho$ defined by

$\displaystyle\rho$ $\displaystyle=-\frac{1}{r+i\Sigma}\,,$ $\displaystyle
2i\Sigma$ $\displaystyle=P^{2}(\bar{\partial}L-\partial\bar{L})\,,$
$\displaystyle W$
$\displaystyle=\frac{\partial_{u}L}{\rho}+\partial(i\Sigma)\,,$
$\displaystyle\partial$ $\displaystyle=\partial_{z}-L\partial_{u}\,,$
$\displaystyle H$
$\displaystyle=\frac{K}{2}-r\partial_{u}(\log(P))-\frac{Mr+N\Sigma}{r^{2}+\Sigma^{2}}\,,$
$\displaystyle K$ $\displaystyle=2P^{2}\real(\partial(\bar{\partial}\log
P-\partial_{u}\bar{L}))\,,$ $\displaystyle N$ $\displaystyle=\Sigma
K+P^{2}\real(\partial\bar{\partial}\Sigma-2\partial_{u}\bar{L}\partial\Sigma-\Sigma\partial_{u}\partial\bar{L})\,.$
(12)

Consider the line element given by

$\differential{s}^{2}=-2\omega^{l}\omega^{n}+2\omega^{m}\omega^{\bar{m}}\,,$
(13)

In this setup, $M(u,z,\bar{z})$ and $P(u,z,\bar{z})$ are real functions and
$L(u,z,\bar{z})$ is a complex function. Let us describe the main kinematical
properties of the geometries belonging to this family. By construction, one
has $\Psi_{0}=\Psi_{1}=0$ such that the geometry is algebraically special. The
most general examples are of Petrov type II. The only non-vanishing Weyl
scalars are given by

$\displaystyle\Psi_{2}$ $\displaystyle=(m+iN)\rho^{3}$ (14)
$\displaystyle\Psi_{3}$ $\displaystyle=-P^{3}\rho^{2}\partial
I+{\mathcal{O}}(\rho^{3})$ (15) $\displaystyle\Psi_{4}$
$\displaystyle=P^{2}\rho\partial_{u}I+{\mathcal{O}}(\rho^{2})$ (16)

The Newman-Penrose spin coefficients (NPSC) satisfy

$\displaystyle\kappa=\sigma=\tau=0$ (17)

while the other NPSC are non-vanishing. In particular, one has
$\text{Im}(\rho)\neq 0$ which implies that the geometry is twisting. It
follows that the null tetrad describes a geodesic, non-shearing but twisting
and expanding congruence of null rays. This family of solutions describes the
twisting algebraically special vacuum radiative solutions of GR. When one
further imposes that the twist vanishes, i.e, that $L=0$, this family of
solutions reduces to the Robinson-Trautman geometries of Petrov type III.

One can show that under the above assumptions (choice of tetrad), the Einstein
field equations reduce to

$\displaystyle i\partial N-3(M+iN)\partial_{u}L=0\,,$ (18) $\displaystyle
P\quantity[\partial+2(\partial\log
P-\partial_{u}\bar{L})]\partial\quantity[\bar{\partial}(\bar{\partial}\log
P-\partial_{u}\bar{L})+(\bar{\partial}\log
P-\partial_{u}\bar{L})^{2}]-\partial_{u}\quantity[P^{-3}(M+iN)]=0\,,$ (19)

which provides the dynamical equations relating the three functions $(M,P,L)$.
When $L=0$, the twist vanishes and this set of solutions reduces to the
Robinson-Trautman family of exact radiative solutions in which case the mass
$M$ can only depend on $u$. How can this family of exact non-linear radiative
solutions be useful to investigate black hole perturbations ?

When investigating black hole perturbations, one is interested in
perturbations about a Petrov type D geometry (being either Kerr or
Schwarzschild). Without further restriction, general perturbations will break
the algebraically special character of the background geometry resulting in a
Petrov type I geometry Araneda:2015gsa . However, it is interesting to focus
on the subset of these perturbations which, while spoiling the Petrov type D
character of the background, still allows the perturbed background to remain
algebraically special. Since the perturbed black hole geometrie remains at
most of Petrov type II, the above family of exact solutions provides a
framework to analyze these algebraically special perturbations without
imposing any boundary conditions.

This strategy was used by Couch and Newman in Couch:1973zc  to derive the
linear ASP of the Schwarzschild black hole and soon after, the same approach
was generalized to the linear ASP of the Kerr black hole by Wald Wald:1973wwa
. In the following, we shall focus on the Schwarzschild perturbations. For
completeness, we will first review the analysis of Couch and Newman for the
linear ASP. Then, we will generalize their result to second order in
perturbation theory and present the exact solutions for the quadratic ASP.

##  III Linear algebraically special perturbations

We start by reviewing the derivation of the linear ASP for the Schwarzschild
background. Within this large family of twisting vacuum radiative solutions
described by the two functions $(M,P,L)$, the Schwarzschild geometry
corresponds

$P=P_{0}=\frac{1}{\sqrt{2}}(1+z\bar{z})\mbox{\quad and\quad}L=0\,.$ (20)

while $M$ is a constant corresponding to the Schwarzschild mass. The “usual”
$(t,r,\theta,\varphi)$ coordinates are related to the coordinates
$(u,r,z,\bar{z})$ by the change of variables

$u=t-r-2M\log(r-2M)\mbox{\quad and\quad}z=\cot\frac{\theta}{2}e^{i\varphi}\,.$

In the following, we will keep working with the coordinates $(u,r,z,\bar{z})$.

###  III.1 Dynamics

To describe the ASP of the Schwarzschild geometry, it is natural to introduce
the perturbed $(P,L)$ which read

$P(u,z,\bar{z})=P_{0}e^{\varepsilon F_{1}}\,,\quad L(u,z,\bar{z})=\varepsilon
L_{1}\,,$ (21)

where $\epsilon$ is a bookkeeping parameter encoding the order of the
expansion in perturbation333In principle, one could also consider a
perturbation of the mass function $M$, but one can show that such a
perturbation cannot be consistently introduced, such that one has to work with
$M$ constant from the start.. Here, $F_{1}$ encodes the polar perturbations
while $L_{1}$ encodes the axial perturbations. It is further useful to define
$N_{1}$ and $\Sigma_{1}$ by

$\Sigma=\varepsilon\Sigma_{1}+\mathcal{O}(\varepsilon^{2})\mbox{\quad
and\quad}N=\varepsilon N_{1}+\mathcal{O}(\varepsilon^{2})\,.$ (22)

Furthermore, we define the operator $\Delta_{0}$ by

$\Delta_{0}=2P_{0}^{2}\partial_{z}\partial_{\bar{z}}\,.$ (23)

One can note that $\Delta_{0}$ is such that $\Delta_{0}\log P_{0}=1$ which
corresponds to the Schwarzschild background value (i.e. to the purely
spherically symmetric case).

Consider the first dynamical equation (18). Linearizing this equation gives

$\partial_{z}N_{1}=-3iM\partial_{u}L_{1}\,,\mbox{\quad
with\quad}N_{1}=\Sigma_{1}+\frac{1}{2}\Delta_{0}\Sigma_{1}\mbox{\quad
and\quad}\Sigma_{1}=-P_{0}^{2}\imaginary(\partial_{z}\bar{L}_{1})\,.$ (24)

As expected, this equation for $L_{1}$ does not contain $F_{1}$ at first order
in $\varepsilon$ and the two axial and polar perturbations decouple. It is
convenient to introduce a real function $f_{1}$ such that
$L_{1}=i\partial_{z}f_{1}$ which allows one to recast $(\Sigma,N)$ as

$\Sigma=\frac{1}{2}\Delta_{0}f_{1}\mbox{\quad
and\quad}N=\frac{1}{2}\Delta_{0}f_{1}+\frac{1}{4}\Delta_{0}\Delta_{0}f_{1}\,.$
(25)

With this new field $f_{1}$ to describe the axial perturbation, the equation
(24) takes the form

$\boxed{\Delta_{0}\Delta_{0}f_{1}+2\Delta_{0}f_{1}-12M\partial_{u}f_{1}=0}\,.$
(26)

It describes the dynamics of the linear axial ASP of the Schwarzschild
background.

We can now turn to the second dynamical equation (19). This equation being
complex, it can be separated into two parts. The imaginary part yields

$\partial_{u}(\Delta_{0}\Delta_{0}f_{1}+2\Delta_{0}f_{1}-4N_{1})=0\,,$ (27)

which is always verified given (25) while the real part gives

$\boxed{\Delta_{0}\Delta_{0}F_{1}+2\Delta_{0}F_{1}+12M\partial_{u}F_{1}=0}\,.$
(28)

It describes the dynamics of the linear polar ASP on the Schwarzschild black
hole. Since both perturbations decouple at the linear order, if one focus
solely on the linear polar sector, one can use the subset of Robinson-Trautman
geometries to derive the same result. The dynamical equation (28) is nothing
else than the well known fourth order RT equation linearized around the
Schwarzschild case $P=P_{0}$ Qi:1993ey .

Having derived the dynamical equations for the linear ASP of the Schwarzschild
black hole, let us see how they can be analytically solved. These solutions
are needed to construct the quadratic perturbations and in particular the
source terms.

###  III.2 Exact solutions

A natural ansätze to solve these equations is to decompose the perturbations
$(F_{1},f_{1})$ onto spherical harmonics defined by the eigenvalue equation

$\Delta_{0}Y_{\ell}^{m}=-\ell(\ell+1)Y_{\ell}^{m}\,.$ (29)

More details about their definition are given in appendix A. The ansätze reads

$F_{1}(u,z,\bar{z})={E}_{\mathrm{p}}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}h_{{\ell}_{\mathrm{p}}}(u)\mbox{\quad
and\quad}f_{1}(u,z,\bar{z})={E}_{\mathrm{a}}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}h_{{\ell}_{\mathrm{a}}}(u)\,,$
(30)

where ${E}_{\mathrm{p}}$ and ${E}_{\mathrm{a}}$ are constant amplitude
factors. Notice that ${E}_{\mathrm{p}}$ has no dimension while
${E}_{\mathrm{a}}$ is of dimension $M$. Plugging this in equations (26) and
(28) gives

$\displaystyle({\ell}_{\mathrm{a}}-1){\ell}_{\mathrm{a}}({\ell}_{\mathrm{a}}+1)({\ell}_{\mathrm{a}}+2)h_{{\ell}_{\mathrm{a}}}-12M\dot{h}_{{\ell}_{\mathrm{a}}}=0\,,$
(31)
$\displaystyle({\ell}_{\mathrm{p}}-1){\ell}_{\mathrm{p}}({\ell}_{\mathrm{p}}+1)({\ell}_{\mathrm{p}}+2)h_{{\ell}_{\mathrm{p}}}+12M\dot{h}_{{\ell}_{\mathrm{p}}}=0\,.$
(32)

Defining $\kappa$ by

$\kappa(\ell)=\frac{(\ell-1)\ell(\ell+1)(\ell+2)}{12M}\,,$ (33)

the general solutions for the ASP are given by

$F_{1}={E}_{\mathrm{p}}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}e^{-{\kappa}_{\mathrm{p}}u}\mbox{\quad
and\quad}f_{1}={E}_{\mathrm{a}}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}e^{+{\kappa}_{\mathrm{a}}u}\,,$
(34)

where ${\kappa}_{\mathrm{a}}=\kappa({\ell}_{\mathrm{a}})$ and
${\kappa}_{\mathrm{p}}=\kappa({\ell}_{\mathrm{p}})$. One should note that,
since perturbations $F_{1}$ and $f_{1}$ are decoupled, ${\ell}_{\mathrm{a}}$
and ${\ell}_{\mathrm{p}}$ are different in general. If one inverts the time
direction, the above solutions are mapped to

$F_{1}={E}_{\mathrm{p}}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}e^{+{\kappa}_{\mathrm{p}}v}\mbox{\quad
and\quad}f_{1}={E}_{\mathrm{a}}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}e^{-{\kappa}_{\mathrm{a}}v}\,,$
(35)

where $v$ is the advanced null coordinate. We have thus recovered the
expressions for the linear ASP of the Schwarzschild black hole first derived
in Couch:1973zc  and revisited more recently in Qi:1993ey . These ASP
reproduce the well-known solutions derived independently by Chandrasekhar in
Chandrasekhar1984 .

###  III.3 Zero modes

With these exact solutions, we can now revisit the question of the zero modes.
We refer the reader to Nakamura:2021mfv ; Nakamura:2021ftr ; Nakamura:2021jlc
for interesting discussion on their roles. From (33), we see that for $\ell=0$
and $\ell=1$, the ASP are static. Let us describe explicitly the two relevant
cases.

First, consider the case where ${\ell}_{\mathrm{a}}={\ell}_{\mathrm{p}}=0$
which corresponds to a sum of axial and polar linear monopole perturbations.
The metric is given by

$\displaystyle\mathrm{d}s^{2}=-\left(f-\frac{\epsilon{\mathcal{E}}_{p}}{3}\right)\mathrm{d}u^{2}-2\mathrm{d}u\mathrm{d}r-\left(1-\frac{\epsilon{\mathcal{E}}_{p}}{3}\right)\frac{2r^{2}\mathrm{d}z\mathrm{d}\bar{z}}{(1+z\bar{z})^{2}}$
(36)

where we have used the shorthand notation

$f=1-\frac{2M}{r}\,,\qquad{\mathcal{E}}_{a,p}=\sqrt{\frac{3}{\pi}}E_{a,p}\,.$
(37)

Notice that the $\ell=0$ axial zero mode does not appear in the solution, i.e.
the metric does not depend on $E_{a}$. Changing the coordinates to

$\displaystyle u$ $\displaystyle=t-r-2M\log(r-2M)\,,$ (38) $\displaystyle z$
$\displaystyle=\cot\frac{\theta}{2}e^{i\varphi}\,,$ (39)

leads to the metric

$\displaystyle\mathrm{d}s^{2}=-f\left(1-\frac{K}{f}\right)\mathrm{d}t^{2}-\frac{2K}{f}\mathrm{d}t\mathrm{d}r-f^{-1}\left(1-\frac{K}{f}\right)\mathrm{d}r^{2}+r^{2}(1+K)\mathrm{d}\Omega^{2}$
(40)

where the constants $K$ are given by

$K=\frac{\epsilon{\mathcal{E}}_{p}}{\sqrt{3}}\,.$ (41)

To remove the off diagonal terms and bring this metric to the Schwarzschild
form, one can perform the following change of time coordinate given by

$\displaystyle t$
$\displaystyle=\left(1-\frac{K}{2}\right)\tilde{t}+\frac{\epsilon{\mathcal{E}}_{p}}{\sqrt{3}}\left(\tilde{r}+4M\log{(\tilde{r}-2M)}-\frac{4M^{2}}{\tilde{r}-2M}\right)\,,$
(42) $\displaystyle r$ $\displaystyle=\left(1+\frac{K}{2}\right)\tilde{r}\,,$
(43)

which brings the metric to the standard Schwarzschild form

$\displaystyle\mathrm{d}s^{2}=-\left(1-\frac{2M(1+\epsilon
M_{1})}{r}\right)\mathrm{d}t^{2}+\left(1-\frac{2M(1+\epsilon
M_{1})}{r}\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}\,,$ (44)

where the shift in mass is given by

$M_{1}=-\frac{3K}{\sqrt{3}}\,.$ (45)

As expected, the $\ell=0$ polar zero mode simply shifts the Schwarzschild
mass.

Next, let us consider the zero mode corresponding to
${\ell}_{\mathrm{a}}={\ell}_{\mathrm{p}}=1$. The resulting geometry is
stationary and given by

$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}-2\epsilon{\mathcal{E}}_{a}\sin^{2}{\theta}\left(\frac{M}{r}\mathrm{d}t-\frac{\mathrm{d}r}{2f}\right)\mathrm{d}\varphi+\frac{\mathrm{d}r^{2}}{f}+r^{2}\left(1-\epsilon{\mathcal{E}}_{p}\cos{\theta}\right)\mathrm{d}\Omega^{2}\,.$
(46)

Despite the non standard off diagonal terms and the presence of the three
constants $(M,{\mathcal{E}}_{a},{\mathcal{E}}_{p})$, this line element
describes the linearized Kerr black hole solution. To see that, one can apply
the following change of the angular coordinates:

$\displaystyle\theta=\tilde{\theta}+\frac{1}{2}\epsilon{\mathcal{E}}_{p}\sin{\tilde{\theta}}\,,\qquad\qquad\varphi=\tilde{\varphi}-\frac{\epsilon{\mathcal{E}}_{p}}{4M}\log{f}\,.$
(47)

Within these new coordinates, the metric takes the more familiar form

$\mathrm{d}s^{2}=-f\mathrm{d}t^{2}-\frac{2\epsilon{\mathcal{E}}_{a}M}{r}\sin^{2}{\tilde{\theta}}\mathrm{d}t\mathrm{d}\tilde{\varphi}+\frac{\mathrm{d}r^{2}}{f}+r^{2}\mathrm{d}\tilde{\Omega}^{2}\,,$
(48)

which one recognizes as being the well-known Kerr black hole geometry
linearized w.r.t its angular momentum. We see that at the linear level, the
dipolar axial perturbation introduces a (small) angular momentum controlled by
its amplitude ${\mathcal{E}}_{a}$. However, the dipolar polar perturbation can
always be absorbed by the change of coordinates (47) and the additional
constant ${\mathcal{E}}_{p}$ completely disappears, showing that it is a pure
gauge, in agreement with the standard result Martel:2005ir .

##  IV Quadratic algebraically special perturbations

In this section, we generalize the results of Couch and Newman and we analyze
the dynamics and present the analytic expression for the profiles of quadratic
ASP of the Schwarzschild black hole.

###  IV.1 Dynamics

Following the same procedure as for the linear perturbations, we now expand
the two functions $(P,L)$ up to second order in perturbations

$P(u,z,\bar{z})=P_{0}e^{\varepsilon F_{1}+\varepsilon^{2}F_{2}}\mbox{\quad
and\quad}L(u,z,\bar{z})=\varepsilon L_{1}+\varepsilon^{2}L_{2}\,.$ (49)

Furthermore, we introduce the function $f_{2}$ such that

$L_{2}=i\partial_{z}f_{2}+{\kappa}_{\mathrm{a}}\partial_{z}(f_{1}^{2})\,.$
(50)

where we recall that $L_{1}$ is related to $f_{1}$ via
$L_{1}=i\partial_{z}f_{1}$. Now, using the expressions for the linear
perturbations $(F_{1},f_{1})$, we expand up to second order the dynamical
equations (18) and (19) which encdes the dynamics of the second order
perturbations $(F_{2},f_{2})$.

After a lengthy computation, the dynamics for the second order axial
perturbation $f_{2}$ splits into

$\boxed{\Delta_{0}\Delta_{0}f_{2}+2\Delta_{0}f_{2}-12M\partial_{u}f_{2}=\mathcal{A}}\,,$
(51)

where ${\mathcal{A}}$ stands for the source term generated by the polar-axial
mode interaction. Its expression reads

$\mathcal{A}=-2F_{1}\Delta_{0}\Delta_{0}f_{1}-2\Delta_{0}F_{1}\Delta_{0}f_{1}-2\Delta_{0}(F_{1}\Delta_{0}f_{1})-8F_{1}\Delta_{0}f_{1}\,.$
(52)

After expanding up to second order the equation (19), and having split its
imaginary and real parts, the dynamical equation for the second order polar
perturbation is given by

$\boxed{\Delta_{0}\Delta_{0}F_{2}+2\Delta_{0}F_{2}+12M\partial_{u}F_{2}=\mathcal{P}+\mathcal{Q}+\mathcal{R}}\,,$
(53)

where the source term is split into three different contributions given
explicitly by

$\displaystyle\mathcal{P}$
$\displaystyle=-2\Delta_{0}(F_{1}\Delta_{0}F_{1})-2\Delta_{0}(F_{1}^{2})-2F_{1}\Delta_{0}\Delta_{0}F_{1}-4F_{1}\Delta_{0}F_{1}\,,$
(54) $\displaystyle\mathcal{Q}$
$\displaystyle={\kappa}_{\mathrm{a}}^{2}\Big{[}\Delta_{0}(f_{1}^{2})-f_{1}\Delta_{0}\Delta_{0}f_{1}+4(\Delta_{0}f_{1})^{2}+6f_{1}\Delta_{0}f_{1}+\frac{3}{2}\Delta_{0}\Delta_{0}(f_{1}^{2})+2\Delta_{0}(f_{1}\Delta_{0}f_{1})\Big{]}\,,$
(55) $\displaystyle\mathcal{R}$
$\displaystyle=8{\kappa}_{\mathrm{p}}\imaginary[\partial_{\bar{z}}(P_{0}^{2}\partial_{\bar{z}}F_{1})\partial_{z}(P_{0}^{2}\partial_{z}f_{1})]-8({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})\imaginary[P_{0}^{2}\partial_{z}f_{1}(\partial_{\bar{z}}\Delta_{0}F_{1}+2\partial_{\bar{z}}F_{1})]\,.$
(56)

It is direct to see that $\mathcal{P}$ encodes the polar-polar interaction,
$\mathcal{Q}$ encodes the axial-axial interaction and finally $\mathcal{R}$
encodes the polar-axial interaction generating the source term. Let us further
point that the differential operator acting on the second order perturbations
$(F_{2},f_{2})$ is exactly the same as the one entering in the wave equation
verified by the linear perturbations $(F_{1},f_{1})$. We now turn to the
resolution of these equations.

###  IV.2 Exact solutions

Despite the rather complicated form of these equations, let us show how one
can construct exact analytical solutions for the quadratic ASP. This is the
first main result of this work.

####  IV.2.1 Axial perturbations

Consider first the quadratic axial ASP encoded in $f_{2}$. In order to solve
Eqs. (51), one can follow a procedure close to the one used to solve the
linear perturbation equations. Given that $f_{2}$ is sourced only by axial-
polar interactions, we can take the following ansatze:

$f_{2}={E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\sum_{\ell,m}a^{\ell}_{m}Y_{\ell}^{m}\,,$
(57)

where the $a^{\ell}_{m}$ are constant coefficients. These can be obtained from
Eqs. (51) by making use of the decomposition of any product
$Y_{\ell_{1}}Y_{\ell_{2}}$ onto the basis of $Y_{\ell}$. This decomposition
uses the Clebsch-Gordan coefficients $k_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}$,
whose definition is given in App. A.3 for convenience. Using this ansatze, the
source term decomposes as

$\displaystyle\mathcal{A}$
$\displaystyle=-{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\Big{[}2Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\Delta_{0}\Delta_{0}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}+2\Delta_{0}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\Delta_{0}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}+2\Delta_{0}(Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\Delta_{0}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}})+8Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\Delta_{0}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}\Big{]}\,,$
$\displaystyle={E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\sum_{\ell}\mathcal{A}_{m}^{\ell}Y_{\ell}^{m}\,,$
(58)

where the coefficients $\mathcal{A}_{m}^{\ell}$ are given explicitly by

$\mathcal{A}_{m}^{\ell}=-8({\lambda}_{\mathrm{a}}+1)({\lambda}_{\mathrm{p}}+{\lambda}_{\mathrm{a}}+\lambda+1)k^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}\,,$
(59)

In these expressions, $\lambda$ is defined from $\ell$ (and similarly,
${\lambda}_{\mathrm{a}}$ and ${\lambda}_{\mathrm{p}}$ from
${\ell}_{\mathrm{a}}$ and ${\ell}_{\mathrm{p}}$ respectively) using Eq. (6).
Using these expressions and plugging them in the dynamical equation (51), one
obtains the following constraint

$\sum_{\ell,m}\Big{[}(\ell-1)\ell(\ell+1)(\ell+2)a^{\ell}_{m}-12M({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})a^{\ell}_{m}-\mathcal{A}^{\ell}_{m}\Big{]}Y_{\ell}^{m}=0\,.$
(60)

which holds for all $\ell$. Since the $Y_{\ell}^{m}$ form a basis of the space
of real functions on the 2-sphere, every coefficient must be put to zero,
leading to

$12M(\kappa-{\kappa}_{\mathrm{a}}+{\kappa}_{\mathrm{p}})a^{\ell}_{m}-\mathcal{A}^{\ell}_{m}=0\,.$
(61)

If ${\kappa}_{\mathrm{a}}\neq{\kappa}_{\mathrm{p}}$, this equation can be
solved by imposing

$a^{\ell}_{m}=\frac{1}{12M}\frac{\mathcal{A}^{\ell}_{m}}{\kappa-{\kappa}_{\mathrm{a}}+{\kappa}_{\mathrm{p}}}\,.$
(62)

If one has $\kappa={\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}}$ then the only
solution is $a^{\ell}_{m}=0$, except when $\ell=0$ or $\ell=1$. The question
oof the zero modes will be discussed at the end of the section.

####  IV.2.2 Polar perturbations

Let us now consider the quadratic polar perturbations encoded in $F_{2}$. In
this case, the source term is much more involved. Nevertheless, a good ansatze
to this equations is given by

$F_{2}={E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\sum_{\ell,m}p^{\ell}_{m}Y_{\ell}^{m}+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\sum_{\ell,m}q^{\ell}_{m}Y_{\ell}^{m}+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\sum_{\ell,m}r^{\ell}_{m}Y_{\ell}^{m}\,,$
(63)

with $p^{\ell}_{m}$, $q^{\ell}_{m}$ and $r^{\ell}_{m}$ constant coefficients.
Decomposing the source terms $\mathcal{P}$ and $\mathcal{Q}$, one obtains the
following expression for $\mathcal{P}$

$\displaystyle\mathcal{P}$
$\displaystyle=-2{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\Big{\\{}\Big{[}-{\ell}_{\mathrm{p}}({\ell}_{\mathrm{p}}+1)+1\Big{]}\Delta_{0}(Y_{{\ell}_{\mathrm{p}}}^{2})+\Big{[}{\ell}_{\mathrm{p}}^{2}({\ell}_{\mathrm{p}}+1)^{2}-2{\ell}_{\mathrm{p}}({\ell}_{\mathrm{p}}+1)\Big{]}Y_{{\ell}_{\mathrm{p}}}^{2}\Big{\\}}\,,$
$\displaystyle={E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\sum_{\ell,m}\mathcal{P}^{\ell}_{m}Y_{\ell}^{m}\,.$
(64)

while $\mathcal{Q}$ is given by

$\displaystyle\mathcal{Q}$
$\displaystyle={\kappa}_{\mathrm{a}}^{2}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\Big{\\{}\frac{3}{2}\Delta_{0}\Delta_{0}(Y_{{\ell}_{\mathrm{a}}}^{2})+\Big{[}-2{\ell}_{\mathrm{a}}({\ell}_{\mathrm{a}}+1)+1\Big{]}\Delta_{0}(Y_{{\ell}_{\mathrm{a}}}^{2})+\Big{[}3{\ell}_{\mathrm{a}}^{2}({\ell}_{\mathrm{a}}+1)^{2}-6{\ell}_{\mathrm{a}}({\ell}_{\mathrm{a}}+1)\Big{]}Y_{{\ell}_{\mathrm{a}}}^{2}\Big{\\}}\,,$
$\displaystyle={E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\sum_{\ell,m}\mathcal{Q}^{\ell}_{m}Y_{\ell}^{m}\,.$
(65)

In both expressions, the coefficients $\mathcal{P}^{\ell}_{m}$ and
$\mathcal{Q}^{\ell}_{m}$ are defined by

$\displaystyle\mathcal{P}^{\ell}_{m}$
$\displaystyle=-4\Big{[}1+\lambda+4{\lambda}_{\mathrm{p}}+2\lambda{\lambda}_{\mathrm{p}}+2{\lambda}_{\mathrm{p}}^{2}\Big{]}k^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}\,,$
(66) $\displaystyle\mathcal{Q}^{\ell}_{m}$
$\displaystyle=2{\kappa}_{\mathrm{a}}^{2}\Big{[}6+3\lambda(\lambda+3)+10{\lambda}_{\mathrm{a}}+4\lambda{\lambda}_{\mathrm{a}}+6{\lambda}_{\mathrm{a}}^{2}\Big{]}k^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}\,.$
(67)

In order to decompose the source term $\mathcal{R}$, it is convenient to
introduce the coefficients $h$ and $l$ defined in App. A.3. Using these
definitions, one can write

$\mathcal{R}={E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\sum_{\ell,m}\mathcal{R}^{\ell}_{m}Y_{\ell}^{m}\,,$
(68)

where the coefficients $\mathcal{R}^{\ell}_{m}$ reads

$\mathcal{R}^{\ell}_{m}=8{\kappa}_{\mathrm{p}}\imaginary(l_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{)}+16{\lambda}_{\mathrm{p}}({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})\imaginary(h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{)}\,.$
(69)

In the case ${m}_{\mathrm{a}}={m}_{\mathrm{p}}=0$, one has $\mathcal{R}=0$,
because $h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{00\ell}$ and
$l_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{00\ell}$ are reals (see App. A).
This means that for ${m}_{\mathrm{a}}={m}_{\mathrm{p}}=0$, quadratic polar
perturbations cannot be sourced by axial-polar interaction; this is expected
for parity conservation reasons.

Using the above decomposition of the source terms, the dynamical equation (53)
for the quadratic polar perturbations splits into three conditions given by

$\displaystyle\sum_{\ell}\Big{[}(\ell-1)\ell(\ell+1)(\ell+2)p^{\ell}_{m}-24M{\kappa}_{\mathrm{p}}p^{\ell}_{m}-\mathcal{P}^{\ell}_{m}\Big{]}Y_{\ell}=0\,,$
(70)
$\displaystyle\sum_{\ell}\Big{[}(\ell-1)\ell(\ell+1)(\ell+2)q^{\ell}_{m}+24M{\kappa}_{\mathrm{a}}q^{\ell}_{m}-\mathcal{Q}^{\ell}_{m}\Big{]}Y_{\ell}=0\,,$
(71)
$\displaystyle\sum_{\ell}\Big{[}(\ell-1)\ell(\ell+1)(\ell+2)r^{\ell}_{m}+12M({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})r^{\ell}_{m}-\mathcal{R}^{\ell}_{m}\Big{]}Y_{\ell}=0\,.$
(72)

These equations are solved by taking

$p^{\ell}_{m}=\frac{1}{12M}\frac{\mathcal{P}^{\ell}_{m}}{\kappa-2{\kappa}_{\mathrm{p}}}\,,\quad
q^{\ell}_{m}=\frac{1}{12M}\frac{\mathcal{Q}^{\ell}_{m}}{\kappa+2{\kappa}_{\mathrm{a}}}\,,\quad
r^{\ell}_{m}=\frac{1}{12M}\frac{\mathcal{R}^{\ell}_{m}}{\kappa+{\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}}}\,.$
(73)

These expressions provide an analytic exact solutions for the quadratic ASP of
the Schwarzschild black hole.

In order to demonstrate the usefulness of the exact analytical solutions
presented here, we have computed the profiles of the quadratic perturbations
$(F_{2},f_{2})$ sourced by linear perturbations $(F_{1},f_{1})$ with quantum
numbers ${\ell}_{\mathrm{a}}=2$, ${m}_{\mathrm{a}}=1$ and
${\ell}_{\mathrm{p}}=3$, ${m}_{\mathrm{p}}=2$. The detailed derivation is
presented in appendix C. This solution describes the quadratic perturbation
sourced by the interaction of a linear axial quadrupole with a linear polar
hexapole which have a decay time given respectively by

${\kappa}_{\mathrm{a}}=\frac{2}{M}\,,\quad{\kappa}_{\mathrm{p}}=\frac{10}{M}$
(74)

The analytic profile of the quadratic axial ASP sourced by the polar-axial
interaction is given by

$f_{2}={E}_{\mathrm{p}}{E}_{\mathrm{a}}\Big{(}-\sqrt{\frac{3}{7\pi}}Y_{1}^{1}-\frac{13}{18\sqrt{6\pi}}Y_{3}^{1}-\frac{13}{54}\sqrt{\frac{5}{2\pi}}Y_{3}^{3}+\frac{2}{39}\sqrt{\frac{55}{21\pi}}Y_{5}^{1}-\frac{2}{117}\sqrt{\frac{110}{\pi}}Y_{5}^{3}\Big{)}e^{-8u/M}\,.$
(75)

while the quadratic polar ASP has additional contributions coming from the
polar-polar, axial-axial and axial-polar interactions. It reads

$\displaystyle F_{2}$
$\displaystyle={E}_{\mathrm{p}}^{2}\Big{(}\frac{1}{2\sqrt{\pi}}Y_{0}^{0}+\frac{119}{66\sqrt{\pi}}Y_{4}^{0}-\frac{17}{66}\sqrt{\frac{35}{\pi}}Y_{4}^{4}-\frac{97}{264\sqrt{13\pi}}Y_{6}^{0}-\frac{97}{264}\sqrt{\frac{7}{13\pi}}Y_{6}^{4}\Big{)}e^{-20u/M}$
(76)
$\displaystyle+\frac{{E}_{\mathrm{a}}^{2}}{M^{2}}\Big{(}\frac{3}{\sqrt{\pi}}Y_{0}^{0}+\frac{16}{21}\sqrt{\frac{5}{\pi}}Y_{2}^{0}+\frac{16}{7}\sqrt{\frac{5}{3\pi}}Y_{2}^{2}-\frac{892}{357\sqrt{\pi}}Y_{4}^{0}+\frac{446}{357}\sqrt{\frac{5}{\pi}}Y_{4}^{2}\Big{)}e^{4u/M}$
(77)
$\displaystyle+\frac{{E}_{\mathrm{p}}{E}_{\mathrm{a}}}{M}\Big{(}\frac{25}{33}\sqrt{\frac{5}{2\pi}}Y_{4}^{-3}-\frac{25}{33}\sqrt{\frac{35}{2\pi}}Y_{4}^{-1}\Big{)}e^{-8u/M}\,.$
(78)

Notice that the contribution coming from the axial-polar interaction is non-
vanishing because we have chosen linear ASP with non-vanishing magnetic
numbers.

###  IV.3 Quadratic zero modes

Now, we can use our exact solutions for both polar and axial quadratic ASP to
analyze the quadratic zero modes. From (57) and (63), we see that in order to
have time-independent quadratic perturbations, one has to fix

${\kappa}_{\mathrm{a}}={\kappa}_{\mathrm{p}}=0$ (79)

which further select the time-independent linear perturbations. They are of
three types:

  * •

${\ell}_{\mathrm{a}}={\ell}_{\mathrm{p}}=1$

  * •

${\ell}_{\mathrm{a}}=0$ and ${\ell}_{\mathrm{p}}=1$

  * •

${\ell}_{\mathrm{a}}=1$ and ${\ell}_{\mathrm{p}}=0$

For each of them, three subcases have to be distinguish depending on the
values of the magnetic numbers ${m}_{\mathrm{a}}$ and ${m}_{\mathrm{p}}$.

####  IV.3.1 Monopole-monopole zero mode

Consider the case where ${\ell}_{\mathrm{a}}={\ell}_{\mathrm{p}}=0$. The
associated metric is given by

$\displaystyle\mathrm{d}s^{2}=-f\left(1-\frac{K_{+}}{f}\right)\mathrm{d}t^{2}+\frac{2K_{+}}{f}\mathrm{d}t\mathrm{d}r-f^{-1}\left(1-\frac{K_{+}}{f}\right)\mathrm{d}r^{2}+r^{2}(1-K_{-})\mathrm{d}\Omega^{2}$
(80)

where the constants $K_{\pm}$ are given by

$K_{\pm}=\frac{\epsilon{\mathcal{E}}_{p}}{\sqrt{3}}\left(\frac{\epsilon{\mathcal{E}}_{p}}{2\sqrt{3}}\pm
1\right)$ (81)

To compare this to the Schwarzschild metric, we need to remove the off
diagonal terms. This is achieved by performing the change

$\displaystyle t$
$\displaystyle=\left(1+\frac{K_{-}}{2}\right)\tilde{t}+\frac{\epsilon{\mathcal{E}}_{p}}{\sqrt{3}}\left(\tilde{r}+4M\log{(\tilde{r}-2M)}-\frac{4M^{2}}{\tilde{r}-2M}\right)+\frac{4\epsilon^{2}{\mathcal{E}}^{2}_{p}M}{\sqrt{3}}\left(\frac{2M(2\tilde{r}-3M)}{(\tilde{r}-2M)^{2}}-\log{(\tilde{r}-2M)}\right)$
(82) $\displaystyle r$ $\displaystyle=\left(1+\frac{K_{+}}{2}\right)\tilde{r}$
(83)

which bring the metric to the standard Schwarzschild form

$\displaystyle\mathrm{d}s^{2}=-\left(1-\frac{2M(1+\epsilon
M_{1}+\epsilon^{2}M_{2})}{r}\right)\mathrm{d}t^{2}+\left(1-\frac{2M(1+\epsilon
M_{1}+\epsilon^{2}M_{2})}{r}\right)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}$
(84)

The new constants $(M_{1},M_{2})$ are given in term of the polar perturbation
by

$M_{1}=-\frac{3}{2\sqrt{3}}{\mathcal{E}}_{p}\mbox{\quad
and\quad}M_{2}=\frac{3}{8}{\mathcal{E}}^{2}_{p}\,.$ (85)

As expected, it corresponds to a quadratic shift of the Schwarzschild mass.

####  IV.3.2 Dipole-dipole zero mode

Let us focus on the simplest one corresponding to
${\ell}_{\mathrm{a}}={\ell}_{\mathrm{p}}=1$ and
${m}_{\mathrm{a}}={m}_{\mathrm{p}}=0$. This geometry does not depend on time
nor on the angular coordinate $\varphi$ since the magnetic numbers have been
chosen to vanish. It is therefore axi-symmetric and the associated metric is
given by

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=\left(1-\frac{K_{1}(r,\theta)}{r^{3}f}\right)\left(-f\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f}\right)+\frac{2K_{1}(r,\theta)}{r^{3}f}\mathrm{d}t\mathrm{d}r+2K_{4}(r,\theta)\left(\frac{M}{r}\mathrm{d}t-\frac{\mathrm{d}r}{2f}\right)\mathrm{d}\varphi$
(86)
$\displaystyle\;\;\;+r^{2}\left(K_{2}(r,\theta)\mathrm{d}\theta^{2}+K_{3}(r,\theta)\sin^{2}{\theta}\mathrm{d}\varphi^{2}\right)$
(87)

where the four functions are given by

$\displaystyle K_{1}(r,\theta)$
$\displaystyle=\frac{1}{4}\left(\epsilon^{2}{\mathcal{E}}^{2}_{a}M(1+\cos{2\theta})-\frac{2}{3}\epsilon^{2}{\mathcal{E}}^{2}_{p}r^{3}\right)$
(88) $\displaystyle K_{2}(r,\theta)$
$\displaystyle=1-\epsilon{\mathcal{E}}_{p}\cos{\theta}+\frac{\epsilon^{2}}{8}\left({\mathcal{E}}^{2}_{a}+\frac{7}{3}{\mathcal{E}}^{2}_{p}r^{2}+({\mathcal{E}}^{2}_{a}+3{\mathcal{E}}^{2}_{p}r^{2})\cos{2\theta}\right)$
(89) $\displaystyle K_{3}(r,\theta)$
$\displaystyle=1-\epsilon{\mathcal{E}}_{p}\cos{\theta}+\frac{\epsilon^{2}}{8}\left(\frac{7}{3}{\mathcal{E}}^{2}_{p}r^{3}+2{\mathcal{E}}^{2}_{a}(M+r)+({\mathcal{E}}^{2}_{p}r^{3}-2{\mathcal{E}}^{2}_{a}M)\cos{2\theta}\right)$
(90) $\displaystyle K_{4}(r,\theta)$
$\displaystyle=-\left(1-\epsilon{\mathcal{E}}_{p}\cos{\theta}\right)\epsilon{\mathcal{E}}_{a}M\sin^{2}{\theta}$
(91)

By virtue of the no-hair theorem, this metric is expected to be related to the
slowly rotating Kerr metric up to quadratic order in the spin. Let us write
down the explicit diffemorphism which allows to transform the above metric to
the more familiar form. To proceed, we introduce the following change of
coordinates

$\displaystyle t$
$\displaystyle=\left(1+\frac{\epsilon^{2}{\mathcal{E}}^{2}_{p}}{12}\right)\tilde{t}+\frac{\epsilon^{2}}{8}\left[\frac{2\tilde{r}}{f(\tilde{r})}\left(\frac{8}{3}{\mathcal{E}}^{2}_{p}M^{2}-{\mathcal{E}}^{2}_{a}\right)-\frac{{\mathcal{E}}^{2}_{a}}{M}\log{f(\tilde{r})}-\frac{16}{3}{\mathcal{E}}^{2}_{p}M\log{(\tilde{r}-2M)}-\frac{{\mathcal{E}}^{2}_{p}\tilde{r}}{12}\right]\,,$
(92) $\displaystyle r$
$\displaystyle=\left(1-\frac{\epsilon^{2}{\mathcal{E}}^{2}_{p}}{12}\right)\tilde{r}\,,$
(93) $\displaystyle\theta$
$\displaystyle=\tilde{\theta}+\frac{1}{2}\epsilon{\mathcal{E}}_{p}\sin{\tilde{\theta}}+\frac{\epsilon^{2}}{16}{\mathcal{E}}^{2}_{p}\sin{2\tilde{\theta}}\,,$
(94) $\displaystyle\varphi$
$\displaystyle=\tilde{\varphi}-\frac{\epsilon{\mathcal{E}}_{a}}{4M}\log{f(\tilde{r})}\,.$
(95)

Within these new coordinates, the off diagonal $g_{tr}$ and $g_{r\varphi}$ in
(86) disappear and the metric takes the following form

$\displaystyle\mathrm{d}s^{2}$
$\displaystyle=-\left[f(\tilde{r})+\epsilon^{2}\left(\frac{2M_{2}}{r}-\frac{2aM\cos^{2}{\tilde{\theta}}}{\tilde{r}^{3}}\right)\right]\mathrm{d}\tilde{t}^{2}+\left[f^{-1}(\tilde{r})+\frac{\epsilon^{2}}{\tilde{r}f(\tilde{r})}\left(\frac{a^{2}-2\tilde{r}M_{2}}{f(\tilde{r})}+a^{2}\cos^{2}{\tilde{\theta}}\right)\right]\mathrm{d}\tilde{r}^{2}$
$\displaystyle-\frac{4\epsilon
aM\sin^{2}{\tilde{\theta}}}{\tilde{r}}\mathrm{d}\tilde{t}\mathrm{d}\tilde{\varphi}+\left(\tilde{r}^{2}+\epsilon^{2}a^{2}\cos^{2}{\tilde{\theta}}\right)\mathrm{d}\tilde{\theta}^{2}+\left[1+\frac{\epsilon^{2}a^{2}(\tilde{r}+2M\sin^{2}{\tilde{\theta}})}{\tilde{r}^{3}}\right]\tilde{r}^{2}\sin^{2}{\tilde{\theta}}\mathrm{d}\tilde{\varphi}^{2}\,,$
(96)

where the constants $(a,{\mathcal{M}},M_{2})$ are given in terms of the
perturbations by

$M_{2}=\frac{{\mathcal{E}}_{p}^{2}M}{4}\mbox{\quad
and\quad}a=\frac{{\mathcal{E}}_{a}}{2}\,.$ (97)

and where $M$ is the original Schwarzschild mass. As expected, it corresponds
to the slowly rotating Kerr metric up to quadratic order in the spin $a$. We
see that the Schwarzschild mass is shifted by the polar perturbation, while
the spin is generated by the axial perturbation. This geometry admits a new
Killing horizon located at

$r_{+}=2M+\frac{\epsilon^{2}M}{3}\left({\mathcal{E}}^{2}_{p}-\frac{{\mathcal{E}}^{2}_{a}}{8M^{2}}\right)$
(98)

Therefore, contrary to the linear regime where the horizon of the black hole
perturbed by the dipolar zero mode is not shifted, in the non-linear regime,
the Schwarzschild black hole geometry dressed with the quadratic zero modes
admits a shifted horizon at $r_{+}\neq 2M$.

##  V Discussion

In this work, we have revisited the description of the algebraically special
perturbations (ASPs) of the Schwarzschild black hole and extended the analysis
of Couch and Newman to the non-linear regime. The dynamic of the quadratic
ASPs can be derived by expanding up to second order the reduced Einstein
equations (18 – 19) of the most general vacuum twisting family of solutions of
general relativity. Expanding these equations up to second order in the
perturbations of the two fields $(P,L)$ which label this family, we have
obtained the master non-linear wave equations (51) and (53) which describe
respectively the dynamic of the quadratic axial perturbations $f_{2}$ and the
dynamic of quadratic polar perturbations $F_{2}$. By construction, these
solutions of these equations preserve the Petrov type II of the geometry.

A nice feature of these equations is that the sources can be written
explicitly. From these expressions, we see that the quadratic axial
perturbation $f_{2}$ is sourced by the axial-polar interaction encoded in the
source ${\mathcal{A}}$ given by (52), while the quadratic polar perturbation
$F_{2}$ is sourced by three types of interactions: polar-polar (54), axial-
axial (55) and axial-polar (56). This last contribution is only present when
the involved linear perturbations exhibit non-zero magnetic numbers.

Then, we have shown that upon using the ansätze (57) and (63) together with a
suitable decomposition of the sources in terms of spherical harmonics, one can
construct analytic solutions to these non-linear wave equations upon solving
the condition (61) for the axial perturbations and the relations (73) for the
polar perturbations. We have further presented an explicit example, given by
(74) and (75), of an analytic solution corresponding to a quadratic ASP
sourced by a linear axial quadrupole and a linear polar hexapole. This example
demonstrates that i) the damping time of the quadratic ASPs is related as
expected to the damping times of the linear perturbations and ii) that the
known divergence at the horizon of the linear ASPs is not cured by their non-
linear interactions.

Finally, we have treated the case of the quadratic zero modes, focusing on the
$\ell=0$ and $\ell=1$ quadratic ASPs. We have shown that as expected, the
quadratic zero mode sourced by the $\ell=0$ axial and polar linear modes
corresponds to a Schwarzschild black hole with quadratic corrections of its
mass given by (84). These corrections are sourced solely by the polar mode
through its amplitude ${\mathcal{E}}_{p}$. Similarly, we have shown that the
quadratic zero mode sourced by the $\ell=1$ axial and polar zero linear modes
(with ${m}_{\mathrm{a}}={m}_{\mathrm{p}}=0$) deform the Schwarzschild
background to the Kerr geometry up to quadratic corrections in the spin and in
the mass corresponding to the metric (96). As expected, the spin correction
(97) is generated by the axial sector and related to the amplitude
${\mathcal{E}}_{a}$. While these results are expected, we provide here the
explicit diffeomorphisms one has to implement to confirm these results. There
are given by (82 \- 83) and (92 \- 95). To our knowledge, this has not been
presented in that form elsewhere.

This study can serves as the basis to explore several open questions. First,
it would be interesting to determine if, similarly to the linear case
MaassenvandenBrink:2000iwh , one can find well-behaved highly damped QQNMs of
the Schwarzschild black hole which are spectrally close (i.e. numerically
indistinguishable) from the quadratic ASPs described in this work. Second, one
can use the expression for the quadratic ASP to search for possible hidden
symmetry structures in the non-linear regime of black hole perturbations.
Finally, we stress that while pathological, the ASPs provide one of the few
examples corresponding to explicit solutions in the time-domain. Finding other
exact solutions within a closed form in the time-domain but describing instead
well-behaved perturbations remains an interesting goal, as it would surely
allows one to better understand the different phases of the signal during the
different stages of the relaxation process.

##  Appendix A Spherical harmonics in stereographic coordinates

In this appendix, we present the relevant properties of the spherical
harmonics which we use throughout this work.

###  A.1 Definition

Spherical harmonics are eigenfunctions of the Laplacian operator $\Delta_{0}$:

$\Delta_{0}Y_{\ell}^{m}=-\ell(\ell+1)Y_{\ell}^{m}\,.$ (99)

They are indexed by the two integers $\ell$ and $m$, with $\ell\in\mathbb{N}$
and $m\in\llbracket-\ell,\ell\rrbracket$. The most common way to construct
them is the following:

$\underline{Y}_{\ell}^{m}=(-1)^{m}\sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}}P_{\ell}^{m}\Big{(}\frac{z\bar{z}-1}{z\bar{z}+1}\Big{)}\Big{(}\frac{z}{\bar{z}}\Big{)}^{m/2}\,.$
(100)

Here, $P_{\ell}^{m}$ represent an associated Legendre polynomial, and we have
written the eigenfunctions $\underline{Y}$ to emphasize the fact that they
take their values in $\mathbb{C}$.

As defined in (100), spherical harmonics satisfy the following properties
beyond the eigenvalue equation (99):

$\displaystyle\Big{(}z\partialderivative{z}-\bar{z}\partialderivative{\bar{z}}\Big{)}\underline{Y}_{\ell}^{m}=m\underline{Y}_{\ell}^{m}\,,$
(101)
$\displaystyle\big{(}\underline{Y}_{\ell}^{m}\big{)}^{*}=(-1)^{m}\underline{Y}_{\ell}^{-m}\,,$
(102)
$\displaystyle(1+z\bar{z})\partial_{z}\underline{Y}_{\ell}^{m}=\frac{m}{z}\underline{Y}_{\ell}^{m}+\frac{\bar{z}}{z}\sqrt{(\ell-m)(\ell+m+1)}\underline{Y}_{\ell}^{m+1}\,,$
(103)
$\displaystyle(1+z\bar{z})\partial_{\bar{z}}\underline{Y}_{\ell}^{m}=-mz\underline{Y}_{\ell}^{m}+\sqrt{(\ell-m)(\ell+m+1)}\underline{Y}_{\ell}^{m+1}\,,$
(104)
$\displaystyle\int\frac{2\differential{z}\differential{\bar{z}}}{P_{0}^{2}}\underline{Y}_{\ell}^{m}\big{(}\underline{Y}_{\ell^{\prime}}^{m^{\prime}}\big{)}^{*}=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$
(105)

###  A.2 Real spherical harmonics

In this work, we focus on _real_ spherical harmonics. These are built from the
complex harmonics $\underline{Y}_{\ell}^{m}$ by

$Y_{\ell}^{m}=\begin{cases*}\frac{1}{\sqrt{2}}\big{[}\underline{Y}_{\ell}^{m}+(-1)^{m}\underline{Y}_{\ell}^{-m}\big{]}=\sqrt{2}\real(\underline{Y}_{\ell}^{m})&if
$m>0$,\\\ \underline{Y}_{\ell}^{0}&if $m=0$,\\\
\frac{1}{\sqrt{2}i}\big{[}\underline{Y}_{\ell}^{\absolutevalue{m}}-(-1)^{\absolutevalue{m}}\underline{Y}_{\ell}^{-\absolutevalue{m}}\big{]}=\sqrt{2}\imaginary(\underline{Y}_{\ell}^{\absolutevalue{m}})&if
$m<0$.\\\ \end{cases*}$ (106)

These harmonics still satisfy the orthonormality relation:

$\int\frac{2\differential{z}\differential{\bar{z}}}{P_{0}^{2}}Y_{\ell}^{m}Y_{\ell^{\prime}}^{m^{\prime}}=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$
(107)

In table 1, we give the expression of the first real spherical harmonics in
stereographic coordinates.

$\bm{\ell}$ | $\bm{m}$ |  $\displaystyle\bm{Y_{\ell}^{m}}$
---|---|---
0 | 0 |  $\displaystyle\frac{1}{2\sqrt{\pi}}$
1 | -1 |  $\displaystyle\sqrt{\frac{3}{\pi}}\frac{\imaginary(z)}{1+z\bar{z}}$
| 0 |  $\displaystyle\sqrt{\frac{3}{\pi}}\frac{-1+z\bar{z}}{2(1+z\bar{z})}$
| 1 |  $\displaystyle\sqrt{\frac{3}{\pi}}\frac{\real(z)}{1+z\bar{z}}$
2 | -2 |  $\displaystyle\sqrt{\frac{15}{\pi}}\frac{\imaginary(z^{2})}{(1+z\bar{z})^{2}}$
| -1 |  $\displaystyle\sqrt{\frac{15}{\pi}}\imaginary(z)\frac{-1+z\bar{z}}{(1+z\bar{z})^{2}}$
| 0 |  $\displaystyle\frac{1}{2}\sqrt{\frac{5}{\pi}}\frac{1-4z\bar{z}+z^{2}\bar{z}^{2}}{(1+z\bar{z})^{2}}$
| 1 |  $\displaystyle\sqrt{\frac{15}{\pi}}\real(z)\frac{-1+z\bar{z}}{(1+z\bar{z})^{2}}$
| 2 |  $\displaystyle\sqrt{\frac{15}{\pi}}\frac{\real(z^{2})}{(1+z\bar{z})^{2}}$
Table 1: Expressions of the first $Y_{\ell}^{m}$.

###  A.3 Decomposition coefficients

Several times throughout this paper, it is necessary to decompose a product of
$Y_{\ell}^{m}$ functions or their derivatives onto a basis of functions. Here,
we define all the coefficients we use. They can be computed using the scalar
product $\innerproduct{\cdot}{\cdot}$ defined by

$\innerproduct{f}{g}=\int\frac{2\differential{z}\differential{\bar{z}}}{P_{0}^{2}}fg\,.$
(108)

This is possible because the real spherical harmonics form an orthonormal
basis of functions on the 2-sphere.

  * •

The most important coefficients are the ones describing the decomposition of a
product of two spherical harmonics onto the basis of spherical harmonics:

$Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}=\sum_{\ell,m}k^{m_{1}m_{2}\ell}_{\ell_{1}\ell_{2}m}Y_{\ell}^{m}\mbox{\quad
with\quad}k^{m_{1}m_{2}\ell}_{\ell_{1}\ell_{2}m}=\innerproduct{Y_{\ell}^{m}}{Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}}\,.$
(109)

These coefficients are the well-known Clebsch-Gordan coefficients.

  * •

In order to compute the coefficients $\mathcal{R}_{m}^{\ell}$ in (68), the
metric expansions in App. B.1 or the metric reconstruction in Sec. B.3, one
needs to use the coefficients $h$ defined by

$P_{0}^{2}\partial_{z}Y_{\ell_{1}}^{m_{1}}\partial_{\bar{z}}Y_{\ell_{2}}^{m_{2}}=\sum_{\ell,m}h_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}Y_{\ell}^{m}\mbox{\quad
or\quad}h_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}=\innerproduct{Y_{\ell}^{m}}{P_{0}^{2}\partial_{z}Y_{\ell_{1}}^{m_{1}}\partial_{\bar{z}}Y_{\ell_{2}}^{m_{2}}}\,.$
(110)

These coefficients are complex when $(\ell_{1},m_{1})\neq(\ell_{2},m_{2})$: in
general, one has

$\big{(}h_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}\big{)}^{*}=h_{\ell_{2}\ell_{1}m}^{m_{2}m_{1}\ell}\,.$
(111)

One should note that in the case $m_{1}=m_{2}=0$, these coefficients are real.

  * •

The computation of the coefficients $\mathcal{R}^{\ell}_{m}$ also requires the
coefficients $l$ such that

$\partial_{z}(P_{0}^{2}\partial_{z}Y_{\ell_{1}}^{m_{1}})\partial_{\bar{z}}(P_{0}^{2}\partial_{\bar{z}}Y_{\ell_{2}}^{m_{2}})=\sum_{\ell,m}l_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}Y_{\ell}^{m}\,.$
(112)

These coefficients can be computed using

$l_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}=\innerproduct{Y_{\ell}^{m}}{\partial_{\bar{z}}(P_{0}^{2}\partial_{\bar{z}}Y_{\ell_{1}}^{m_{1}})\partial_{z}(P_{0}^{2}\partial_{z}Y_{\ell_{2}}^{m_{2}})}\,.$
(113)

One should note that in the case $m_{1}=m_{2}=0$, these coefficients are real.

  * •

Finally, the metric reconstruction also requires defining the coefficients $j$
such that

$\big{(}\partial_{z}Y_{\ell_{1}}^{m_{1}}\big{)}^{2}=\sum_{\ell,m}j_{\ell_{1}m}^{m_{1}\ell}(\partial_{z}^{2}+2\partial_{z}(\log
P_{0})\partial_{z})Y_{\ell}^{m}\,,$ (114)

and the coefficients $g$ defined by

$Y_{\ell_{1}}^{m_{1}}\partial_{z}Y_{\ell_{2}}^{m_{2}}=\sum_{\ell,m}g_{\ell_{1}\ell_{2}m}^{m_{1}m_{2}\ell}\partial_{z}Y_{\ell}^{m}\,.$
(115)

With these definitions, one can decompose the various appearance of product of
spherical harmonics and their derivatives in a straigtforward way.

##  Appendix B Reconstruction of the perturbed geometry

In this appendix, we present the reconstruction of the full metric up to
second order. We also compute of the metric perturbation functions as defined
by Regge and Wheeler Regge:1957td .

###  B.1 Perturbative expansion of the twisting vacuum radiative metric

Let us first compute the $\mathcal{O}(\varepsilon)$ and
$\mathcal{O}(\varepsilon^{2})$ contributions to the metric perturbations in
the $(u,r,z,\bar{z})$ coordinates. The expansions of the functions appearing
in (12) are written as

$\displaystyle L$ $\displaystyle=\varepsilon
L_{1}+\varepsilon^{2}L_{2}+\mathcal{O}(\varepsilon^{3})\,,$ $\displaystyle P$
$\displaystyle=P_{0}e^{\varepsilon
F_{1}+\varepsilon^{2}F_{2}}+\mathcal{O}(\varepsilon^{3})\,,$ (116)
$\displaystyle\Sigma$
$\displaystyle=\varepsilon\Sigma_{1}+\varepsilon^{2}\Sigma_{2}+\mathcal{O}(\varepsilon^{3})\,,$
$\displaystyle K$ $\displaystyle=1+\varepsilon
K_{1}+\varepsilon^{2}K_{2}+\mathcal{O}(\varepsilon^{3})\,,$ (117)
$\displaystyle W$ $\displaystyle=\varepsilon
W_{1}+\varepsilon^{2}W_{2}+\mathcal{O}(\varepsilon^{3})\,,$ $\displaystyle N$
$\displaystyle=\varepsilon
N_{1}+\varepsilon^{2}N_{2}+\mathcal{O}(\varepsilon^{3})\,,$ (118)
$\displaystyle 2H$ $\displaystyle=A(r)+2\varepsilon
H_{1}+2\varepsilon^{2}H_{2}+\mathcal{O}(\varepsilon^{3})\,,$ (119)

where the linear contributions are given by

$\displaystyle L_{1}$ $\displaystyle=i\partial_{z}f_{1}\,,$ $\displaystyle
P_{1}$ $\displaystyle=P_{0}F_{1}\,,$ $\displaystyle\Sigma_{1}$
$\displaystyle=\frac{1}{2}\Delta_{0}f_{1}\,,$ (120) $\displaystyle K_{1}$
$\displaystyle=\Delta_{0}F_{1}+2F_{1}\,,$ $\displaystyle W_{1}$
$\displaystyle=-r\partial_{u}L_{1}+i\partial_{z}\Sigma_{1}\,,$ $\displaystyle
N_{1}$
$\displaystyle=\frac{1}{4}\Delta_{0}\Delta_{0}f_{1}+\frac{1}{2}\Delta_{0}f_{1}\,,$
(121) $\displaystyle H_{1}$
$\displaystyle=\frac{1}{2}K_{1}-r\partial_{u}F_{1}\,,$ (122)

while the quadratic perturbations read

$\displaystyle L_{2}$ $\displaystyle=i\partial_{z}f_{2}+g\,,$ (123)
$\displaystyle g$
$\displaystyle={\kappa}_{\mathrm{a}}\partial_{z}(f_{1}^{2})\,,$ (124)
$\displaystyle P_{2}$
$\displaystyle=P_{0}\Big{(}F_{2}+\frac{1}{2}F_{1}^{2}\Big{)}\,,$ (125)
$\displaystyle\Sigma_{2}$
$\displaystyle=\frac{1}{2}\Delta_{0}f_{2}+F_{1}\Delta_{0}f_{1}+P_{0}^{2}\imaginary(\partial_{\bar{z}}g+\partial_{z}f_{1}\partial_{u}\partial_{\bar{z}}f_{1})\,,$
(126) $\displaystyle K_{2}$
$\displaystyle=\Delta_{0}F_{2}+2F_{2}+2F_{1}\Delta_{0}F_{1}+2F_{1}^{2}-4P_{0}^{2}\imaginary(\partial_{\bar{z}}f_{1}\partial_{u}\partial_{z}F_{1})+2P_{0}^{2}\real(\partial_{z}f_{1}\partial_{u}^{2}\partial_{\bar{z}}f_{1}-\partial_{u}\partial_{z}\bar{g})\,,$
(127) $\displaystyle W_{2}$
$\displaystyle=-r\partial_{u}L_{2}+i\partial_{z}\Sigma_{2}-i\partial_{u}(L_{1}\Sigma_{1})\,,$
(128) $\displaystyle N_{2}$
$\displaystyle=\\!\begin{multlined}\frac{1}{4}\Delta_{0}\Delta_{0}f_{2}+\frac{1}{2}\Delta_{0}f_{2}+2F_{1}\Delta_{0}f_{1}+\frac{1}{2}F_{1}\Delta_{0}\Delta_{0}f_{1}+\frac{1}{2}\Delta_{0}(F_{1}\Delta_{0}f_{1})+\frac{1}{2}\Delta_{0}f_{1}\Delta_{0}F_{1}\\\
+P_{0}^{2}\imaginary(\partial_{\bar{z}}g+\partial_{z}f_{1}\partial_{u}\partial_{\bar{z}}f_{1})-\frac{1}{2}\imaginary[\Delta_{0}(P_{0}^{2}(\partial_{\bar{z}}g+\partial_{z}f_{1}\partial_{u}\partial_{\bar{z}}f_{1}))\big{]}\\\
-\frac{1}{2}P_{0}^{2}\partial_{u}\imaginary[2\partial_{\bar{z}}f_{1}\partial_{z}\Delta_{0}f_{1}+\Delta_{0}f_{1}\partial_{z}\partial_{\bar{z}}f_{1}\big{]}\,,\end{multlined}\frac{1}{4}\Delta_{0}\Delta_{0}f_{2}+\frac{1}{2}\Delta_{0}f_{2}+2F_{1}\Delta_{0}f_{1}+\frac{1}{2}F_{1}\Delta_{0}\Delta_{0}f_{1}+\frac{1}{2}\Delta_{0}(F_{1}\Delta_{0}f_{1})+\frac{1}{2}\Delta_{0}f_{1}\Delta_{0}F_{1}\\\
+P_{0}^{2}\imaginary(\partial_{\bar{z}}g+\partial_{z}f_{1}\partial_{u}\partial_{\bar{z}}f_{1})-\frac{1}{2}\imaginary[\Delta_{0}(P_{0}^{2}(\partial_{\bar{z}}g+\partial_{z}f_{1}\partial_{u}\partial_{\bar{z}}f_{1}))\big{]}\\\
-\frac{1}{2}P_{0}^{2}\partial_{u}\imaginary[2\partial_{\bar{z}}f_{1}\partial_{z}\Delta_{0}f_{1}+\Delta_{0}f_{1}\partial_{z}\partial_{\bar{z}}f_{1}\big{]}\,,$
(132) $\displaystyle H_{2}$
$\displaystyle=\frac{1}{2}K_{2}-r\partial_{u}F_{2}+\frac{M\Sigma_{1}^{2}}{r^{3}}-\frac{N_{1}\Sigma_{1}}{r^{2}}\,.$
(133)

Injecting expressions for the exact solutions we have obtained, given by Eq
(35) and Eq (57) and (63), one obtains

$\displaystyle L_{1}$
$\displaystyle=i{E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}\partial_{z}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}\,,$
$\displaystyle P_{1}$
$\displaystyle={E}_{\mathrm{p}}P_{0}e^{-{\kappa}_{\mathrm{p}}u}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\,,$
$\displaystyle\Sigma_{1}$
$\displaystyle=-({\lambda}_{\mathrm{a}}+1){E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}\,,$
$\displaystyle K_{1}$
$\displaystyle=-2{\lambda}_{\mathrm{p}}E_{p}e^{-{\kappa}_{\mathrm{p}}u}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}\,,$
$\displaystyle W_{1}$
$\displaystyle=-i({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1){E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}\partial_{z}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}\,,$
$\displaystyle N_{1}$
$\displaystyle=3M{\kappa}_{\mathrm{a}}{E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}Y_{{\ell}_{\mathrm{a}}}^{{m}_{\mathrm{a}}}\,,$
$\displaystyle H_{1}$
$\displaystyle=(r{\kappa}_{\mathrm{p}}-{\lambda}_{\mathrm{p}})E_{p}e^{-{\kappa}_{\mathrm{p}}u}Y_{{\ell}_{\mathrm{p}}}^{{m}_{\mathrm{p}}}$
(134)

for the first-order contributions and

$\displaystyle L_{2}$
$\displaystyle=\sum_{\ell,m}\Big{[}i{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}a_{m}^{\ell}+{\kappa}_{\mathrm{a}}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\Big{]}\partial_{z}Y_{\ell}^{m}\,,$
$\displaystyle P_{2}$
$\displaystyle=P_{0}\sum_{\ell,m}\Big{[}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{(}p_{m}^{\ell}+\frac{1}{2}k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{)}+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}r_{m}^{\ell}+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}q_{m}^{\ell}\Big{]}Y_{\ell}^{m}\,,$
$\displaystyle\Sigma_{2}$
$\displaystyle=\sum_{\ell,m}{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}-(\lambda+1)a_{m}^{\ell}-2({\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{]}Y_{\ell}^{m}\,,$
$\displaystyle K_{2}$
$\displaystyle=\\!\begin{multlined}\sum_{\ell,m}\Big{[}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{\\{}-2\lambda
p_{m}^{\ell}-2(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{\\}}+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}-2\lambda
r_{m}^{\ell}+4{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}-2\lambda
q_{m}^{\ell}+2{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})\big{\\}}\Big{]}Y_{\ell}^{m}\,,\end{multlined}\sum_{\ell,m}\Big{[}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{\\{}-2\lambda
p_{m}^{\ell}-2(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{\\}}+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}-2\lambda
r_{m}^{\ell}+4{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}-2\lambda
q_{m}^{\ell}+2{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})\big{\\}}\Big{]}Y_{\ell}^{m}\,,$
(137) $\displaystyle W_{2}$
$\displaystyle=\\!\begin{multlined}\sum_{\ell,m}\Big{[}-i{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})a_{m}^{\ell}+(\lambda+1)a_{m}^{\ell}+2({\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}-2{\kappa}_{\mathrm{a}}^{2}r-{\kappa}_{\mathrm{a}}({\lambda}_{\mathrm{a}}+1)\big{\\}}k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\Big{]}\partial_{z}Y_{\ell}^{m}\,,\end{multlined}\sum_{\ell,m}\Big{[}-i{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})a_{m}^{\ell}+(\lambda+1)a_{m}^{\ell}+2({\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}-2{\kappa}_{\mathrm{a}}^{2}r-{\kappa}_{\mathrm{a}}({\lambda}_{\mathrm{a}}+1)\big{\\}}k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\Big{]}\partial_{z}Y_{\ell}^{m}\,,$
(140) $\displaystyle N_{2}$
$\displaystyle=\sum_{\ell,m}{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}3M\kappa
a_{m}^{\ell}+2({\lambda}_{\mathrm{a}}+1)(1+\lambda+{\lambda}_{\mathrm{a}}+{\lambda}_{\mathrm{p}})k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}\big{]}Y_{\ell}^{m}\,,$
$\displaystyle H_{2}$
$\displaystyle=\\!\begin{multlined}\sum_{\ell,m}\Big{[}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{\\{}(2r{\kappa}_{\mathrm{p}}-\lambda)p_{m}^{\ell}-(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{\\}}\\\
+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}(-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda)r_{m}^{\ell}+2{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}(-2r{\kappa}_{\mathrm{a}}-\lambda)q_{m}^{\ell}+{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})+\frac{{\lambda}_{\mathrm{a}}+1}{r^{3}}M(3r{\kappa}_{\mathrm{a}}+{\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\big{\\}}\Big{]}Y_{\ell}^{m}\end{multlined}\sum_{\ell,m}\Big{[}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{\\{}(2r{\kappa}_{\mathrm{p}}-\lambda)p_{m}^{\ell}-(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{\\}}\\\
+{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{\\{}(-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda)r_{m}^{\ell}+2{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{\\}}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{\\{}(-2r{\kappa}_{\mathrm{a}}-\lambda)q_{m}^{\ell}+{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})+\frac{{\lambda}_{\mathrm{a}}+1}{r^{3}}M(3r{\kappa}_{\mathrm{a}}+{\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\big{\\}}\Big{]}Y_{\ell}^{m}$
(144)

for the second-order contributions. This provides the explicit expressions for
the different contributions to the perturbative radiative geometry.

###  B.2 Covariant perturbation formalism

Perturbations around a Schwarzschild BH can be decomposed into 10 functions
$B_{1}$, …, $B_{10}$, using the covariant form of perturbations described in
Thompson:2016fxe . For a metric whose line element is

$\differential{s}^{2}=\differential{s}_{\mathrm{Schwa}}^{2}+h_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}\,,$
(145)

the perturbations tensor $h_{\mu\nu}$ is decomposed as

$h_{\mu\nu}=B_{1}k_{\mu}k_{\nu}Y_{\ell}^{m}+2rB_{2}k_{(\mu}\nabla_{\nu)}Y_{\ell}^{m}+2rB_{3}k_{(\mu}\varepsilon\indices{{}^{\rho}_{\nu)}{}^{\sigma}{}^{\lambda}}k_{\rho}n^{\sigma}\nabla_{\lambda}Y_{\ell}^{m}+2B_{4}k_{(\mu}n_{\nu)}Y_{\ell}^{m}+B_{5}\varsigma_{\mu\nu}Y_{\ell}^{m}\\\
+r^{2}B_{6}\big{(}\varsigma\indices{{}_{\mu}^{\rho}}\varsigma\indices{{}_{\nu}^{\sigma}}-\frac{1}{2}\varsigma_{\mu\nu}\varsigma^{\rho\sigma}\big{)}\nabla_{\rho}\nabla_{\sigma}Y_{\ell}^{m}+r^{2}B_{7}\varsigma\indices{{}_{(\mu}^{\rho}}\varepsilon\indices{{}^{\sigma}_{\nu)}{}_{\lambda}^{\kappa}}k_{\sigma}n^{\lambda}\nabla_{\rho}\nabla_{\kappa}Y_{\ell}^{m}+2rB_{8}n_{(\mu}\nabla_{\nu)}Y_{\ell}^{m}\\\
+2rB_{9}n_{(\mu}\varepsilon\indices{{}^{\rho}_{\nu)}{}_{\sigma}^{\lambda}}k_{\rho}n^{\sigma}\nabla_{\lambda}Y_{\ell}^{m}+B_{10}n_{\mu}n_{\nu}Y_{\ell}^{m}\,,$
(146)

where the forms $k_{\mu}$, $n_{\mu}$ and $\varsigma_{\mu\nu}$ are defined such
that

$k_{\mu}\differential{x}^{\mu}=-\differential{u}-\frac{\differential{r}}{A}\,,\quad
n_{\mu}\differential{x}^{\mu}=\differential{r}\,,\quad\varsigma_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}=r^{2}\differential{\Omega}^{2}\,.$
(147)

Here, all functions $B_{1}$, …, $B_{10}$ depend on $u$, $r$, and the integers
$\ell$ and $m$. We do not write this explicitly in order to lighten the
notations.

The functions $B_{1}$, …, $B_{10}$ can be related to the functions used by
Regge and Wheeler in their original work on Schwarzschild perturbations
Regge:1957td . In this work, axial perturbations are represented by $h_{0}$,
$h_{1}$ and $h_{2}$ while polar perturbations are described by444In the
original work Regge:1957td , the notations $h_{0}$ and $h_{1}$ are used twice.
Here, we keep $h_{0}$ and $h_{1}$ for the description of axial perturbations
and we rename them respectively $\beta$ and $\alpha$ for the description of
even perturbations. $H_{0}$, $H_{1}$, $H_{2}$, $\alpha$, $\beta$, $K$ and $G$.
The relations between these functions and the ones appearing in the covariant
description (146) are

$\displaystyle B_{1}$ $\displaystyle=AH_{0}\,,$ $\displaystyle B_{2}$
$\displaystyle=-\frac{1}{r}\beta\,,$ $\displaystyle B_{3}$
$\displaystyle=-\frac{1}{r}h_{0}\,,$ $\displaystyle B_{4}$
$\displaystyle=-H_{1}\,,$ $\displaystyle B_{5}$
$\displaystyle=K-\frac{1}{2}\ell(\ell+1)G\,,$ $\displaystyle B_{6}$
$\displaystyle=G\,,$ $\displaystyle B_{7}$
$\displaystyle=-\frac{1}{r^{2}}h_{2}\,,$ $\displaystyle B_{8}$
$\displaystyle=\frac{1}{r}\alpha\,,$ $\displaystyle B_{9}$
$\displaystyle=\frac{1}{r}h_{1}\,,$ $\displaystyle B_{10}$
$\displaystyle=A^{-1}H_{2}\,.$ (148)

In the $(u,r,z,\bar{z})$ coordinate system, using (146), one obtains

$h_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}=B_{1}Y_{\ell}^{m}\differential{u}^{2}+\big{(}B_{10}-\frac{2}{A}B_{4}+\frac{1}{A^{2}}B_{1}\big{)}Y_{\ell}^{m}\differential{r}^{2}-2\big{(}B_{4}-\frac{B_{1}}{A}\big{)}Y_{\ell}^{m}\differential{u}\differential{r}\\\
-2r\differential{u}\real[(B_{2}-iB_{3})\partial_{z}Y_{\ell}^{m}\differential{z}\big{]}+2r\differential{r}\real[\big{(}(B_{8}-\frac{B_{2}}{A})-i(B_{9}-\frac{B_{3}}{A})\big{)}\partial_{z}Y_{\ell}^{m}\differential{z}\Big{]}\\\
+r^{2}B_{5}Y_{\ell}^{m}\differential{\Omega^{2}}+2r^{2}\real[(B_{6}-2iB_{7})(\partial_{z}^{2}+2\partial_{z}(\log
P_{0})\partial_{z})Y_{\ell}^{m}\differential{z}^{2}\big{]}\,.$ (149)

###  B.3 Reconstructing Regge-Wheeler perturbations

The computation of $f_{1}$, $f_{2}$, $F_{1}$ and $F_{2}$ allows us to write
the line element (13) perturbatively as

$\differential{s}^{2}=\differential{s}_{\mathrm{Schwa}}^{2}+\varepsilon
h^{(1)}_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}+\varepsilon^{2}h^{(2)}_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}+\mathcal{O}(\varepsilon^{3})\,,$
(150)

with

$\differential{s}_{\mathrm{Schwa}}^{2}=-A(r)\differential{u}^{2}-2\differential{r}\differential{u}+r^{2}\differential{\Omega}^{2}\,,\quad
A(r)=1-\frac{2M}{r}\,.$ (151)

We have introduced the first and second-order perturbations to a background
Schwarzschild metric, respectively $h^{(1)}_{\mu\nu}$ and $h^{(2)}_{\mu\nu}$.
These two contributions can be computed using the following expansions for the
functions appearing in (13) and presented in app B.1. The linear contribution
decomposes as

$\displaystyle h^{(1)}_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}$
$\displaystyle=-2H_{1}\differential{u}^{2}+\frac{r}{2}\Big{[}\imaginary(\Sigma_{1})-4F_{1}r\Big{]}\differential{\Omega}^{2}$
$\displaystyle\;\;\;\;\;-4\differential{r}\real(L_{1}\differential{z})-4\differential{u}\real[\big{(}W_{1}+L_{1}A(r)\big{)}\differential{z}\Big{]}$
(152)

while the second order contributions is given by

$\displaystyle h^{(2)}_{\mu\nu}\differential{x}^{\mu}\differential{x}^{\nu}$
$\displaystyle=-2H_{2}\differential{u}^{2}-4\differential{r}\real(L_{2}\differential{z})-4\differential{u}\real[\big{(}W_{2}+L_{2}A(r)+2H_{1}L_{1}\big{)}\differential{z}\Big{]}-4A\real(L_{1}\differential{z})^{2}$
$\displaystyle\;\;\;\;\;-8\real(L_{1}\differential{z})\real(W_{1}\differential{z})+\Big{[}\frac{1}{2}r\imaginary(\Sigma_{2})-2F_{2}r^{2}+2F_{1}^{2}r^{2}-F_{1}r\imaginary(\Sigma_{1})+\Sigma_{1}\bar{\Sigma}_{1}\Big{]}\differential{\Omega}^{2}\,.$
(153)

One can then compute the explicit expressions for the Regge-Wheeler metric
perturbation functions in the case of perturbations given in (152) and (153)
by comparing (134), (144) and (149), using the definitions (148).

At first order, one obtains

$\displaystyle{}^{(1)}H_{0}={}^{(1)}H_{2}=-{}^{(1)}H_{1}=-\frac{2}{A}(r{\kappa}_{\mathrm{p}}-{\lambda}_{\mathrm{p}}){E}_{\mathrm{p}}e^{-{\kappa}_{\mathrm{p}}u}\,,$
(154)
$\displaystyle{}^{(1)}K=-2{E}_{\mathrm{p}}e^{-{\kappa}_{\mathrm{p}}u}\,,\quad{}^{(1)}\alpha={}^{(1)}\beta=0\,,\quad{}^{(1)}G=0$
(155)

for the even perturbations and

$\displaystyle{}^{(1)}h_{0}$
$\displaystyle=-2({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1-A){E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}\,,$
(156) $\displaystyle{}^{(1)}h_{1}$
$\displaystyle=\frac{2}{A}({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1){E}_{\mathrm{a}}e^{{\kappa}_{\mathrm{a}}u}\,,$
(157) $\displaystyle{}^{(1)}h_{2}$ $\displaystyle=0\,$ (158)

for the axial perturbations. We have not written explicitly the $\ell$ and $m$
dependency of these functions since only the
$(\ell={\ell}_{\mathrm{a}},m={m}_{\mathrm{a}})$ functions and
$(\ell={\ell}_{\mathrm{p}},m={m}_{\mathrm{p}})$ are non-zero respectively in
the axial and polar cases. Noticed that at first order, the solution satisfies
the RWZ gauge since

${}^{(1)}G={}^{(1)}h_{2}={}^{(1)}\alpha={}^{(1)}\beta=0$ (159)

However, at second order, one should keep in mind that there are several
contributions from different values of $(\ell,m)$. One obtains

$\displaystyle{}^{(2)}H_{0,\ell}^{m}$
$\displaystyle=\\!\begin{multlined}-{}^{(2)}H_{1,\ell}^{m}={}^{(2)}H_{2,\ell}^{m}=-\frac{2}{A}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{[}(2r{\kappa}_{\mathrm{p}}-\lambda)p_{m}^{\ell}-(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{]}\\\
-\frac{2}{A}{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}(-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda)r_{m}^{\ell}+2{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{]}\\\
-\frac{2}{A}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{[}(-2r{\kappa}_{\mathrm{a}}-\lambda)q_{m}^{\ell}+{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})+\frac{{\lambda}_{\mathrm{a}}+1}{r^{3}}M(3r{\kappa}_{\mathrm{a}}+{\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\big{]}\,,\end{multlined}-{}^{(2)}H_{1,\ell}^{m}={}^{(2)}H_{2,\ell}^{m}=-\frac{2}{A}{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{[}(2r{\kappa}_{\mathrm{p}}-\lambda)p_{m}^{\ell}-(2{\lambda}_{\mathrm{p}}+1)k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{]}\\\
-\frac{2}{A}{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}(-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda)r_{m}^{\ell}+2{\kappa}_{\mathrm{p}}\imaginary(h_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell})\big{]}\\\
-\frac{2}{A}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{[}(-2r{\kappa}_{\mathrm{a}}-\lambda)q_{m}^{\ell}+{\kappa}_{\mathrm{a}}^{2}(\ell(\ell+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}+h_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell})+\frac{{\lambda}_{\mathrm{a}}+1}{r^{3}}M(3r{\kappa}_{\mathrm{a}}+{\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\big{]}\,,$
(163) $\displaystyle{}^{(2)}\beta_{\ell}^{m}$
$\displaystyle=2{\kappa}_{\mathrm{a}}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}(2{\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1-A)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\,,$
(164) $\displaystyle{}^{(2)}\alpha_{\ell}^{m}$
$\displaystyle=-2{\kappa}_{\mathrm{a}}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}(2{\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+2-A)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}\,,$
(165) $\displaystyle{}^{(2)}K_{\ell}^{m}$
$\displaystyle=\\!\begin{multlined}-2{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{(}p^{\ell}_{m}-k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{)}-2r^{\ell}_{m}{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{[}-2q^{\ell}_{m}+\frac{({\lambda}_{\mathrm{a}}+1)^{2}}{r^{2}}k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}-\frac{A-2({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1)}{r^{2}}\big{(}Ah_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}-(\lambda+1)j_{{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}\ell}\big{)}\big{]}\,,\end{multlined}-2{E}_{\mathrm{p}}^{2}e^{-2{\kappa}_{\mathrm{p}}u}\big{(}p^{\ell}_{m}-k_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{p}}\ell}\big{)}-2r^{\ell}_{m}{E}_{\mathrm{p}}{E}_{\mathrm{a}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\\\
+{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}\big{[}-2q^{\ell}_{m}+\frac{({\lambda}_{\mathrm{a}}+1)^{2}}{r^{2}}k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}-\frac{A-2({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1)}{r^{2}}\big{(}Ah_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{a}}\ell}-(\lambda+1)j_{{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}\ell}\big{)}\big{]}\,,$
(168) $\displaystyle{}^{(2)}G_{\ell}^{m}$
$\displaystyle=-\frac{1}{r^{2}}{E}_{\mathrm{a}}^{2}e^{2{\kappa}_{\mathrm{a}}u}j_{{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{a}}\ell}\big{[}-A+2({\kappa}_{\mathrm{a}}r+{\lambda}_{\mathrm{a}}+1)\big{]}\,,$
(169) $\displaystyle{}^{(2)}h_{0,\ell}^{m}$
$\displaystyle=2{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}\big{(}A-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda-1\big{)}a^{\ell}_{m}-2({\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}+2(r{\kappa}_{\mathrm{p}}-{\lambda}_{\mathrm{p}})g_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell}\big{]}\,,$
(170) $\displaystyle{}^{(2)}h_{1,\ell}^{m}$
$\displaystyle=-2{E}_{\mathrm{a}}{E}_{\mathrm{p}}e^{({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})u}\big{[}\big{(}A-r({\kappa}_{\mathrm{a}}-{\kappa}_{\mathrm{p}})-\lambda-2\big{)}a^{\ell}_{m}-2({\lambda}_{\mathrm{a}}+1)k_{{\ell}_{\mathrm{a}}{\ell}_{\mathrm{p}}m}^{{m}_{\mathrm{a}}{m}_{\mathrm{p}}\ell}+2(r{\kappa}_{\mathrm{p}}-{\lambda}_{\mathrm{p}})g_{{\ell}_{\mathrm{p}}{\ell}_{\mathrm{a}}m}^{{m}_{\mathrm{p}}{m}_{\mathrm{a}}\ell}\big{]}\,,$
(171) $\displaystyle{}^{(2)}h_{2,\ell}^{m}$ $\displaystyle=0\,.$ (172)

Notice that the second order perturbations do not satisfy all the RWZ gauge
conditions since we have only ${}^{(2)}h_{2,\ell}^{m}=0$. Thus only the axial
part satisfies this gauge.

##  Appendix C An illustrative concrete example for a quadratic ASP

In this appendix, we present the detailed derivation of one concrete example
in order to illustrate the efficiency of our approach. We consider linear
perturbations with ${\ell}_{\mathrm{a}}=2$, ${m}_{\mathrm{a}}=1$ and
${\ell}_{\mathrm{p}}=3$, ${m}_{\mathrm{p}}=2$ which implies that their decay
time and the associated $\lambda$ are given respectively by

${\kappa}_{\mathrm{a}}=\frac{2}{M}\,,\quad{\kappa}_{\mathrm{p}}=\frac{10}{M}\,,\quad{\lambda}_{\mathrm{a}}=2\mbox{\quad
and\quad}{\lambda}_{\mathrm{p}}=5\,.$ (173)

The first task is to compute explicitly the Clebsch-Gordan coefficients
entering in the sources. For the present case, the relevant ones are given by

$\displaystyle k_{220}^{110}$ $\displaystyle=\frac{1}{2\sqrt{\pi}}\,,$
$\displaystyle k_{220}^{112}$
$\displaystyle=\frac{1}{14}\sqrt{\frac{5}{\pi}}\,,$ $\displaystyle
k_{222}^{112}$ $\displaystyle=\frac{1}{14}\sqrt{\frac{15}{\pi}}\,,$
$\displaystyle k_{220}^{114}$ $\displaystyle=-\frac{2}{7\sqrt{\pi}}\,,$
$\displaystyle k_{222}^{114}$
$\displaystyle=\frac{1}{7}\sqrt{\frac{5}{\pi}}\,.$ $\displaystyle
k_{231}^{121}$ $\displaystyle=\frac{1}{2}\sqrt{\frac{3}{7\pi}}\,,$
$\displaystyle k_{231}^{123}$ $\displaystyle=\frac{1}{2\sqrt{6\pi}}\,,$
$\displaystyle k_{233}^{123}$
$\displaystyle=\frac{1}{6}\sqrt{\frac{5}{2\pi}}\,,$ $\displaystyle
k_{231}^{125}$ $\displaystyle=-\sqrt{\frac{5}{231\pi}}\,,$ $\displaystyle
k_{233}^{125}$ $\displaystyle=\frac{1}{3}\sqrt{\frac{10}{11\pi}}\,.$
$\displaystyle k_{330}^{220}$ $\displaystyle=\frac{1}{2\sqrt{\pi}}\,,$
$\displaystyle k_{330}^{224}$ $\displaystyle=-\frac{7}{22\sqrt{\pi}}\,,$
$\displaystyle k_{334}^{224}$ $\displaystyle=\frac{1}{22}
\sqrt{\frac{35}{\pi}}\,,$ $\displaystyle k_{330}^{226}$
$\displaystyle=\frac{5}{11\sqrt{13\pi}}\,,$ $\displaystyle k_{334}^{226}$
$\displaystyle=\frac{5}{11}\sqrt{\frac{7}{13\pi}}\,.$ (174)

Proceeding to the decomposition of the different source terms using the
explicit expressions (59), (66) and (67), one obtains for the polar-polar and
axial-axial sources

$\displaystyle\mathcal{A}^{1}_{1}$ $\displaystyle=-96\sqrt{\frac{3}{7\pi}}\,,$
$\displaystyle\mathcal{A}^{3}_{1}$ $\displaystyle=-26\sqrt{\frac{6}{\pi}}\,,$
$\displaystyle\mathcal{A}^{3}_{3}$ $\displaystyle=-26\sqrt{\frac{10}{\pi}}\,,$
$\displaystyle\mathcal{A}^{5}_{1}$
$\displaystyle=16\sqrt{\frac{165}{7\pi}}\,,$
$\displaystyle\mathcal{A}^{5}_{3}$
$\displaystyle=-16\sqrt{\frac{110}{\pi}}\,,$
$\displaystyle\mathcal{P}^{0}_{0}$ $\displaystyle=-\frac{120}{\sqrt{\pi}}\,,$
$\displaystyle\mathcal{P}^{4}_{0}$
$\displaystyle=\frac{2380}{11\sqrt{\pi}}\,,$
$\displaystyle\mathcal{P}^{4}_{4}$
$\displaystyle=-\frac{340}{11}\sqrt{\frac{35}{\pi}}\,,$
$\displaystyle\mathcal{P}^{6}_{0}$
$\displaystyle=-\frac{5820}{11\sqrt{13\pi}}\,,$
$\displaystyle\mathcal{P}^{6}_{4}$
$\displaystyle=-\frac{5820}{11}\sqrt{\frac{7}{13\pi}}\,,$ $\displaystyle
M^{2}\mathcal{Q}^{0}_{0}$ $\displaystyle=\frac{144}{\sqrt{\pi}}\,,$
$\displaystyle M^{2}\mathcal{Q}^{2}_{0}$
$\displaystyle=\frac{384}{7}\sqrt{\frac{5}{\pi}}\,,$ $\displaystyle
M^{2}\mathcal{Q}^{2}_{2}$
$\displaystyle=\frac{384}{7}\sqrt{\frac{15}{\pi}}\,,$ $\displaystyle
M^{2}\mathcal{Q}^{4}_{0}$ $\displaystyle=-\frac{7136}{7\sqrt{\pi}}\,,$
$\displaystyle M^{2}\mathcal{Q}^{4}_{2}$
$\displaystyle=\frac{3568}{7}\sqrt{\frac{5}{\pi}}\,,$ (175)

while for the coefficients $\mathcal{R}^{\ell}_{m}$ describing the polar-axial
source, one finds

$M\mathcal{R}^{4}_{-3}=100\sqrt{\frac{10}{\pi}}\,,\qquad
M\mathcal{R}^{4}_{-1}=100\sqrt{\frac{70}{\pi}}\,.$ (176)

Gathering these different pieces, one can now evaluate explicitly the profile
of the quadratic ASP $(F_{2},f_{2})$. They are given in the present case by

$f_{2}={E}_{\mathrm{p}}{E}_{\mathrm{a}}\Big{(}-\sqrt{\frac{3}{7\pi}}Y_{1}^{1}-\frac{13}{18\sqrt{6\pi}}Y_{3}^{1}-\frac{13}{54}\sqrt{\frac{5}{2\pi}}Y_{3}^{3}+\frac{2}{39}\sqrt{\frac{55}{21\pi}}Y_{5}^{1}-\frac{2}{117}\sqrt{\frac{110}{\pi}}Y_{5}^{3}\Big{)}e^{-8u/M}\,.$
(177)

and

$\displaystyle F_{2}$
$\displaystyle={E}_{\mathrm{p}}^{2}\Big{(}\frac{1}{2\sqrt{\pi}}Y_{0}^{0}+\frac{119}{66\sqrt{\pi}}Y_{4}^{0}-\frac{17}{66}\sqrt{\frac{35}{\pi}}Y_{4}^{4}-\frac{97}{264\sqrt{13\pi}}Y_{6}^{0}-\frac{97}{264}\sqrt{\frac{7}{13\pi}}Y_{6}^{4}\Big{)}e^{-20u/M}$
(178)
$\displaystyle+\frac{{E}_{\mathrm{a}}^{2}}{M^{2}}\Big{(}\frac{3}{\sqrt{\pi}}Y_{0}^{0}+\frac{16}{21}\sqrt{\frac{5}{\pi}}Y_{2}^{0}+\frac{16}{7}\sqrt{\frac{5}{3\pi}}Y_{2}^{2}-\frac{892}{357\sqrt{\pi}}Y_{4}^{0}+\frac{446}{357}\sqrt{\frac{5}{\pi}}Y_{4}^{2}\Big{)}e^{4u/M}$
(179)
$\displaystyle+\frac{{E}_{\mathrm{p}}{E}_{\mathrm{a}}}{M}\Big{(}\frac{25}{33}\sqrt{\frac{5}{2\pi}}Y_{4}^{-3}-\frac{25}{33}\sqrt{\frac{35}{2\pi}}Y_{4}^{-1}\Big{)}e^{-8u/M}\,.$
(180)

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