# Quasinormal modes of slowly-spinning horizonless compact objects

M. V. S. Saketh<EMAIL_ADDRESS>Department of Physics, University of
Maryland, College Park, MD 20742, USA  Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany
Elisa Maggio<EMAIL_ADDRESS>Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), Am Mühlenberg 1, Potsdam 14476, Germany

###### Abstract

One of the main predictions of general relativity is the existence of black
holes featuring a horizon beyond which nothing can escape. Gravitational waves
from the remnants of compact binary coalescences have the potential to probe
new physics close to the BH horizons. This prospect is of particular interest
given several quantum-gravity models that predict the presence of horizonless
and singularity-free compact objects. The membrane paradigm is a generic
framework that allows one to parametrise the interior of compact objects in
terms of the properties of a fictitious fluid located at the object’s radius.
It has been used to derive the quasinormal mode spectrum of static horizonless
compact objects. Extending the membrane paradigm to rotating objects is
crucial to constrain the properties of the spinning merger remnants. In this
work, we extend the membrane paradigm to linear order in spin and use it to
analyse the relationships between the quasi-normal modes, the object’s
reflectivity, and the membrane parameters. We find a breaking of
isospectrality between axial and polar modes when the object is partially
reflecting or the compactness differs from the black hole case. We also find
that in reflective ultracompact objects some of the modes tend towards
instability as the spin increases. Finally, we show that the spin enhances the
deviations from the black-hole quasinormal mode spectrum as the compactness
decreases. This implies that spinning horizonless compact objects may be more
easily differentiated than nonspinning ones in the prompt ringdown.

##  I Introduction

Gravitational waves (GWs) provide a unique opportunity to test gravity and
investigate the nature of compact objects. The ground-based detectors LIGO and
Virgo have detected ninety GW candidates from the coalescence of compact
binaries up to the end of their third observing run Abbott  _et al._ (2019a,
2021a, 2024, 2021b). Some important discoveries include the first observation
of a black-hole (BH) coalescence Abbott  _et al._ (2016a), the multi-messenger
observation of a binary-neutron-star merger Abbott  _et al._ (2017), and an
intermediate-mass BH Abbott  _et al._ (2020). Parametrised tests of general
relativity (GR) can constrain the degree to which the data agree with GR by
introducing fractional deviations to the inspiral and postmerger stages of the
gravitational waveform Li  _et al._ (2012); Agathos  _et al._ (2014); Ghosh
_et al._ (2016, 2018); Brito  _et al._ (2018); Carullo  _et al._ (2019); Ghosh
_et al._ (2021); Isi and Farr (2021); Mehta  _et al._ (2023); Maggio  _et al._
(2023); Gennari  _et al._ (2024). No evidence in support of physics beyond GR
has been found by the LIGO-Virgo-KAGRA collaboration, and increasingly
stringent limits have been set on GW data Abbott  _et al._ (2016b, 2019b,
2021c, 2021d).

GWs offer the promising prospect of testing one of the main predictions of GR,
namely the presence of horizons in BHs. On the theoretical side, BHs feature
horizons beyond which nothing – not even light – can escape. Moreover, BHs
contain a curvature singularity with infinite tidal forces where Einstein
equations break down. In the semiclassical approximation, when a massless
field is quantized in the Schwarzschild background, BHs are thermodynamically
unstable and radiate a thermal spectrum at the Hawking temperature Hawking
(1974). They are characterised by the information-loss paradox Mathur (2009)
and their entropy is far over the one of a typical stellar progenitor
Bekenstein (1973); Hawking (1976).

Several models of horizonless and singularity-free compact objects have been
proposed to address the BH paradoxes Giudice  _et al._ (2016); Cardoso and
Pani (2017, 2019); Maggio  _et al._ (2021a). New physics can prevent the
formation of horizons in quantum-gravity extensions of GR (e.g., fuzzballs
Mathur (2005); Bena and Warner (2008); Balasubramanian  _et al._ (2008),
gravastars Mazur and Mottola (2004), nonlocal stars Buoninfante  _et al._
(2018); Buoninfante and Mazumdar (2019)) and in GR with the presence of dark
matter and exotic fields (e.g., boson stars Kaup (1968); Ruffini and Bonazzola
(1969); Seidel and Suen (1991); Liebling and Palenzuela (2012); Brito  _et
al._ (2016), wormholes Morris and Thorne (1988); Visser (1995); Damour and
Solodukhin (2007)). Horizonless compact objects can mimic the observational
features of BHs – since they can be as compact as BHs – and quantify the
existence of horizons in astrophysical sources. Horizonless compact objects
can deviate from BHs for their compactness, i.e., the inverse of the effective
radius in units of the total mass, and reflectivity, which is generically
complex and frequency-dependent and differs from the totally absorbing BH
case.

The absence of horizons in compact objects affects the gravitational waveform
in the inspiral and postinspiral stages. In the inspiral, the energy flux at
the radius of horizonless compact objects differs from the energy flux at the
horizon of BHs Datta  _et al._ (2020); Mukherjee  _et al._ (2022); Saketh  _et
al._ (2023). Furthermore, resonances are excited when the orbital frequency
matches the characteristic oscillation frequencies of the inspiralling compact
objects Cardoso  _et al._ (2019); Asali  _et al._ (2020); Maggio  _et al._
(2021b). Also, the tidal Love numbers, which measure the deformability of a
compact object when immersed in an external tidal field, are nonzero for
horizonless compact objects Pani (2015); Uchikata  _et al._ (2016); Cardoso
_et al._ (2017); Sennett  _et al._ (2017); Raposo  _et al._ (2019); Nair  _et
al._ (2023); Chakraborty  _et al._ (2023). During the ringdown, the
quasinormal mode (QNM) frequencies of the perturbed remnant deviate from the
BH QNM spectrum in the absence of horizons. If the remnant is an ultracompact
horizonless object, the prompt ringdown signal would be nearly the same as the
one emitted by a BH since it is excited at the light ring Cardoso  _et al._
(2016a, b). Additional signal would be emitted in the form of GW echoes at
late times due to the absence of horizons in the remnant Abedi  _et al._
(2017); Wang  _et al._ (2020); Mark  _et al._ (2017); Testa and Pani (2018);
Maggio  _et al._ (2019a); Xin  _et al._ (2021); Ma  _et al._ (2022); Vellucci
_et al._ (2023); Siemonsen (2024). The QNM spectrum would also deviate from
the BH one as a function of the compactness of the object Maggio  _et al._
(2020). Future observatories, such as the Einstein Telescope, Cosmic Explorer
and LISA, will allow us to detect or rule out the imprints of horizonless
compact objects in the gravitational waveform with unprecedented accuracy
Branchesi  _et al._ (2023); Arun  _et al._ (2022); Cardoso  _et al._ (2017);
Maselli  _et al._ (2018); Piovano  _et al._ (2023); Barsanti  _et al._ (2023).

Here, we study the behaviour of generic horizonless compact objects to
external gravitational perturbations to distinguish them from BHs. It is
useful and convenient to incorporate the study of various compact objects into
a single framework. This can also lead to an understanding of the relations
between the parameters describing compact objects – such as compactness and
reflectivity – and their GW observables – such as QNMs and tidal Love numbers.
One way to effectively parametrise a generic compact object is by using the
membrane paradigm Damour (1982); Thorne  _et al._ (1986); Price and Thorne
(1986). The membrane paradigm is a theoretical framework originally developed
to describe BHs, according to which the BH interior can be replaced by a
three-dimensional membrane located at the event horizon. The membrane is made
of a viscous fluid whose density, pressure and viscosity are fixed by the
Israel-Darmois junction conditions Darmois (1927); Israel (1966); Visser
(1996).

The membrane paradigm was recently applied to compact objects other than BHs
by locating the membrane at the object’s radius Abedi  _et al._ (2020); Maggio
_et al._ (2020); Chakraborty  _et al._ (2023). The fluid parameters may be
freely varied to parametrise different models of compact objects, and only a
specific choice of these parameters describes BHs.

The membrane paradigm was first used to derive the QNM spectrum of generic
nonspinning compact objects in Ref. Maggio  _et al._ (2020), along with a
comparison with the measurement accuracy of the fundamental QNM of GW150914
Abbott  _et al._ (2016b). However, due to the initial orbital angular momentum
of the binary, the final merged remnant is expected to be spinning Abbott  _et
al._ (2019a, 2021a, 2024, 2021b). This necessitates the importance of
incorporating the spin into the membrane paradigm for generic compact objects.

In this work, we present an extension of the membrane paradigm for generic
compact objects to the linear order in spin. In Sec. II, we present the
theoretical details involved in constructing the membrane paradigm to linear
order in spin. Particularly, we discuss the background metric, the linear
perturbations, and outline the derivation of the boundary conditions that the
metric perturbations must obey at the membrane surface from the Israel-Damour
junction conditions. In Sec. III, we analyse the reflectivity of compact
objects as a function of the fluid viscous parameters. We also show the
resolution to an apparent paradox that arises in the linear-in-spin
approximation associated with the superradiant scattering. Furthermore, we
derive the QNM frequencies in the BH limit, and compare them with known Kerr
QNM spectrum to quantify the upper limit (in spin) to which the linear-in-spin
approximation may be trusted. Finally, we compute the QNM frequencies of
generic horizonless compact objects as a function of the compactness and the
fluid viscous parameters. We discuss the breaking of isospectrality between
axial and polar modes, and potential constraints that can be put on the
object’s compactness based on the current measurement accuracies of the
fundamental QNM. In Sec. IV, we summarise the results and conclude with a
brief discussion of the related future extensions. We assume $G=c=1$ units
throughout.

##  II The membrane paradigm

The membrane paradigm allows one to derive the QNM spectrum of compact objects
with a given exterior geometry and different interior solutions. In its
standard formulation, a static observer can replace the interior of a compact
object with a fictitious membrane located at the boundary of the object. The
Israel-Darmois junction conditions set the properties of the membrane as
Darmois (1927); Israel (1966)

$[[K_{ab}-Kh_{ab}]]=-8\pi T_{ab}\,,\qquad[[h_{ab}]]=0\,,$ (1)

where $h_{ab}$ is the induced metric on the membrane, $K_{ab}$ is the
extrinsic curvature, $K=K_{ab}h^{ab}$, and $T_{ab}$ is the stress-energy
tensor of the fluid on the membrane. The notation
$[[...]]=(...)^{+}-(...)^{-}$ denotes the jump of a quantity across the
membrane, where $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$ are the exterior and
interior spacetimes to the membrane, and $a,b$ are the indices of a 3-D
coordinate system laid on the membrane.

The fictitious membrane is such that the extrinsic curvature of the interior
spacetime vanishes, i.e., $K_{ab}^{-}=0$. This constraint allows one to map
different geometries for the interior spacetime into the properties of the
fictitious membrane. The Israel-Darmois junction conditions impose that the
membrane is a viscous fluid with stress-energy tensor Thorne  _et al._ (1986)

$\displaystyle T_{ab}=\rho
u_{a}u_{b}+(p-\zeta\Theta)\gamma_{ab}-2\eta\sigma_{ab}\,,$ (2)

where $\rho$ is the surface energy density, $p$ is the surface pressure,
$\zeta$ and $\eta$ are the bulk and shear viscosities, $u_{a}$ is the
3-velocity of the fluid, $\Theta=u^{a}_{;a}$ is the expansion,
$\gamma_{ab}=h_{ab}+u_{a}u_{b}$ is the projector tensor,
$\sigma_{ab}=\frac{1}{2}\left(u_{a;c}\gamma_{b}^{c}+u_{b;c}\gamma_{a}^{c}-\Theta\gamma_{ab}\right)$
is the shear tensor, and the semicolon is a covariant derivative with the
induced metric.

The membrane paradigm was originally developed to describe perturbed BHs
Damour (1982); Thorne  _et al._ (1986); Price and Thorne (1986); Abedi  _et
al._ (2020) in terms of the shear and bulk viscosities of a fictitious viscous
fluid located at the horizon, where

$\eta_{\rm{BH}}=\frac{1}{16\pi}\,,\quad\zeta_{\rm{BH}}=-\frac{1}{16\pi}\,.$
(3)

The membrane paradigm was then extended to perturbed horizonless compact
objects with a Schwarzschild exterior Maggio  _et al._ (2020), where the
fictitious fluid is located at the object’s radius, and the shear and bulk
viscosity are generically complex and frequency-dependent. For each model of
horizonless compact object, the shear and bulk viscosity are uniquely
determined and are related to the reflective properties of the object.

###  II.1 Background metric

Motivated by the fact that the remnants of compact binary coalescences have
large values of the spin Abbott  _et al._ (2019a, 2021a, 2024, 2021b), here we
extend the membrane paradigm to spinning horizonless compact objects which are
generally described by the Kerr metric at the first order in the spin. It is
worth mentioning that the Kerr geometry does not necessarily describe the
vacuum region outside a spinning object due to the absence of Birkhoff’s
theorem beyond spherical symmetry. Here, we assume that GR works sufficiently
well on the exterior of the compact object, and no specific model is
considered for the object’s interior. This assumption is valid in modified
theories of gravity where the corrections to the metric are suppressed by
powers of $\ell_{P}/r_{0}\ll 1$, being $r_{0}$ the object’s radius and
$\ell_{P}$ the Planck length or the scale of new physics Berti  _et al._
(2015).

The background is the Kerr metric at the linear order in spin, which in Boyer-
Lindquist coordinates is given by Kojima (1992)

$ds^{2}-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Sigma^{2}-4\frac{J}{r}\sin^{2}\theta
dtd\phi\,,$ (4)

where $f(r)=1-2M/r$, $d\Sigma^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$,
$J=aM$ is the angular momentum, $\chi=a/M$ is the dimensionless spin, and $M$
is the total mass of the compact object. Note that for BHs, the magnitude of
the spin parameter is $|\chi|\leq 1$ to prevent naked singularities, but no
such constraint exists for generic compact objects. A generic spinning body is
also characterized by the spin-induced multipole moments, which are relevant
from the second order in spin Vines and Steinhoff (2018); Saketh and Vines
(2022). For the Kerr metric, all the multipole moments are known as unique
functions of $M$ and $J$. In this work, we truncate to linear order in spin
and thus remove contribution of spin-induced multipole moments to the metric.
Moreover, the exterior spacetime has no ergoregion, i.e., the region outside
the horizon where a static observer cannot exist and is forced to corotate
with the compact object.

From the linear-in-spin metric given above, the BH horizon is located at
$r_{+}=2M$. We analyse horizonless compact objects whose radius is located at

$\displaystyle r_{0}=r_{+}(1+\epsilon)\,,$ (5)

where $\epsilon>0$ and lies in the range
$\mathcal{O}(10^{-40})-\mathcal{O}(1)$ for compact objects with Planckian
corrections at the horizon scale and whose compactness is comparable to that
of a neutron star, respectively.

In the unperturbed case, the fictitious membrane is located at the object’s
radius, and the normal vector to the membrane has components

$n_{t}=0,\ n_{r}=1/\sqrt{f(r)},\ n_{\theta}=0,\ n_{\phi}=0\,.$ (6)

The Israel-Darmois junction conditions in Eq. (1) determine the density,
pressure, and 3-velocity of the viscous fluid as

$\displaystyle\rho_{0}$ $\displaystyle=-\frac{\sqrt{f(r_{0})}}{4\pi
r_{0}},~{}p_{0}=\frac{1+f(r_{0})}{16\pi r_{0}\sqrt{f(r_{0})}},$ $\displaystyle
u^{a}_{0}$
$\displaystyle=\left(\frac{1}{\sqrt{f(r_{0})}},0,\frac{2J}{r_{0}^{3}\left(1-3f(r_{0})\right)\sqrt{f(r_{0})}}\right).$
(7)

The detailed derivation of the membrane parameters is given in Appendix A. As
expected, the presence of the angular momentum induces a non-zero azimuthal
velocity to the fluid.

###  II.2 Linear perturbations

Let us now perturb the background metric as

$g_{\mu\nu}=g_{\mu\nu}^{0}+\epsilon_{p}\delta g_{\mu\nu}\,,$ (8)

where $g_{\mu\nu}^{0}$ is the metric in Eq. (4) and $\epsilon_{p}$ is a small
parameter controlling the strength of perturbation. The metric perturbation
$\delta g_{\mu\nu}$ can be decomposed in the spherical harmonic basis as Regge
and Wheeler (1957); Zerilli (1970)

$\delta g_{\mu\nu}=\delta g_{\mu\nu}^{\rm{axial}}+\delta
g_{\mu\nu}^{\rm{polar}}\,,$ (9)

where under parity transformations ($\theta\rightarrow\pi-\theta$,
$\phi\rightarrow\phi+\pi$), the axial sector transforms as $(-1)^{\ell+1}$ and
the polar sector as $(-1)^{\ell}$. In the absence of spin, the two sectors
evolve independently without mixing. However, in the presence of spin, they
mix in the field equations and the boundary conditions. In the slow-rotation
limit, it can be shown that the coupling terms between the axial and polar
sectors can be dropped for the computation of th QNM frequencies to linear
order in spin Pani (2013). The axial and polar perturbations are given by
Kojima (1992); Pani (2013)

$\displaystyle\delta g_{\mu\nu}^{\text{axial}}$ $\displaystyle=2\sum_{\ell
m}\big{\\{}h_{0}(r,t)\left[S_{\theta}(\theta,\phi)d\theta+S_{\phi}(\theta,\phi)d\phi\right]dt$
$\displaystyle+h_{1}(r,t)\left[S_{\theta}(\theta,\phi)d\theta+S_{\phi}(\theta,\phi)d\phi\right]dr\big{\\}}\,,$
(10) $\displaystyle\delta g_{\mu\nu}^{\text{polar}}=$ $\displaystyle\sum_{\ell
m}\left[H_{0}(r,t)dt^{2}+H_{2}(r,t)dr^{2}+r^{2}K(r,t)d\Sigma^{2}\right.$
$\displaystyle\left.+2H_{1}(r,t)dtdr\right]Y^{\ell m}(\theta,\phi)\,,$ (11)

where $S_{\theta}(\theta,\phi)=-\partial_{\phi}Y_{\ell
m}(\theta,\phi)/\sin\theta$,
$S_{\phi}(\theta,\phi)=\sin\theta\partial_{\theta}Y_{\ell m}(\theta,\phi)$,
$Y_{\ell m}(\theta,\phi)$ are the tensor spherical harmonics, and $\ell$ and
$m$ are the angular and azimuthal numbers of the perturbation. The time
dependence can be factorised as

$h_{0}(r,t)=h_{0}(r)e^{-i\omega t}\,,$ (12)

where the same holds for $h_{1}(r,t)$, $H_{0}(r,t)$, $H_{1}(r,t)$,
$H_{2}(r,t)$, $K(r,t)$, and the explicit dependence of the radial functions on
$(\ell,m)$ is dropped for simplicity of notation.

In response to the metric perturbation, the density, pressure and 3-velocity
of the membrane stress-energy tensor are perturbed as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\rho_{0}+\epsilon_{p}\delta\rho(\theta,\phi)e^{-i\omega t}\,,$
$\displaystyle p$ $\displaystyle=$ $\displaystyle p_{0}+\epsilon_{p}\delta
p(\theta,\phi)e^{-i\omega t}\,,$ (13) $\displaystyle u^{a}$ $\displaystyle=$
$\displaystyle u^{a}_{0}+\epsilon_{p}\delta u^{a}(\theta,\phi)e^{-i\omega
t}\,,$

where $\rho_{0}$, $p_{0}$ and $u^{a}_{0}$ are given in Eq. (II.1), and their
perturbations can be decomposed in the spherical harmonic basis. In the
spinless case ($J=0$) the axial perturbations couple with the azimuthal
component of the velocity perturbation of the fluid, whereas the polar
perturbations couple with the density perturbation, the pressure perturbation,
and the radial and polar deformations of the membrane location defined in Eq.
(66). At the first order in spin, the perturbations with a given parity and
angular number $\ell$ are coupled to those with opposite parity and angular
number $\ell\pm 1$. However, it can be shown that the couplings to the
$\ell\pm 1$ terms do not contribute to the QNM spectrum to the linear order in
spin Pani (2013). Thus, we neglect these terms in the following, and work with
axial and polar modes separately. See Appendix C for the derivation of the
equations relating the metric perturbations to the fluid perturbations of the
membrane.

Outside the membrane, the Einstein equations governing the metric
perturbations can be reduced to two Schrödinger-like equations, namely the
Regge-Wheeler and Zerilli equations describing axial and polar perturbations,
respectively Regge and Wheeler (1957); Zerilli (1970). They are of the form:

$\frac{d^{2}\psi(r)}{dr_{*}^{2}}+\left[\omega^{2}-\frac{4mJ\omega}{r^{3}}-V(r)\right]\psi(r)=0\,,$
(14)

where the Regge-Wheeler effective potential at linear order in the spin is
Kojima (1992); Pani (2013)

$V_{\rm{RW}}=f(r)\left[\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}+\frac{24Jm(3r-7M)}{\ell(\ell+1)\omega
r^{6}}\right]\,,$ (15)

and the Zerilli effective potential is given in Eq. (56), where the tortoise
coordinate is defined as $dr_{*}/dr=1/f(r)$. The Regge-Wheeler function
$\psi_{\text{RW}}(r)$ is related to the metric perturbation function
$h_{1}(r)$ as

$\displaystyle\psi_{\text{RW}}(r)=\left(1-\frac{2mJ}{r^{3}\omega}\right)^{-1}\frac{f(r)}{r}h_{1}(r)\,,$
(16)

and $h_{0}(r)$ is related to $h_{1}(r)$ via Eq. (53). Similarly, the Zerilli
function $\psi_{\text{Z}}(r)$ can be written as a linear combination of
$H_{0}(r)$, $H_{1}(r)$, $H_{2}(r)$ and $K(r)$ as in Eqs. (57)-(61) at linear
order in spin.

###  II.3 Boundary conditions

By imposing boundary conditions at infinity and the radius of the compact
object, Eq. (14) defines an eigenvalue problem. From the junction condition in
Eq. (1), one finds a constraint on the metric perturbation at the membrane
surface, i.e., a boundary condition at the radius of the compact object.
Different models of horizonless compact objects are described by different
boundary conditions, which depend on the membrane parameters such as the shear
and bulk viscosities of the fluid. The boundary condition at membrane surface
for axial perturbations turns out to be

$\displaystyle\frac{1}{\psi_{\text{RW}}(r)}\frac{d\psi_{\text{RW}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}=-\frac{i\omega}{\nu}+\frac{My^{2}V_{\text{RW}}(r_{0})}{2w(3w-y)}+$
$\displaystyle\frac{m\chi}{2\ell(\ell+1)My^{5}(y-3w)^{2}\nu}\Big{\\{}w^{3}\Big{[}36w^{2}y\Bigl{(}i\left(\ell^{2}+\ell+9\right)y+$
$\displaystyle 16\nu+\nu
y^{2}\Bigl{)}+6wy^{2}\Bigl{(}\left(\ell^{2}+\ell-7\right)\nu
y^{2}-2iy\left(2\ell^{2}+2\ell+13\right)$
$\displaystyle-42\nu\Bigl{)}+y^{3}\Bigl{(}-3\left(\ell^{2}+\ell-4\right)\nu
y^{2}+4i\left(\ell^{2}+\ell+6\right)y+$ $\displaystyle
36\nu\Bigl{)}-216iw^{3}\left(y-2i\nu\right)\Big{]}-3w^{3}(2w-y)\left(6w-\ell(\ell+1)y\right)$
$\displaystyle\Bigl{[}\left(4w-2y+y^{3}\right)\nu-2iy(3w-y)\Bigl{]}\Big{\\}},$
(17)

where $\nu=16\pi\eta$, $y=\omega r_{0}$, and $w=M\omega$. The polar boundary
condition turns out to be

$\displaystyle\frac{1}{\psi_{\text{Z}}(r)}\frac{d\psi_{\text{Z}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}$
$\displaystyle=-16\pi\eta i\omega+G(r_{0},\omega,\eta,\zeta)$
$\displaystyle+m\chi H(r_{0},\omega,\eta,\zeta)\,,$ (18)

where $G(r_{0},\omega,\eta,\zeta)$ is given in Ref. Maggio  _et al._ (2020),
and the supplemental Mathematica notebook in git  contains the complete
expression for the polar boundary condition including
$H(r_{0},\omega,\eta,\zeta)$. The detailed derivation of the above boundary
conditions starting from the junction conditions in Eq. (1) is provided in
Appendix C.

Note that the above boundary conditions are invariant under the transformation
$(m\rightarrow-m$, $\chi\rightarrow-\chi)$. It is also worth noting that the
axial boundary condition does not depend on the bulk viscosity $\zeta$ of the
fluid membrane.

The BH boundary condition is recovered in the $\epsilon\rightarrow 0$ limit,
for which $V(r_{0})=0$ and $y=2w$, and by setting the membrane shear and bulk
viscosities to BH values as in Eq. (3). In this limit, the axial and polar
boundary conditions in Eqs. (II.3) and (II.3) reduce to

$\displaystyle\frac{d\psi_{\rm{RW}}(r)/dr_{*}}{\psi_{\rm{RW}}(r)}\Big{|}_{r=r_{+}}=\frac{d\psi_{\rm{Z}}(r)/dr_{*}}{\psi_{\rm{Z}}(r)}\Big{|}_{r=r_{+}}=-i\tilde{\omega}\,,$
(19)

where $\tilde{\omega}=\omega-m\Omega_{H}$ and $\Omega_{H}=J/(4M^{3})$ is the
horizon angular velocity. This is consistent with the expected BH boundary
condition of a purely ingoing perturbation near the horizon, i.e.,

$\psi_{\rm{RW}}(r)\sim\psi_{\rm{Z}}(r)\sim
e^{-i\tilde{\omega}r_{*}}\,,\quad\text{as}\ r_{*}\to-\infty\,.$ (20)

Finally, we check that in the spinless limit, $\chi\rightarrow 0$, the
boundary conditions in Eqs. (II.3) and (II.3) reduce to those obtained in Ref.
Maggio  _et al._ (2020) for a horizonless compact object with a Schwarzschild
exterior.

##  III Observables

###  III.1 Reflectivity of compact objects

Figure 1: Time-independent scattering of gravitational perturbations off a
compact object, which is described by a membrane located at the object’s
radius. At infinity, the Regge-Wheeler and Zerilli wave functions asymptote
towards free-wave solutions. At the membrane surface, the boundary conditions
in Eqs. (II.3) and (II.3) are obeyed. The reflectivity of the compact object
at infinity is defined as $\mathcal{R}=A_{\mathrm{out}}/A_{\mathrm{in}}$.

One way to differentiate between compact objects is based on their
reflectivity as measured from infinity. Given an ingoing wave packet sent from
infinity in the distant past, the reflectivity is defined as the resultant
scattered waveform at infinity in the distant future. It is convenient to
study a time-independent scattering, where both ingoing and outgoing spherical
waves are simultaneously present. This is schematically represented in Fig. 1,
where the ratio of the amplitudes of the outgoing and ingoing waves at
infinity defines the reflectivity as

$\psi(r)\sim e^{-i\omega r_{*}}+\mathcal{R}e^{i\omega r_{*}}\,,\quad\text{as}\
r_{*}\to\infty\,,$ (21)

after the wave interacts with the compact object and is subjected to the
boundary conditions in Eqs. (II.3) and (II.3) for axial and polar
perturbations, respectively.

For non-spinning BHs, with a horizon but no ergoregion, the absolute value of
reflectivity, $|\mathcal{R}|$, is bounded from above by 1. For small
frequencies, or equivalently large wavelengths, the wave packets larger than
the BH size are only negligibly absorbed, yielding $|\mathcal{R}|\approx 1$.
As the wavelength becomes comparable with the BH size, absorption is more
relevant and $|\mathcal{R}|<1$.

For Kerr BHs possessing an ergoregion, the reflectivity can exceed $1$ when
the frequency of the perturbation is smaller than the BH horizon angular
velocity, i.e., $\omega<m\Omega_{H}$. This is due to the phenomenon of
superradiance, in which the energy associated with the object’s rotation may
be transmitted to the waves/particles scattered by it. In this case, the BH
loses its rotational energy, leading to an amplification of the outgoing
radiation. The analytical expression for the reflectivity of Kerr BHs at low
frequencies to the next-to-leading order in $M\omega$ was derived in Refs.
Starobinsky (1973); Starobinskij and Churilov (1973); Creci  _et al._ (2021);
Saketh  _et al._ (2023); Dolan (2008); Saketh  _et al._ (2024).

For generic compact objects, the reflectivity $|\mathcal{R}|$ is related to
the membrane parameters (i.e., shear and bulk viscosity of the fluid) through
the boundary conditions in Eqs. (II.3) and (II.3) for axial and polar
perturbations, respectively. The computation of the reflectivity for arbitrary
frequencies is done via numerical integration of the perturbation equations in
Eq. (14) in Sec III.1. However, some features of the reflectivity may be
extracted analytically and are discussed in the next few paragraphs. We will
also discuss the limitations of the linear-in-spin approximation in the form
of spurious effects, specifically superradiance, leading to $|\mathcal{R}|>1$
when the spin parameter $\chi$ becomes large enough. This is physically not
allowed as superradiance requires the presence of an ergoregion, which is
absent at the linear order in spin. Nevertheless, this spuriousness is
understood in terms of the limits of validity of the linear-in-spin
approximation.

For an ultracompact object ($\epsilon\ll 1$), in the limits $r\rightarrow 2M$
and $r\rightarrow\infty$, the Regge-Wheeler and Zerilli equations at the
linear order in spin reduce to free-wave equations

$\displaystyle\frac{d^{2}\psi(r)}{dr_{*}^{2}}+\left[\omega^{2}-\frac{m\chi\omega}{2M}\right]\psi(r)=0,$
$\displaystyle\quad r\rightarrow 2M\,,$ (22)
$\displaystyle\frac{d^{2}\psi(r)}{dr_{*}^{2}}+\omega^{2}\psi(r)=0,$
$\displaystyle\quad r\rightarrow\infty,$ (23)

In both these limits, the asymptotic solutions can be written as

$\displaystyle\psi(r)\sim C_{\text{trans}}\
e^{-i\omega^{\prime}r_{*}}+C_{\text{ref}}\ e^{i\omega^{\prime}r_{*}},\quad$
$\displaystyle r\to 2M\,,$ (24) $\displaystyle\psi(r)\sim A_{\text{in}}\
e^{-i\omega r_{*}}+A_{\text{out}}\ e^{i\omega r_{*}},\quad$ $\displaystyle
r\to\infty\,,$ (25)

where $\omega^{\prime}=\omega\sqrt{1-m\chi/(2M\omega)}$ and
$\omega\in\text{Reals}$. For BHs, the purely ingoing boundary condition at the
horizon fixes $C_{\text{ref}}=0$. In order to provide an analytical
description of the reflectivity, we use the following conservation law valid
for the solution of a Schrödinger-like equation
$\psi^{\prime\prime}+(\omega^{2}-V)~{}\psi=0$ with real potential and
frequency:

$\displaystyle\frac{d}{dr_{*}}\left(\psi^{\prime}\bar{\psi}-\bar{\psi}^{\prime}\psi\right)=\psi^{\prime\prime}\bar{\psi}-\bar{\psi}^{\prime\prime}\psi=-\tilde{V}\psi\bar{\psi}+\tilde{V}\bar{\psi}\psi=0$
$\displaystyle\implies\left(\psi^{\prime}\bar{\psi}-\bar{\psi}^{\prime}\psi\right)\Big{|}_{r=2M}=\left(\psi^{\prime}\bar{\psi}-\bar{\psi}^{\prime}\psi\right)\Big{|}_{r\to\infty},$
(26)

where the prime denotes a derivative with respect to the tortoise coordinate,
$\bar{\psi}$ is the complex conjugate of $\psi$, and
$\tilde{V}=-\omega^{2}+4mJ\omega/r^{3}+V(r)$. Eq. (26) together with Eqs.
(22),(23) implies in turn the relation

$\displaystyle\frac{\text{Re}(\omega^{\prime})}{\omega}|C_{\text{trans}}|^{2}e^{2\text{Im}(\omega^{\prime})r_{*}}=|A_{\text{in}}|^{2}-|A_{\text{out}}|^{2}\,,$
(27)

where we note that $\omega^{\prime}\in\text{Reals}$, provided
$\omega>m\chi/2M$, and furthermore $\omega^{\prime}>0$. Thus, for sufficiently
large frequencies, i.e., $\omega>m\chi/2M$, we have
$|A_{\rm{in}}|^{2}>|A_{\rm{out}}|^{2}$ or $|\mathcal{R}|^{2}<1$ as expected
from BHs. When the frequency is small, i.e., $\omega<m\chi/2M$,
$\omega^{\prime}$ becomes purely imaginary and the right hand side of Eq. (27)
vanishes, yielding $|\mathcal{R}|^{2}=1$. This is different from the behaviour
of Kerr BHs where one observes a superradiant amplification, i.e.,
$|\mathcal{R}|^{2}>1$, when $\omega<m\Omega_{H}$ Penrose and Floyd (1971);
Zel’Dovich (1971); Blandford and Znajek (1977); Brito  _et al._ (2015). This
can be explained by the fact that we are using a linear-in-spin approximation
of the Kerr metric, which does not possess an ergoregion and cannot reproduce
a superradiant scattering.

Something interesting happens, however, when we expand $\omega^{\prime}$ to
linear order in spin, i.e.,
$\omega^{\prime}=\omega-m\chi/(4M)+\mathcal{O}(\chi^{2})$, which modifies Eq.
(27) to

$\displaystyle\frac{\tilde{\omega}}{\omega}|C_{\text{trans}}|^{2}=|A_{\text{in}}|^{2}-|A_{\text{out}}|^{2},$
(28)

where $\tilde{\omega}$ is defined below Eq. (19) and
$\tilde{\omega}\in\text{Reals}$. Now, we find a superradiant behaviour when
$\omega<m\Omega_{H}$, which is analogous to the Kerr BH case. Nevertheless,
since we do not expect superradiance at linear order in spin due to the
absence of an ergoregion, we interpret this as a spurious result. Indeed, a
consistent study of the slow rotation approximation requires going to at least
second order in spin Pani  _et al._ (2012); Pani (2013). This argument
requires extra care when dealing with membranes as their boundary condition is
valid up to leading order in spin (unlike the expression in Eqs. (24)). We
thus interpret any observation of effects related to the presence of an
ergoregion as a spurious result. This will become relevant for the analysis of
the stability of QNMs as shown in Sec. III.2.

For the compact objects analysed here, the asymptotic behaviour of the
perturbations at infinity in Eq. (25) is still valid, however the asymptotic
solution near the object’s radius does not have a simple analytical form. The
following conservation law holds

$\displaystyle\left(\psi^{\prime}\bar{\psi}-\bar{\psi}^{\prime}\psi\right)\Big{|}_{r=r_{0}}=(\psi^{\prime}\bar{\psi}-\bar{\psi}^{\prime}\psi)\Big{|}_{r=\infty},$
(29)

where $r_{0}$ is the location of the membrane. We now use the boundary
conditions in Eqs. (II.3) and (II.3) for axial and polar perturbations to
rewrite the conservation law as

$\displaystyle-|\psi|^{2}\frac{\text{Im}[\text{BC}(\omega,r_{0},\eta,\zeta)]}{i\omega}=|A_{\text{in}}|^{2}-|A_{\text{out}}|^{2}\,$
(30)

where $\text{BC}(\omega,r_{0},\eta,\zeta)=\psi^{\prime}/\psi|_{r=r_{0}}$ is
the boundary condition to be satisfied by the relevant perturbation functions
(Regge-Wheeler or Zerilli). In the BH limit, i.e., $\epsilon\rightarrow 0$,
$\eta\rightarrow\eta_{\rm{BH}}and\zeta\rightarrow\zeta_{\rm{BH}}$, Eq. (30)
reduces to Eq. (28) as expected. Since the exterior spacetime is the Kerr
metric at the linear order in spin, which does not have an ergoregion, we
cannot trust the regime in which $|\mathcal{R}|^{2}>1$. It is also worth
noting that a vanishing of the imaginary part of the boundary condition on the
membrane leads to a perfect reflection where
$|A_{\rm{out}}|^{2}=|A_{\rm{in}}|^{2}$. This can also be achieved by setting
$\psi=0$ on the membrane, corresponding to a Dirichlet boundary condition.

####  III.1.1 Numerical computation of the reflectivity : Methods

The reflectivity can be computed numerically by integrating the perturbation
equations in Eq. (14) with the boundary conditions represented in Fig. 1. The
second-order Regge-Wheeler and Zerilli equations are integrated outwards
starting from the membrane radius $r_{0}$, where the wave function and its
first derivative need to be specified at the membrane surface. Among the two
unknown functions, one is simply a scale, therefore we can set
$\psi(r_{0})=1$. Then, $\psi^{\prime}(r_{0})$ is specified by the boundary
conditions in Eqs. (II.3) and (II.3) for axial and polar perturbations,
respectively.

The equations are numerically integrated outwards up to a point sufficiently
far away (say $r_{\infty}=50M$), which may be regarded as “infinity”. We check
that the result of the integration is numerically stable by changing the
numerical value for the infinity. The asymptotic solutions at infinity are
given in Eq. (25). Then, given the (numerically obtained) wavefunction and its
first derivative, we compute the reflectivity as

$\displaystyle|\mathcal{R}|=\Bigg{|}\frac{A_{\text{out}}}{A_{\text{in}}}\Bigg{|}=\Bigg{|}\frac{i\omega\psi(r_{\infty})+\psi^{\prime}(r_{\infty})}{i\omega\psi(r_{\infty})-\psi^{\prime}(r_{\infty})}\Bigg{|}.$
(31)

The accuracy of the integration can be improved by making use of series
solutions at infinity or close to the horizon in the BH case. This modifies
the relation in Eq. (31) between the wavefunction and the reflectivity at
infinity accordingly. However, concerning the computation of the reflectivity,
series solutions at infinity do not affect significantly the result. For
instance, for a compact object with $\epsilon=10^{-10}$ and
$\eta=\eta_{\rm{BH}}$, the absolute value of the reflectivity obtained without
asymptotic series solutions at infinity and using a nineteenth-order
asymptotic series solution in powers of $1/r$ differ by less than $0.1\%$ at
the frequency $M\omega=0.37$. For this reason, we use Eq. (31) for the
computation of the reflectivity, where $\psi(r_{\infty})$ and
$\psi^{\prime}(r_{\infty})$ are obtained from a direct integration.

####  III.1.2 Numerical computation of the reflectivity : Results

Figure 2: Reflectivity of a compact object for axial perturbations in terms of
the shear viscosity, $\eta$, of a fictitious fluid located at the object’s
radius, $r_{0}=2M(1+\epsilon)$, with $\epsilon=10^{-10}$, $\chi=0.1$ and
$\ell=m=2$. The shear viscosity varies as $\eta\in(0,\eta_{\rm{BH}})$ in the
top panel and $\eta\in(\eta_{\rm{BH}},\infty)$ in the bottom panel. Note that
the limits $\eta\to 0$ and $\eta\to\infty$ lead to a perfectly reflecting
compact object, described by Dirichlet and Neumann boundary conditions,
respectively. When $\eta\neq\eta_{\rm{BH}}$, the reflectivity has an
oscillatory structure, which is related to the QNMs of the object.

Let us discuss the variation of reflectivity of the compact object with
respect to the fluid parameters. The shear viscosity $\eta$, appearing in both
the axial and polar boundary conditions, controls the reflectivity of the
object in the ultracompact limit. This can be seen from the
$\epsilon\rightarrow 0$ limit of Eqs. (II.3) and (II.3), which is

$\displaystyle\frac{1}{\psi_{\rm{RW}}(r)}\frac{d\psi_{\rm{RW}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}$
$\displaystyle=-\frac{i}{16\pi\eta}\Big{(}\omega-m\frac{\chi}{4M}\Big{)},$
(32)
$\displaystyle\frac{1}{\psi_{\rm{Z}}(r)}\frac{d\psi_{\rm{Z}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}$
$\displaystyle=-16\pi i\eta\Big{(}\omega-m\frac{\chi}{4M}\Big{)}+$ (33)
$\displaystyle\frac{\left(\ell^{2}+\ell-1\right)m\left[\left(16\pi\eta\right)^{2}-1\right]\chi\omega}{2l\left(\ell+1\right)\left(\ell^{2}+\ell+1\right)}.$

In the BH limit, $\eta\rightarrow\eta_{\rm{BH}}$, the above equations reduces
to the purely ingoing boundary condition as expected from BHs. In the spinless
limit ($\chi=0$), the limit $\eta\to 0$ corresponds to a Dirichlet ($\psi_{\rm
RW}=0$) and a Neumann ($\psi^{\prime}_{\rm{Z}}=0$) boundary condition on axial
and polar perturbations, respectively. Whereas the limit $\eta\to\infty$
corresponds to a Dirichlet ($\psi_{\rm Z}=0$) and a Neumann
($\psi^{\prime}_{\rm{RW}}=0$) boundary condition on polar and axial
perturbations, respectively. In both of these limits, the compact object is
purely reflecting at any frequency, i.e., $|\mathcal{R}|=1$. Note that when
the spin is turned on ($\chi\neq 0$), the limit $\eta\rightarrow 0$ no longer
corresponds to a Neumann boundary condition for polar perturbations. However,
this limit still describes a purely reflective compact object with
$|\mathcal{R}|=1$. This can be seen from Eq. (30), where the imaginary part of
the boundary condition (i.e., the right hand side of Eq. (33)) vanishes in the
limit $\eta\rightarrow 0$ while $\psi_{\rm{Z}}(r_{0})$ remains finite.

Fig. 2 shows the absolute value of the reflectivity of a compact object with
$\epsilon\ll 1$ and different values of the shear viscosity, particularly
$\eta\in(0,\eta_{\rm{BH}})$ in the top panel and
$\eta\in(\eta_{\rm{BH}},+\infty)$ in the bottom panel. We have set $\chi=0.1$
for simplicity. The plots are for axial perturbations, but they look
qualitatively similar in the polar case, yielding to the same results. The
reflectivity profile coincides with the BH one when $\eta=\eta_{\rm{BH}}$, and
describes a perfectly reflecting object in the $\eta\to 0$ and
$\eta\to+\infty$ limits. It is interesting to note that for generic values of
the shear viscosity, the reflectivity shows an oscillatory structure that
becomes resonant when $\eta\to 0$ and $\eta\to+\infty$. This behaviour is due
to the QNMs of the compact object, where the frequencies at which the dips (or
resonances) occur coincide with the real part of the QNMs of the object. Using
the analytical formulae for the QNMs of ultracompact objects in the small-
frequency regime Maggio  _et al._ (2019b, 2021b), we check that the distance
between subsequent dips (or resonances) is given by

$\Delta\omega=\frac{\pi}{|r_{*}^{0}|}\sim\frac{\pi}{2M|\log\epsilon|}\,.$ (34)

The resonance width is proportional to the imaginary part of the QNMs Maggio
_et al._ (2019b, 2021b), which becomes smaller in the perfectly reflecting
cases compared to the partially absorbing ones. This explains the sharpness of
the resonances in the $\eta\to 0$ and $\eta\to\infty$ limits.

Let us now analyse the variation of the reflectivity with the object’s
compactness for a fixed viscosity. For this purpose, we fix the viscosity in
the axial boundary condition in Eq. (II.3) to the BH value. The purely ingoing
boundary condition is recovered for $\epsilon\rightarrow 0$, as seen from Eq.
(32). However, in the limit $\epsilon\rightarrow 1/2$, corresponding to the
membrane radius located at the light ring, $r_{0}\rightarrow 3M$, the
reflectivity becomes unity as the axial boundary condition tends to the
Dirichlet boundary condition, i.e., $\psi_{\rm{RW}}(r_{0})=0$. This happens
regardless of the spin as the second and third terms in the right hand side of
Eq. (32) are divergent. The top panel of Fig. 3 shows the variation of the
reflectivity with the object’s compactness for axial modes in the slowly
spinning case where $\chi=0.1$. The reflectivity coincides with the BH one
when $\eta=\eta_{\rm{BH}}$, and describes a perfectly reflecting object in the
$\epsilon\to 1/2$ limit.

Figure 3:  Reflectivity of a compact object for axial (top panel) and polar
(bottom panel) perturbations in terms of the shear ($\eta=\eta_{\rm{BH}}$)
viscosity of a fictitious fluid located at the object’s radius,
$r_{0}=2M(1+\epsilon)$, for $\chi=0.1$ and $\ell=m=2$. The object’s
compactness varies as $\epsilon\in(10^{-10},0.5)$. Note that the reflectivity
approaches 1 as $\epsilon\rightarrow 0.5$, which corresponds to the object’s
radius approaching the light ring.

In the polar case, interestingly, the boundary condition in the spinless case
($\chi=0$) does not reach the perfectly reflecting limit as
$\epsilon\rightarrow 1/2$. Moreover, the reflectivity depends on the choice of
the bulk viscosity of the membrane fluid. However, the linear-in-spin term
$H(r_{0},\omega,\eta,\zeta)$ in Eq. (II.3) is divergent in the limit
$\epsilon\rightarrow 1/2$, thus changing the behaviour even for a small value
of spin. The bottom panel of Fig. 3 shows the variation of the reflectivity
with the object’s compactness for polar modes in the slowly spinning case
where $\chi=0.1$. Note that the small-spin approximation breaks down for axial
and polar modes in the limit $\epsilon\rightarrow 1/2$. This is because the
linear-in-spin term of the boundary condition scales as $(\epsilon-1/2)^{-2}$,
whereas the spinless term of the boundary condition scales as
$(\epsilon-1/2)^{-1}$, as shown in Eq. (II.3). Nevertheless, due to the
boundary condition diverging in the limit $\epsilon\rightarrow 1/2$ both in
the nonspinning and spinning cases, for $\epsilon=1/2$ the compact object is
purely reflecting.

###  III.2 Quasinormal mode spectrum

Another important observable, which is useful for differentiating between
compact objects, is the QNM spectrum. By imposing boundary conditions at
infinity and the radius of the compact object, the complex eigenvalues of Eq.
(14) are the QNM frequencies of the system, $\omega=\omega_{R}+i\omega_{I}$.
In our convention, a stable mode has $\omega_{I}<0$ and corresponds to an
exponentially damped sinusoidal signal with frequency $f=\omega_{R}/(2\pi)$
and damping time $\tau_{\text{damp}}=-1/\omega_{I}$. Conversely, an unstable
mode has $\omega_{I}>0$ with instability timescale
$\tau_{\rm{inst}}=1/\omega_{I}$. At infinity, we impose that the perturbation
is a purely outgoing wave, i.e.,

$\psi(r)\sim e^{i\omega r_{*}}\,,\quad\text{as}\ r_{*}\to\infty\,.$ (35)

At the object’s radius, we impose the boundary conditions in Eqs. (II.3) and
(II.3) for axial and polar perturbations, respectively. In the following, we
discuss the differences from the BH QNM spectrum for different values of the
membrane parameters and the object’s compactness.

####  III.2.1 Numerical methods for the QNMs

In this work, we employ the direct integration and continued fractions methods
for the numerical computation of the QNMs Pani (2013); Leaver (1985); Pani
_et al._ (2009). In the direct integration method, the perturbation equations
are integrated numerically starting from the membrane location, where the
boundary conditions in Eqs. (II.3) and (II.3) are imposed, to a sufficiently
large value of $r$. Then at infinity, the purely outgoing boundary condition
in Eq. (35) is imposed. The latter condition is satisfied for certain values
of the frequency, which are the QNM frequencies. The equations are solved
numerically using root-finding algorithms, along with an initial guess for the
frequency. This approach needs to use high-order asymptotic solutions
sufficiently far from the compact object (at $r\rightarrow\infty$) and at the
horizon of BHs ($r\rightarrow 2M$) to guarantee numerical accuracy. A
limitation of the direct integration method is that is does not give accurate
results for overtones, and is mainly useful for computing fundamental mode
frequencies.

The continued fractions method was originally developed in Ref. Leaver (1985)
to compute the QNMs as roots of implicit equations. The eigenfunction is
written as a series expansion whose coefficients satisfy a finite-term
recurrence relation. This method was generalized to work with boundary
conditions at the radius of compact objects in Refs. Pani  _et al._ (2009);
Maggio  _et al._ (2020).

####  III.2.2 Validity of the linear-in-spin approximation

Let us compute the fundamental QNM of a Kerr BH at linear order in spin, and
compare it with the fundamental QNM of a Kerr BH as a function of the spin.
This comparison allows one to set the maximum spin to which the linear-in-spin
approximatiom can be trusted. Since axial and polar modes are isospectral in
BHs regardless of the spin, we restrict our analysis to the axial QNMs.

To compute the QNM frequencies for a BH to linear order in spin, we impose
either a purely ingoing boundary condition at the horizon as in Eq. (19) or
the membrane boundary condition in Eq. (II.3) in the BH limit, i.e.,
$\epsilon\rightarrow 0$, $\eta\rightarrow\eta_{\rm{BH}}$ and
$\zeta\rightarrow\zeta_{\rm{BH}}$. From the numerical integration,
$\epsilon=0$ leads to a coordinate singularity, however it is sufficient to
choose a sufficiently small value of $\epsilon$, for example
$\epsilon=10^{-10}$, for which the integration is well defined. We checked
that the BH QNM does not depend on the choice of the numerical value for
$\epsilon$.

Since our analysis is valid in the slow-spin approximation, we expand the
numerically computed QNM frequencies as Pani (2013); Pierini and Gualtieri
(2022)

$\displaystyle\omega(\chi)=\omega^{(0)}+m\chi\omega^{(1)}+\mathcal{O}(\chi^{2}),$
(36)

and we discard the second-order-in-spin terms. The numerically solved QNMs
even using the linearised boundary conditions have a general dependence on
$\chi$. Thus, we Taylor-expand the numerically computed QNMs and truncate them
to linear order in spin. The quantity $\omega^{(0)}$ is computed when
$\chi=0$, and the quantity $\omega^{(1)}$ is computed as the slope of the
linear-in-spin expansion by simply using the finite difference formula

$\displaystyle\omega^{(1)}=\frac{\omega(\chi_{2})-\omega(\chi_{1})}{m(\chi_{2}-\chi_{1})},$
(37)

where $\chi_{1}=0$ and we choose $\chi_{2}=10^{-4}$. Alternatively, we can
construct a polynomial fit for $\omega(\chi)$ and truncate the fitted
polynomial to the linear order in spin. Both methods yield the same result for
$\omega^{(1)}$ within $10^{-3}\%$.

Figure 4: Real (top panel) and imaginary (bottom panel) part of the
fundamental $\ell=m=2$ QNM of a slowly-spinning BH with respect to a Kerr BH
as a function of the spin. The shaded band shows the 1% range beyond which the
linear-in-spin approximation is not valid.

To estimate the maximum value of the spin for which the linear-in-spin
approximation is valid, we plot the ratio $\omega/\omega^{\text{Kerr}}$ as a
function of the spin in Fig. 4. The plot shows that the linear-in-spin BH QNMs
are consistent with the Kerr QNMs within $1\%$ (shaded region) up to $\chi\sim
0.2$ for the real part of the frequency and $\chi\sim 0.35$ for the imaginary
part of the frequency. Based on this argument, we restrict to $\chi\leq 0.2$
as the region in which the linear-in-spin approximation is valid in the rest
of this work.

An additional constraint applies if clear unphysical features appear in the
results. For instance, as discussed in Sec. III.1, the linear-in-spin
approximation can lead to spurious superradiant effects for sufficiently small
frequencies, i.e., $m\chi/(4M\omega)>1$. In the next section, we show that
certain modes become unstable in purely reflecting ultracompact objects with
large values of the spin. This is explained as an ergoregion instability,
which is a spurious effect at linear order in spin Pani (2013); Maggio  _et
al._ (2019b).

####  III.2.3 QNMs with the membrane paradigm

Let us now investigate the QNM frequencies of slowly spinning horizonless
compact objects, and their variation with the membrane parameters.

First, we analyse a horizonless compact object with $\epsilon\ll 1$, whose
reflectivity is related to the shear viscosity of the membrane as in Fig. 2.
In this limit, the axial boundary condition approaches the expression in Eq.
(32). We derive numerically the variation of the QNM frequencies with the
shear viscosity $\eta$. This corresponds to the study of how the fundamental
mode evolves by varying the reflectivity of the compact object away from the
BH case.

Figure 5: Real (top panel) and imaginary (bottom panel) part of the
fundamental $\ell=m=2$ QNM of an ultracompact object as a function of the
viscosity parameter $\eta$ for different values of the spin. For
$\eta=1/(16\pi)$, the compact object is totally absorbing; whereas for
$\eta\to 0$ and $\eta\to\infty$, the compact object is perfectly reflecting.

Fig. 5 shows the real (top panel) and imaginary (bottom panel) part of the
axial QNMs of an ultracompact object as a function of the shear viscosity for
different values of the spin. The nonspinning case was derived earlier in Ref.
Maggio  _et al._ (2020).

When $\eta=1/(16\pi)$, the compact object is totally absorbing and we recover
the BH fundamental QNM. Then, the variation of the viscosity affects
drastically the mode, with a sharp behaviour around the the BH limit. For
$\eta\to 0$ and $\eta\to\infty$, the compact object is perfectly reflecting,
as shown in Fig. 2.

It is interesting to note the increase in the imaginary part of the QNMs when
the spin increases in the limits $\eta\rightarrow 0,~{}\infty$. This shows
that the $\ell=m=2$ mode decays faster with the spin. Moreover, according to
Eq. (36), the $m=-2$ modes should become longer lived. In particular, the
imaginary part of these modes can cross 0 indicating an instability (this
phenomenon was already observed in Refs. Maggio  _et al._ (2019b); Zhong  _et
al._ (2023) for totally reflecting boundary conditions). Fig. 6 shows the
imaginary part of the fundamental $(\ell,m)=(2,\pm 2)$ QNMs as a function of
the spin for the limits of $\eta\to 0$ and $\eta\to\infty$, corresponding to
total reflection.

Figure 6: Imaginary part of the $\ell=2$, $m=\pm 2$ QNMs of an ultracompact
object as a function of the spin for a perfectly reflecting compact object
with viscosity $\eta\gg\eta_{\rm{BH}}$ and $\eta\ll\eta_{\rm{BH}}$. Note that
the imaginary part for $m=-2$ changes sign above a critical value of the spin
rendering the mode unstable.

We note that the imaginary part of the $m=-2$ QNMs indeed changes sign,
signalling an instability in the totally reflecting limits. The critical value
of the spin where this happens ($\chi\sim 0.11,0.15$ for $\eta\to 0,\infty$,
respectively) is compatible to Ref. Maggio  _et al._ (2019b), where the
instability is associated with the ergoregion instability for spinning and
totally reflective ultracompact objects. However, from the earlier discussion
in Sec. III.1, we know that the presence of ergoregion-related behaviour is an
artefact of the expansion at the linear order in spin. Thus, it is merely a
coincidence that the linear-in-spin expansion makes the result more consistent
with the behaviour for arbitrary values of the spin. Nevertheless, it is clear
from Fig. 6 that the modes become longer lived as the spin increases in the
purely reflecting limit.

The variation of the polar fundamental QNM mode as a function of the shear
viscosity is qualitatively similar to the axial case. Thus, we do not show it
here. However, due to the differing boundary conditions in the
$\epsilon\rightarrow 0$ limit in Eqs. (32) and (33), the isospectrality
between axial and polar QNMs is broken when $\eta$ differs from the BH value
in both spinning and nonspinning cases. We also note that, for $\epsilon\to
0$, the polar boundary condition does not depend on the bulk viscosity of the
membrane. Fig. 7 shows the breaking of isospectrality between axial and polar
modes for $\chi=0,0.1$ as a function of the shear viscosity of the fluid.

Figure 7: Real (top panel) and imaginary (bottom panel) part of the $\ell=m=2$
QNM of an ultracompact object as a function of the viscosity parameter $\eta$
for axial and polar perturbations, and different values of the spin. Note that
the isospectrality between axial and polar modes is broken in both the
spinless and spinning cases.

Let us now analyse the variation of the QNMs with the compactness of the
object for a fixed shear viscosity. Fig. 8 shows the real (top panel) and
imaginary (bottom panel) part of the axial and polar QNMs of a spinning
horizonless compact object as a function of $\epsilon$, where $\chi=0.2$ and
$\eta=\eta_{\text{BH}}$. For $\epsilon\to 0$ we recover fundamental $\ell=m=2$
QNM of a Kerr BH at the first order in spin. For larger values of $\epsilon$,
the compactness of the object decreases and the QNMs depart from the BH
picture. The axial and polar QNMs are not isospectral; moreover, the polar
QNMs depend on the bulk viscosity of the membrane fluid.

The variation of the fundamental QNM in the spinless case for axial and polar
modes was derived in Ref. Maggio  _et al._ (2020). It was estimated that
horizonless compact objects with $\epsilon\lesssim 0.1$ are compatible with
the measurement accuracy of the fundamental QNM in GW150914 Abbott  _et al._
(2016b), which is approximately 10% and 15% for the real and imaginary part of
the frequency, respectively (corresponding to the shaded area in Fig. 8).
Here, we reconsider this argument by including the spin at the linear order.

Figure 8:  Real (top panel) and imaginary (bottom panel) part of the
fundamental $\ell=m=2$ QNM of a compact object with effective shear viscosity
$\eta=\eta_{\rm{BH}}$, with respect to their corresponding value for the BH
case. The QNMs are functions of the parameter $\epsilon$, which is related to
the compactness of the object. For $\epsilon\to 0$, we recover the fundamental
BH QNM and axial and polar modes are isospectral. The shaded band correspond
to the error bars (10% and 15%, respectively) for the real and the imaginary
part of the fundamental QNM for the merger event GW150914 Abbott  _et al._
(2016b).

By comparing Fig. 8 with Ref. Maggio  _et al._ (2020), we find that the
variation of the real part of the fundamental mode as a function of $\epsilon$
is suppressed by the spin. However, the variation of the imaginary part of the
fundamental mode is enhanced by the spin and it crosses the shaded band for
smaller values of $\epsilon$ than in the nonspinning case. Fig. 8 shows that
spinning ($\chi=0.2$) horizonless compact objects with $\epsilon\lesssim 0.05$
are compatible with measurement accyracy of the fundamental QNM in GW150914.
This suggests that spinning horizonless compact objects may be more easily
differentiated than nonspinning ones in the prompt ringdown. Moreover, the
breaking of isospectrality between axial and polar modes may be more readily
seen for increasing the values of the compactness.

##  IV Summary and conclusions

In this work, we have extended the membrane paradigm to linear order in spin,
to investigate slowly spinning horizonless compact objects. We have derived
the boundary conditions describing axial and polar perturbations of generic
compact objects to linear order in spin, and presented their expressions in
the Appendix and the supplemental Mathematica document git . We have discussed
the effect of spin in observables such as the object’s reflectivity at
infinity and the fundamental QNM frequencies in the ringdown. In particular,
we numerically computed the reflectivity as a function of the shear $\eta$ and
bulk $\zeta$ viscosities of the membrane and the object’s compactness. We
discussed the relation between the viscosity and the reflectivity in the limit
of the BH compactness ($\epsilon\rightarrow 0$), where the limits
$\eta\rightarrow 0,~{}\infty$ lead to the cases of purely reflecting compact
objects. In the spinning case, the reflectivity changes discontinuously when
the object’s radius approaches the light ring ($\epsilon\rightarrow 1/2$) even
for small values of the spin, both for axial and polar perturbations.

We analysed the variation of the fundamental QNM frequency with the membrane
parameters in the presence of spin. We computed the fundamental $\ell=m=2$
mode for the membrane parameters of a Kerr BH at linear order in spin
($\epsilon\rightarrow 0$, $\eta=1/16\pi$, $\zeta=-1/16\pi$) as a function of
the spin, and compared it with the fundamental QNM of a Kerr BH. We found that
the linear-in-spin QNM frequency is within 1% of the Kerr value up to
$\chi\sim 0.2$. We thus restricted the upper limit on the spin in this work at
$\chi=0.2$. However, considering the presence of spurious superradiance at
linear order in spin, we cautioned that one must keep a look out for such
unphysical effects, and setting $\chi<0.2$ as the regime of validity for
generic compact objects can lead to incorrect conclusions.

We analysed the shift in the fundamental QNM frequency starting from the BH
case ($\eta=1/16\pi$) and going towards the purely reflecting cases
($\eta\rightarrow 0,~{}\infty$). We noticed that for $m=-2$, the fundamental
mode frequencies become longer lived (i.e., the imaginary part tends to 0) as
the spin increases in the purely reflecting limits ($\eta\rightarrow
0,~{}\infty$). In particular, for $\chi\sim 0.1-0.15$, the imaginary part of
the fundamental mode switches sign signalling an instability. While this is
consistent with previous literature on QNMs of purely reflecting spinning
objects, we argued that the linear-in-spin approximation cannot exhibit such
effects lacking an ergosphere. Thus, we cannot claim the presence of an
instability from the linear-in-spin approach, but the trend of longer lived
fundamental mode as the spin is increased for purely reflecting bodies can be
taken as a result. We then considered the variation of the fundamental mode
frequency as the compactness approaches the BH value, for both axial and polar
modes. The polar modes have an additional dependence on the bulk viscosity
$\zeta$ when $\epsilon\neq 0$. Similarly to the nonspinning case analysed
earlier, we found that the axial and polar modes are not isospectral when
$\epsilon\neq 0$. Moreover, the spin enhances the variation of the imaginary
part of the fundamental mode with respect to the BH value. Therefore, spinning
horizonless compact objects may be more easily differentiated than nonspinning
ones in the prompt ringdown.

Overall, the inclusion of spin causes the QNM spectrum of a horizonless
compact object to deviate more strongly from the one of a BH with the same
spin parameter, even at linear order when the spin-induced multipole moments
are neglected. However, the remnants of compact binary coalescences have large
values of the spins, $\chi\gtrsim 0.6$. Therefore, it is important to extend
this framework to higher orders in spin as a future work. Finally, the
development of a Bayesian framework to map current constraints from
parametrised tests of GR into constraints on the properties of compact objects
is left for future work.

## Acknowledgements

We thank Hector O. Silva and Sebastian Völkel for useful discussions. E.M.
acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) - project
number: 386119226. E.M. is supported by the European Union’s Horizon Europe
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 101107586.

##  Appendix A Membrane paradigm at the linear order in spin: unperturbed
background

The unperturbed metric outside the membrane to linear order in spin is given
by :

$ds^{2}\Big{|}_{(0)}=g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}d\Sigma^{2}-4\frac{J}{r}\sin^{2}\theta
dtd\phi,$ (38)

where $f(r)=1-2M/r$, $d\Sigma^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, and
$J=Ma=M^{2}\chi$ is the angular momentum of the compact object (being modelled
by the membrane). The unperturbed membrane is located at the radius $r=r_{0}$.
We will use $x^{\mu}=\\{t,r,\theta,\phi\\}$ to refer to the global coordinates
and $y^{a}=\\{T=t,\Theta=\theta,\Phi=\phi\\}$ for the membrane coordinates.
The basis vectors for the membrane coordinates are defined as
$e_{a}^{\mu}=\partial x^{\mu}/\partial y^{a}$, where

$\displaystyle
e_{T}^{\mu}=\\{1,0,0,0\\},~{}e_{\Theta}^{\mu}=\\{0,0,1,0\\},~{}e_{\Phi}^{\mu}=\\{0,0,0,1\\}.$
(39)

The normal vector to the membrane, $n^{\mu}$, is fixed by the requirements
$n_{\mu}e_{a}^{\mu}=0$ and $g^{\mu\nu}n_{\mu}n_{\nu}=1$ to be
$n_{\mu}=\\{0,1/\sqrt{f(r)},0,0\\}$. The induced metric on the membrane
$h_{ab}$ is then given by

$\displaystyle
ds^{2}\Big{|}_{\text{membrane}}=h_{ab}dy^{a}dy^{b}=g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}dy^{a}dy^{b}=-f(r_{0})dT^{2}+r_{0}^{2}d\Theta^{2}+r_{0}^{2}\sin^{2}\Theta
d\Phi^{2}-4\frac{J}{r_{0}}\sin^{2}\Theta dTd\Phi\,.$ (40)

The extrinsic curvature is defined as
$K_{ab}=e_{a}^{\mu}e_{b}^{\nu}\nabla_{(\mu}n_{\nu)}$ and is given by

$\displaystyle
K_{ab}=\sqrt{f(r_{0})}\begin{bmatrix}-\frac{M}{r_{0}^{2}}&0&\frac{J\sin^{2}\Theta}{r_{0}^{2}}\\\
0&r_{0}&0\\\
\frac{J\sin^{2}\Theta}{r_{0}^{2}}&0&r_{0}\sin^{2}\Theta\end{bmatrix},$ (41)

where the trace is
$K=K_{ab}h^{ab}=\frac{2r_{0}-3M}{\sqrt{f(r_{0})}r_{0}^{2}}$. Thus the left
hand side of the junction condition in Eq. (1) is given by

$\displaystyle
K_{ab}-Kh_{ab}=\begin{bmatrix}\frac{2f(r_{0})^{3/2}}{r_{0}}&0&\frac{J(5r_{0}-8M)\sin^{2}\Theta}{\sqrt{f(r_{0})}r_{0}^{3}}\\\
0&\frac{M-r_{0}}{\sqrt{f(r_{0})}}&0\\\
\frac{J(5r_{0}-8M)\sin^{2}\Theta}{\sqrt{f(r_{0})}r_{0}^{3}}&0&\frac{M-r_{0}}{\sqrt{f(r_{0})}}\sin^{2}\Theta\end{bmatrix}.$
(42)

To relate the extrinsic curvature to the membrane fluid parameters, we also
need to compute the energy-momentum tensor in Eq. (2). This depends on the
fluid velocity $u^{a}$ in membrane coordinates. Intuitively, we expect the
fluid to be rotating in the global coordinates, while remaining static on the
membrane surface. Thus, we expect the radial and polar components to vanish
$u^{\mu}=\\{u^{t},0,0,Jv^{\phi}\\}$, where we factor out $J$ from the
azimuthal component of the fluid velocity. Using the metric in Eq. (38), we
can see this is also true for the contravariant components
$u_{\mu}=\\{u_{t},0,0,Jv_{\phi}\\}$. We also expect the components to be
independent of $t$ and $\phi$ due to stationarity and axial symmetry. In the
membrane coordinates, we can write

$u_{a}=u_{\mu}e_{a}^{\mu}=\\{u_{T}=u_{t}(\Theta),0,u_{\Phi}=Jv_{\phi}(\Theta)\\}\,.$
(43)

Now, demanding normalization $u^{\mu}u_{\mu}=-1$, we obtain
$u^{t}=1/\sqrt{f(r)}$ and for the contravariant components,

$\displaystyle u_{T}$ $\displaystyle=$ $\displaystyle
g_{t\mu}u^{\mu}=g_{tt}u^{t}+\mathcal{O}(J^{2})\approx-\sqrt{f(r_{0})}\,,$ (44)
$\displaystyle u_{\Phi}$ $\displaystyle=$ $\displaystyle
g_{\phi\mu}u^{\mu}=-2J\sin^{2}\theta/(r_{0}\sqrt{f(r_{0})})+r_{0}^{2}v^{\phi})+\mathcal{O}(J^{2})\,.$
(45)

Plugging this into Eq. (2), we obtain the stress energy tensor to be

$\displaystyle-8\pi T_{ab}=-8\pi\begin{bmatrix}\rho
f(r_{0})&0&\frac{J\sin^{2}\Theta}{r_{0}}\left[2\rho-f(r_{0})r_{0}^{3}(\rho+p)v^{\phi}\right]\\\
0&pr_{0}^{2}&-r_{0}^{2}J\sin^{2}\Theta\eta\partial_{\Theta}v^{\phi}\\\
\frac{J\sin^{2}\Theta}{r_{0}}\left[2\rho-f(r_{0})r_{0}^{3}(\rho+p)v^{\phi}\right]&-r_{0}^{2}J\sin^{2}\Theta\eta\partial_{\Theta}v^{\phi}&pr_{0}^{2}\sin^{2}\Theta\end{bmatrix}.$
(46)

Now, comparing Eq. (46) with Eq (42), and imposing the junction condition in
Eq. (1), we get

$\displaystyle\rho=-\frac{\sqrt{f(r_{0})}}{4\pi
r_{0}},~{}p=\frac{1+f(r_{0})}{16\pi
r_{0}\sqrt{f(r_{0})}},~{}Jv^{\Phi}=\frac{2J}{r_{0}^{3}\left(1-3f(r_{0})\right)\sqrt{f(r_{0})}}\,.$
(47)

The membrane fluid parameters with $J\to 0$ are consistent with the ones
obtained in the spinless case in Ref. Maggio  _et al._ (2020) .

##  Appendix B Perturbation equations

The perturbations can be expanded in the spherical harmonic basis as

$\displaystyle g_{\mu\nu}$ $\displaystyle=g_{\mu\nu}^{0}+\epsilon_{p}\delta
g_{\mu\nu},$ (48) $\displaystyle\delta g_{\mu\nu}$
$\displaystyle=\sum_{\ell,m}\Big{\\{}\left[H_{0}^{\ell
m}(r,t)dt^{2}+H_{2}^{\ell m}(r,t)dr^{2}+r^{2}K^{\ell
m}(r,t)d\Sigma^{2}+2H_{1}^{\ell m}(r,t)drdt\right]Y_{\ell m}(\theta,\phi)+$
$\displaystyle 2h_{0}^{\ell m}(r,t)\left[S_{\theta}^{\ell
m}(\theta,\phi)d\theta+S_{\phi}^{\ell
m}(\theta,\phi)d\phi\right]dt+2h_{1}^{\ell m}(r,t)\left[S_{\theta}^{\ell
m}(\theta,\phi)d\theta+S_{\phi}^{\ell m}(\theta,\phi)d\phi\right]dr\Big{\\}},$
(49)

where $S_{\theta}^{\ell m}(\theta,\phi)=-\partial_{\phi}Y_{\ell
m}(\theta,\phi)/\sin\theta$, $S_{\phi}^{\ell
m}(\theta,\phi)=\sin\theta\partial_{\theta}Y_{\ell m}(\theta,\phi)$,
$\epsilon_{p}$ is a small parameter controlling the strength of perturbation,
and we restrict to linear order in $\epsilon_{p}$. The metric perturbation can
be split into the axial and polar sectors considering their transformation
under parity. In the absence of spin, perturbations with different parity do
not mix, and satisfy the Regge-Wheeler and Zerilli equations for axial and
polar perturbations, respectively. To the linear order in spin, the coupling
between axial and polar modes can be neglected as far as the computation of
the QNM frequencies is concerned.

###  B.1 Axial sector

The axial sector is the part of the metric perturbation that transforms as
$(-1)^{\ell+1}$ under parity transformation, i.e.,
($\theta\rightarrow\pi-\theta$, $\phi\rightarrow\phi+\pi$). This is given by

$\displaystyle\delta g_{\mu\nu}^{\text{axial}}$
$\displaystyle=2\sum_{\ell,m}\left\\{h^{\ell m}_{0}(r,t)\left[S_{\theta}^{\ell
m}(\theta,\phi)d\theta+S_{\phi}^{\ell
m}(\theta,\phi)d\phi\right]dt+h_{1}(r,t)\left[S_{\theta}^{\ell
m}(\theta,\phi)d\theta+S_{\phi}^{\ell m}(\theta,\phi)d\phi\right]dr\right\\}.$
(50)

The Einstein equations in vacuum, $G_{\mu\nu}=0$, for axial perturbations to
the linear order in spin and in the perturbation, can be reduced to a modified
Regge-Wheeler equation to first order in spin given by Pani (2013)

$\displaystyle\frac{d^{2}\psi_{\text{RW}}(r)}{dr_{*}^{2}}+\left[\omega^{2}-\frac{4mJ\omega}{r^{3}}-f(r)\left(\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}+\frac{24Jm(-7M+3r)}{\ell(\ell+1)r^{6}\omega}\right)\right]\psi_{\text{RW}}(r)=0,$
(51)

where

$\displaystyle\psi_{\text{RW}}(r)=h_{1}(r)\frac{f(r)}{r}\left(1+\frac{2mJ}{r^{3}\omega}\right),$
(52)

and we have suppressed the $\ell,m$ indices for simplicity of notation. The
other function in the axial sector, $h_{0}(r)$, is related to $h_{1}(r)$
through the relation

$\displaystyle h_{0}(r)=\frac{if(r)\left[\ell(\ell+1)r^{4}\omega
h_{1}(r)f^{\prime}(r)+f(r)(-12Jqmh_{1}(r)+\tilde{q}r^{4}\omega
h_{1}^{\prime}(r)\right]}{\tilde{q}\omega r(-2Jm+r^{3}\omega)},$ (53)

where $q=(\ell-1)(\ell+2)/2$, $\tilde{q}=\ell(\ell+1)$, and the prime stands
for radial derivative ($d/dr$). Note that in the above equations, we have
factorized the time-dependence as $h_{1}(r,t)=h_{1}(r)e^{-i\omega t}$ and
similarly for $h_{0}(r,t)$.

###  B.2 Polar sector

The polar sector transforms as $(-1)^{\ell}$ under parity transformations, and
is given by

$\displaystyle\delta g_{\mu\nu}^{\text{polar}}=$
$\displaystyle\sum_{\ell,m}\left[H_{0}^{\ell m}(r,t)dt^{2}+H_{2}^{\ell
m}(r,t)dr^{2}+r^{2}K(r,t)d\Sigma^{2}+2H_{1}^{\ell m}(r,t)drdt\right]Y_{\ell
m}(\theta,\phi).$ (54)

The Einstein equations for the polar perturbations can be reduced to a
modified Zerilli equation to the first order in spin given by Pani (2013)

$\displaystyle\frac{d^{2}\psi_{\text{Z}}(r)}{dr_{*}^{2}}+\left[\omega^{2}-\frac{4mJ\omega}{r^{3}}-f(r)V_{\text{Z}}(r)\right]\psi_{\text{Z}}(r)=0,$
(55)

where

$\displaystyle
V_{\text{Z}}(r)=\frac{2M}{r^{3}}+\frac{(\ell-1)(\ell+2)}{3}\left[\frac{2(\ell-1)(\ell+2)(\ell^{2}+\ell+1)}{(6M+r(\ell^{2}+\ell-2))^{2}}+\frac{1}{r^{2}}\right]+$
(56)
$\displaystyle\frac{4mJ}{r^{7}\omega\ell(\ell+1)(6M+r(\ell^{2}+\ell-2))^{4}}\Big{\\{}27648M^{6}+2592M^{5}r(6\ell(\ell+1)-19)+144M^{4}r^{2}\Big{[}6r^{2}\omega^{2}+\ell(\ell+1)\times$
$\displaystyle(21\ell(\ell+1)-148)+230\Big{]}+12M^{3}r^{3}(\ell^{2}+\ell-2)\Big{[}72r^{2}\omega^{2}+\ell(\ell+1)(29\ell(\ell+1)-200)+374\Big{]}+12M^{2}r^{4}\times$
$\displaystyle(\ell^{2}+\ell-2)^{2}\Big{[}28r^{2}\omega^{2}+\ell(\ell+1)(5\ell(\ell+1)-12)-4\Big{]}+Mr^{5}(\ell^{2}+\ell-2)^{2}\Big{[}24r^{2}\omega^{2}(2\ell(\ell+1)-5)+(\ell-1)\times$
$\displaystyle(\ell+2)(\ell^{2}+\ell+2)(7\ell(\ell+1)-38)\Big{]}+r^{6}(\ell^{2}+\ell-2)^{3}\Big{[}2r^{2}\omega^{2}(\ell^{2}+\ell-4)-3(\ell^{2}+\ell-2)(\ell^{2}+\ell+2)\Big{]}\Big{\\}}.$

The Zerilli function $\psi_{\text{Z}}(r)$ is related to the radial functions
in the polar sector via the relations

$\displaystyle\begin{bmatrix}K(r)\\\
H_{1}(r)\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&A_{22}\end{bmatrix}.\begin{bmatrix}\tilde{\psi}_{\text{Z}}(r)\\\
\tilde{\psi}_{\text{Z}}^{\prime}(r)\end{bmatrix},$ (57)

where

$\displaystyle\mathbf{A}=\begin{bmatrix}\frac{12M^{2}+6Mqr+\ell(\ell+1)qr^{2}}{2r^{2}(3M+qr)}&1\\\
-i\omega\left[1+M\left(\frac{1}{2M-r}-\frac{6}{2M+2qr}\right)\right]&\frac{i\omega
r^{2}}{2M-r}\end{bmatrix}\,,$ (58)

and

$\displaystyle\tilde{\psi}_{\text{Z}}(r)=\psi_{\text{Z}}(r)\left[1+\frac{2mJ(12M^{3}-6M^{2}r+qr^{3}(q+r^{2}\omega^{2})+M(-2q^{2}r^{2}+3r^{4}\omega^{2})}{\ell(\ell+1)r^{4}(3M+qr)^{2}\omega}\right].$
(59)

The other functions describing polar metric perturbrations, $H_{0}$ and
$H_{2}$, can be written in terms of $H_{1}$ and $K$ via the relations

$\displaystyle H_{2}(r)$
$\displaystyle=\frac{r^{2}}{(r-2M)^{2}}H_{0}(r)+4mJ\omega
r\frac{K(r)}{\ell(\ell+1)(r-2M)^{2}},$ (60) $\displaystyle H_{0}(r)$
$\displaystyle=\frac{i(2M-r)(\tilde{q}M-2r^{3}\omega^{2})}{2r^{2}(3M+qr)\omega}H_{1}(r)-\frac{3M^{2}+M(-1+2q)r-qr^{2}+r^{4}\omega^{2})}{r(3M+qr)}K(r)$
(61)
$\displaystyle+\frac{mJ}{2\tilde{q}r^{5}(3M+qr)^{2}\omega^{2}}\\{i(2M-r)\left[\tilde{q}^{2}M(3M+qr)-2(\tilde{q}M-2r^{3}\omega^{2})(\tilde{q}M+r^{3}\omega^{2})\right]H_{1}(r)$
$\displaystyle+r\omega[4(\tilde{q}M+r^{3}\omega^{2})(3M^{2}+M(-1+2q)r-qr^{2}+r^{4}\omega^{2})+(6M+2qr)(2(3M-2r)r^{3}\omega^{2}$
$\displaystyle+\tilde{q}(-3M^{2}+Mr+2r^{4}\omega^{2}))]K(r)\\}.$

Although the Zerilli equation appears different than the Regge-Wheeler
equation, they both yield the same spectrum of QNMs when the boundary
condition is that of BHs. This can be shown in the non-spinning case via the
Chandrashekar transformation, and at the linear order in spin with a more
general transformation. For a generic compact object described by the membrane
paradigm, the boundary condition at the membrane surface can break the
isospectrality.

##  Appendix C Membrane paradigm at the linear order in spin: gravitational
perturbations

In accordance with the junction conditions, the extrinsic curvature and the
induced metric from either side of the membrane surface must satisfy

$\displaystyle(K^{+}_{ab}-h^{+}_{ab}K^{+})-(K^{-}_{ab}-h^{-}_{ab}K^{-})=-8\pi
T_{ab},\qquad h^{+}_{ab}=h^{-}_{ab},$ (62)

where $T_{ab}$ is the surface stress-energy tensor of the membrane. Normally,
to proceed further, one needs to fix the spacetime in the interior region to
the membrane. However, in this work, the membrane is intended to be an
effective way to parametrise an arbitrary compact object. This choice does not
break any symmetries and lets the stress-energy tensor do all the heavy
lifting. A simple way to implement this choice is to demand $K^{-}_{ab}=0$,
which is also the standard choice when the membrane is used to describe BHs.
Then, the junction condition reduces to

$\displaystyle K_{ab}-h_{ab}K=-8\pi T_{ab},$ (63)

where we have removed the $+$ signs denoting that quantities are evaluated
outside the membrane for convenience. The surface stress energy tensor is
given by Maggio  _et al._ (2020)

$\displaystyle T_{ab}=\rho
u_{a}u_{b}+(p-\zeta\Theta)\gamma_{ab}-2\eta\sigma_{ab},$ (64)

where $\Theta=h^{ab}\nabla_{b}u_{a}$, is the expansion rate,
$\sigma_{ab}=\nabla_{(a}u_{b)}-\frac{1}{2}\Theta\gamma_{ab}$ is the shear
tensor, and $\gamma_{ab}=h_{ab}+u_{a}u_{b}$ is the projection tensor
orthogonal to the fluid velocity. The derivation of the density, pressure and
fluid velocity when the membrane is unperturbed is given in Appendix A. Here,
we derive the boundary conditions for the metric perturbations at the surface
of the membrane to the linear order in spin.

When the perturbation is added to the background, its contribution appears
both on the membrane parameters and the metric perturbations in Eq. (63).
Thus, the outside metric is given by Eq. (48), and the membrane surface is
parametrised as

$\displaystyle x^{\mu}(y^{a})$
$\displaystyle=\left(t=T,r=r_{0}+\epsilon_{p}\delta r(\Theta,\Phi)e^{-i\omega
T},\theta=\Theta,\phi=\Phi\right),$ (65) $\displaystyle\delta r(\Theta,\Phi)$
$\displaystyle=\sum_{\ell,m}\xi^{\ell m}_{r}Y_{\ell m}(\Theta,\Phi),$ (66)

where $(T,\Theta,\Phi)$ are intrinsic coordinates to parametrise the
(perturbed) membrane, and $\delta r(\Theta,\Phi)$ is the radial deformation of
the membrane surface. We can derive the perturbed tangent vectors as
$e_{a}^{\mu}=dx^{\mu}/dy^{a}=(e_{a}^{\mu})^{0}+\epsilon_{p}\delta e_{a}^{\mu}$
where

$\displaystyle\delta e_{a}^{\mu}\equiv\begin{bmatrix}\delta e_{t}^{\mu}\\\
\delta e_{\Theta}^{\mu}\\\ \delta e_{\phi}^{\mu}\end{bmatrix}=\sum_{\ell
m}\begin{bmatrix}0&-i\omega\xi_{r}^{\ell m}Y_{\ell m}(\Theta,\Phi)&0&0\\\
0&\xi^{\ell m}_{r}\partial_{\Theta}Y_{\ell m}(\Theta,\Phi)&0&0\\\
0&\xi_{r}^{\ell m}\partial_{\Phi}Y_{\ell
m}(\Theta,\Phi)&0&0\end{bmatrix}e^{-i\omega T}\,,$ (67)

and the basis vectors in the unperturbed case, $(e_{a}^{\mu})^{0}$, is given
in Eq. (39). We can now derive the perturbations in the induced metric
$h_{ab}=g_{\mu\nu}e_{a}^{\mu}e_{b}^{\nu}=h_{ab}^{0}+\epsilon_{p}\delta
h_{ab}e^{-i\omega T}$, with $h_{ab}^{0}$ given in Eq. (40) and

$\displaystyle\delta h_{ab}=\sum_{\ell
m}\begin{bmatrix}\frac{r_{0}^{2}H_{0}^{\ell m}(r_{0})-2M\xi_{r}^{\ell
m}}{r_{0}^{2}}&-\csc\Theta h^{\ell m}_{0}(r_{0})\partial_{\Phi}&h^{\ell
m}_{0}(r_{0})\sin\Theta\partial_{\Theta}+\frac{2J\sin^{2}\Theta}{r_{0}^{2}}\xi_{r}^{\ell
m}\\\ -\csc\Theta h^{\ell m}_{0}(r_{0})\partial_{\Phi}&r_{0}\left(r_{0}K^{\ell
m}(r_{0})+2\xi_{r}^{\ell m}\right)&0\\\ h_{0}^{\ell
m}(r_{0})\sin\Theta\partial_{\Theta}+\frac{2J\sin^{2}\Theta}{r_{0}^{2}}\xi_{r}^{\ell
m}&0&r_{0}\left(r_{0}K^{\ell m}(r_{0})+2\xi_{r}^{\ell
m}\right)\sin^{2}\Theta\end{bmatrix}Y_{\ell m}(\Theta,\Phi).$ (68)

The normal vector $n_{\mu}=n_{\mu}^{0}+\epsilon_{p}\delta n_{\mu}e^{-i\omega
T}$, defined through the relations $n_{\mu}e_{a}^{\mu}=0$, and
$n_{\mu}n_{\nu}g^{\mu\nu}=1$, is perturbed as

$\displaystyle\delta n_{\mu}=\sum_{\ell
m}\begin{bmatrix}i\omega\frac{\xi_{r}^{\ell m}}{\sqrt{f(r_{0})}}\\\
\frac{H_{2}^{\ell m}(r_{0})}{2}\sqrt{f(r_{0})}\\\ -\frac{\xi_{r}^{\ell
m}}{\sqrt{f(r_{0})}}\partial_{\Theta}\\\ -\frac{\xi_{r}^{\ell
m}}{\sqrt{f(r_{0})}}\partial_{\Phi}\end{bmatrix}Y_{\ell m}(\Theta,\Phi).\,,$
(69)

where $n_{\mu}^{0}=\\{0,1/\sqrt{f(r_{0})},0,0\\}$. Subsequently, we can
compute the perturbation to the extrinsic curvature
$K_{ab}=K_{ab}^{0}+\epsilon_{p}\delta K_{ab}e^{-i\omega T}$, using
$K_{ab}=e_{a}^{\mu}e_{b}^{\nu}\nabla_{(\mu}n_{\nu)}$. $K_{ab}^{0}$ is defined
in Eq. (41), and

$\displaystyle\delta K_{TT}$ $\displaystyle=\sum_{\ell
m}\frac{\left(-5f^{2}+6f+4r_{0}^{2}\omega^{2}-1\right)\xi_{r}^{\ell
m}+r_{0}f\left(-f^{2}H_{2}^{\ell m}+fH_{2}^{\ell m}+4i\omega r_{0}H_{1}^{\ell
m}+2r_{0}(H_{0}^{\ell m})^{\prime}\right)}{4r_{0}^{2}\sqrt{f}}Y_{\ell
m}(\Theta,\Phi)\,,$ $\displaystyle\delta K_{\Theta\Theta}$
$\displaystyle=\sum_{\ell m}\frac{f\left(2\partial_{\Phi}h_{1}^{\ell
m}\csc\Theta\left(\cot\Theta-\partial_{\Theta}\right)+r_{0}^{2}(K^{\ell
m})^{\prime}+2r_{0}K^{\ell m}+\xi_{r}^{\ell m}\right)-r_{0}f^{2}H_{2}^{\ell
m}+\left(1-2\partial_{\Theta}^{2}\right)\xi_{r}^{\ell m}}{2\sqrt{f}}Y_{\ell
m}(\Theta,\Phi)\,,$ $\displaystyle\delta K_{T\Theta}$
$\displaystyle=\sum_{\ell m}\left[\frac{f\left[\partial_{\Theta}H_{1}^{\ell
m}+i\partial_{\Phi}\csc\Theta\left(\omega h_{1}^{\ell m}+i(h_{0}^{\ell
m})^{\prime}\right)\right]+2i\omega\partial_{\Theta}\xi_{r}^{\ell
m}}{2\sqrt{f}}+\frac{2J\cot\Theta\left(fh_{1}^{\ell
m}\sin\Theta\partial_{\Theta}-\partial_{\Phi}\xi_{r}^{\ell
m}\right)}{r_{0}^{3}\sqrt{f}}\right]Y_{\ell m}(\Theta,\Phi)\,,$
$\displaystyle\delta K_{T\Phi}$ $\displaystyle=\sum_{\ell
m}\left\\{\frac{f\left(i\omega h_{1}^{\ell
m}\sin\Theta\partial_{\Theta}+\sin\Theta\partial_{\Theta}(h_{0}^{\ell
m})^{\prime}-\partial_{\Phi}H_{1}^{\ell
m}\right)+2i\omega\partial_{\Phi}\xi_{r}^{\ell m}}{2\sqrt{f}}+\right.$
$\displaystyle\left.\frac{J\left[-f\left(4\partial_{\Phi}h_{1}^{\ell
m}\cos\Theta+5\sin^{2}\Theta\xi_{r}^{\ell m}\right)+r_{0}f^{2}H_{2}^{\ell
m}\sin^{2}\Theta+\sin\Theta\xi_{r}^{\ell
m}\left(4\cos\Theta\partial_{\Theta}+\sin\Theta\right)\right]}{2r_{0}^{3}\sqrt{f}}\right\\}Y_{\ell
m}(\Theta,\Phi)\,,$ $\displaystyle\delta K_{\Theta\Phi}$
$\displaystyle=\sum_{\ell
m}\frac{2\partial_{\Phi}\left(\cot\Theta-\partial_{\Theta}\right)\xi_{r}^{\ell
m}-fh_{1}^{\ell
m}\sin\Theta\left(\partial_{\Phi}^{2}\csc^{2}\Theta-\partial_{\Theta}^{2}+\cot\Theta\partial_{\Theta}\right)}{2\sqrt{f}}Y_{\ell
m}(\Theta,\Phi)\,,$ $\displaystyle\delta K_{\Phi\Phi}$ $\displaystyle=\delta
K_{\Theta\Theta}\sin^{2}\Theta+\sum_{\ell m}\frac{2\partial_{\Phi}fh_{1}^{\ell
m}\left(\sin\Theta\partial_{\Theta}-\cos\Theta\right)-\xi_{r}^{\ell
m}\left(-\sin^{2}\Theta\partial_{\Theta}^{2}+\sin\Theta\cos\Theta\partial_{\Theta}+\partial_{\Phi}^{2}\right)}{\sqrt{f}}Y_{\ell
m}(\Theta,\Phi)\,,$ (70)

where we have suppressed the radial argument in the perturbation variables for
simplicity of notation. All the radial functions and their derivatives are to
be evaluated at $r=r_{0}$. Also note that $\partial_{\Theta}$
($\partial_{\Phi}$) denote a differentiation with respect to $\Theta$
($\Phi$), acting on the spherical harmonic $Y_{\ell m}(\Theta,\Phi)$, and the
prime denotes a radial derivative.

The fluid membrane parameters are also perturbed so that Eq. (63) continues to
hold. We parametrise the fluid perturbations as

$\displaystyle\rho$
$\displaystyle=\rho^{0}+\epsilon_{p}\delta\rho(\Theta,\Phi)e^{-i\omega T},$
(71) $\displaystyle p$ $\displaystyle=p^{0}+\epsilon_{p}\delta
p(\Theta,\Phi)e^{-i\omega T},$ (72) $\displaystyle u^{a}$
$\displaystyle=(u^{a})^{0}+\epsilon_{p}\delta u^{a}(\Theta,\Phi)e^{-i\omega
T}.$ (73)

Furthermore, we expand the membrane perturbations in spherical harmonics
(analogous to stellar perturbation theory in Ref. Kojima (1992)) as

$\displaystyle\delta u^{\Theta}$ $\displaystyle=\sum_{\ell m}\Big{[}V^{\ell
m}\frac{S^{\ell m}_{\phi}(\Theta,\Phi)}{\sin\Theta}+U^{\ell m}S^{\ell
m}_{\theta}(\Theta,\Phi)\Big{]},$ (74) $\displaystyle\delta u^{\Phi}$
$\displaystyle=\sum_{\ell m}\Big{[}-V^{\ell m}\frac{S^{\ell
m}_{\theta}(\Theta,\Phi)}{\sin\Theta}+U^{\ell m}S^{\ell
m}_{\phi}(\Theta,\Phi)\Big{]},$ (75) $\displaystyle\delta\rho(\Theta,\Phi)$
$\displaystyle=\sum_{\ell m}\delta\rho^{\ell m}Y_{\ell m}(\Theta,\Phi),$ (76)
$\displaystyle\delta p(\Theta,\Phi)$ $\displaystyle=\sum_{\ell m}\delta
p^{\ell m}Y_{\ell m}(\Theta,\Phi).$ (77)

This makes the process of deriving the boundary conditions more convenient,
and splits the perturbations based on their transformation under parity. The
perturbed stress energy tensor is then given by

$\displaystyle\delta T_{TT}$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}\left[\frac{\sqrt{f}\left((f-1)\xi_{r}^{\ell m}+r_{0}H_{0}^{\ell
m}\right)}{4\pi r_{0}^{2}}+f\delta\rho^{\ell m}-\frac{J(\sin\Theta U^{\ell
m}\partial_{\Theta}+\partial_{\Phi}U^{\ell m})}{4\pi r_{0}^{2}}\right]Y_{\ell
m}(\Theta,\Phi),$ $\displaystyle\delta T_{\Theta\Theta}$ $\displaystyle=$
$\displaystyle\sum_{\ell m}\Big{\\{}r_{0}^{2}\Big{[}\text{$\delta p$}^{\ell
m}-2\eta\partial_{\Phi}\cot\Theta\csc\Theta U^{\ell
m}+\left(2\eta\partial_{\Phi}\csc\Theta U^{\ell m}-(\zeta-\eta)\cot\Theta
V^{\ell m}\right)\partial_{\Theta}-$ $\displaystyle V^{\ell
m}\Big{(}\partial_{\Phi}^{2}(\zeta-\eta)\csc^{2}\Theta+\partial_{\Theta}^{2}(\zeta+\eta)\Big{)}\Big{]}+\frac{\left(r_{0}K^{\ell
m}+2\xi_{r}^{\ell m}\right)\left(-M+8i\pi\zeta
r_{0}^{2}\omega+r_{0}\right)}{8\pi r_{0}\sqrt{f}}+$
$\displaystyle\frac{J\left[-\partial_{\Phi}\left(2\zeta r_{0}K^{\ell
m}+(3\zeta+\eta)\xi_{r}^{\ell m}\right)-6ir_{0}\omega(\zeta-\eta)h_{0}^{\ell
m}\sin\Theta\partial_{\Theta}\right]}{2r_{0}\sqrt{f}\left(r_{0}-3M\right)}+\frac{J(\zeta-\eta)}{f^{\frac{3}{2}}(3f-1)r_{0}^{2}}\times$
$\displaystyle\Big{[}4ir_{0}\omega h_{0}^{\ell
m}\sin\Theta\partial_{\Theta}-\partial_{\Phi}\left(\xi_{r}^{\ell
m}-r_{0}H_{0}^{\ell m}\right)-6ir_{0}^{3}\omega f^{3/2}\left(\sin\Theta
U^{\ell m}\partial_{\Theta}+\partial_{\Phi}V^{\ell
m}\right)+4ir_{0}^{3}\omega\sqrt{f}\times$ $\displaystyle\left(\sin\Theta
U^{\ell m}\partial_{\Theta}+\partial_{\Phi}V^{\ell
m}\right)\Big{]}\Big{\\}}Y_{\ell m}(\Theta,\Phi)\,,$ $\displaystyle\delta
T_{T\Theta}$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}\Big{\\{}\frac{r_{0}^{2}\partial_{\Theta}\left(3f-1\right)V^{\ell
m}-\partial_{\Phi}\csc\Theta\left(4\sqrt{f}h_{0}^{\ell m}+3r_{0}^{2}fU^{\ell
m}-r_{0}^{2}U^{\ell m}\right)}{16\pi r_{0}}-$ $\displaystyle\frac{2\eta
J\sin\Theta\left[-U^{\ell
m}\partial_{\Theta}^{2}+\partial_{\Phi}\csc^{2}\Theta\left(\partial_{\Phi}U^{\ell
m}+2\cos\Theta V^{\ell m}\right)+\csc\Theta\left(\cos\Theta U^{\ell
m}\partial_{\Theta}-2\partial_{\Phi}V^{\ell
m}\right)\right]}{r_{0}\left(3f-1\right)}\Big{\\}}Y_{\ell m}(\Theta,\Phi)\,,$
$\displaystyle\delta T_{T\Phi}$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}\Big{\\{}\frac{r_{0}^{2}\partial_{\Phi}\left(3f-1\right)V^{\ell
m}-\sin\Theta\partial_{\Theta}\left(4\sqrt{f}h_{0}^{\ell m}+3r_{0}^{2}fU^{\ell
m}-r_{0}^{2}U^{\ell m}\right)}{16\pi
r_{0}}-\frac{J}{r_{0}\left(r_{0}-3M\right)}\times$
$\displaystyle\Big{[}6M\sin^{2}\Theta\delta\rho^{\ell
m}+r_{0}\Big{(}\sin^{2}\Theta(\text{$\delta p$}^{\ell m}-\delta\rho^{\ell
m})+2\eta\partial_{\Phi}\cos\Theta U^{\ell
m}+\sin\Theta\left(2\eta\partial_{\Phi}U^{\ell m}+(\zeta+\eta)\cos\Theta
V^{\ell m}\right)\partial_{\Theta}$ $\displaystyle-(\zeta-\eta)\sin^{2}\Theta
V^{\ell m}\partial_{\Theta}^{2}-\partial_{\Phi}^{2}(\zeta+\eta)V^{\ell
m}\Big{)}\Big{]}+\frac{J\sin^{2}\Theta}{16\pi
r_{0}^{3}f^{3/2}\left(3f-1\right)}\Big{[}-ir_{0}f\left(-3iH_{0}^{\ell
m}+32\pi\zeta r_{0}\omega K^{\ell m}\right.$ $\displaystyle\left.-2iK^{\ell
m}+64\pi\zeta\omega\xi_{r}^{\ell m}\right)+f^{2}\left(6r_{0}K^{\ell
m}+\xi_{r}^{\ell m}\right)+24f^{3}\xi_{r}^{\ell m}+r_{0}H_{0}^{\ell
m}-\xi_{r}^{\ell m}\Big{]}\Big{\\}}Y_{\ell m}(\Theta,\Phi)\,,$
$\displaystyle\delta T_{\Theta\Phi}$ $\displaystyle=$ $\displaystyle\sum_{\ell
m}\Big{\\{}\eta r_{0}^{2}\csc\Theta\left[-U^{\ell
m}\sin^{2}\Theta\partial_{\Theta}^{2}+\partial_{\Phi}\left(\partial_{\Phi}U^{\ell
m}+2\cos\Theta V^{\ell m}\right)+\sin\Theta\left(\cos\Theta U^{\ell
m}-2\partial_{\Phi}V^{\ell m}\right)\partial_{\Theta}\right]+$
$\displaystyle\frac{J\sin\Theta}{8\pi
r_{0}^{2}f^{3/2}\left(3f-1\right)}\Big{[}f\left(8\pi\eta\sin\Theta\partial_{\Theta}\xi_{r}^{\ell
m}+3\partial_{\Phi}h_{0}^{\ell m}\left(3+16i\pi\eta
r_{0}\omega\right)\right)-9\partial_{\Phi}f^{2}h_{0}^{\ell
m}-3ir_{0}^{2}f^{3/2}\left(16\pi\eta r_{0}\omega-3i\right)$
$\displaystyle\left(\sin\Theta V^{\ell
m}\partial_{\Theta}-\partial_{\Phi}U^{\ell
m}\right)+2r_{0}^{2}\sqrt{f}\left(1+16i\pi\eta
r_{0}\omega\right)\left(\sin\Theta V^{\ell
m}\partial_{\Theta}-\partial_{\Phi}U^{\ell m}\right)+9r_{0}^{2}f^{5/2}\times$
$\displaystyle\left(\sin\Theta V^{\ell
m}\partial_{\Theta}-\partial_{\Phi}U^{\ell
m}\right)-2\partial_{\Phi}h_{0}^{\ell m}\left(1+16i\pi\eta
r_{0}\omega\right)+8\pi\eta r_{0}H_{0}^{\ell
m}\sin\Theta\partial_{\Theta}-8\pi\eta\sin\Theta\xi_{r}^{\ell
m}\partial_{\Theta}\Big{]}\Big{\\}}Y_{\ell m}(\Theta,\Phi),$
$\displaystyle\delta T_{\Phi\Phi}$ $\displaystyle=$ $\displaystyle\delta
T_{\Theta\Theta}\sin^{2}\Theta-\sum_{\ell m}\Big{\\{}2\eta
r_{0}^{2}\left[\partial_{\Phi}\left(\partial_{\Phi}V^{\ell m}-2\cos\Theta
U^{\ell m}\right)+\sin\Theta\left(2\partial_{\Phi}U^{\ell m}+\cos\Theta
V^{\ell m}\right)\partial_{\Theta}-\sin^{2}\Theta V^{\ell
m}\partial_{\Theta}^{2}\right]+$ (78) $\displaystyle\frac{J\sin^{2}\Theta
f}{4\pi
r_{0}^{2}f^{3/2}\left(3f-1\right)}\Big{[}\left(8\pi\eta\partial_{\Phi}\xi_{r}^{\ell
m}-3ih_{0}^{\ell m}\sin\Theta\partial_{\Theta}\left(16\pi\eta
r_{0}\omega-3i\right)\right)+9f^{2}h_{0}^{\ell
m}\sin\Theta\partial_{\Theta}-3ir_{0}^{2}f^{3/2}$
$\displaystyle\left(16\pi\eta r_{0}\omega-3i\right)\left(\sin\Theta U^{\ell
m}\partial_{\Theta}+\partial_{\Phi}V^{\ell
m}\right)+2r_{0}^{2}\sqrt{f}\left(1+16i\pi\eta
r_{0}\omega\right)\left(\sin\Theta U^{\ell
m}\partial_{\Theta}+\partial_{\Phi}V^{\ell m}\right)+9r_{0}^{2}f^{5/2}\times$
$\displaystyle\left(\sin\Theta U^{\ell
m}\partial_{\Theta}+\partial_{\Phi}V^{\ell m}\right)+2h_{0}^{\ell
m}\sin\Theta\partial_{\Theta}\left(1+16i\pi\eta
r_{0}\omega\right)-8\pi\eta\partial_{\Phi}\left(\xi_{r}^{\ell
m}-r_{0}H_{0}^{\ell m}\right)\Big{]}\Big{\\}}Y_{\ell m}(\Theta,\Phi).$

Now, using Eq. (63), we can write the constraints on the perturbations to
these quantities as

$\displaystyle\delta K_{ab}-h_{ab}\delta K-K\delta h_{ab}=-8\pi\delta T_{ab},$
(79)

which is a total of six equations that fix the perturbations to the membrane
as well as the boundary condition. We project the equations onto the spherical
harmonics $Y_{\ell m}(\Theta,\Phi)$ and extract the relations for a given
($\ell,m$) mode (and its interaction with the neighbouring modes) Kojima
(1992); Pani (2013) yielding to

$\displaystyle(ab)=(T\Phi)\rightarrow~{}$
$\displaystyle\ell(\ell+1)\beta^{\ell
m}_{0}+imJ[(\ell-1)(\ell+2)\chi_{0}^{\ell m}+\tilde{\alpha}^{\ell
m}_{0}+\eta_{0}^{\ell m}]-JQ^{\ell
m}(\ell+1)[(\ell-2)(\ell-1)\xi_{0}^{\ell-1,m}$
$\displaystyle-(\ell-1)\tilde{\beta}_{0}^{\ell-1,m}+\zeta_{0}^{\ell-1,m}]+JQ^{\ell+1,m}\ell[(\ell+2)(\ell+3)\xi_{0}^{\ell+1,m}+(\ell+2)\tilde{\beta}^{\ell+1,m}_{0}+\zeta_{0}^{\ell+1,m}]=0,$
(80) $\displaystyle(\Theta\Theta)-(\Phi\Phi)\rightarrow~{}$
$\displaystyle\ell(\ell+1)K_{00}^{\ell m}+imJK_{gg}^{\ell m}-JQ^{\ell
m}(\ell+1)K_{ff}^{\ell-1,m}+JQ^{\ell+1,m}lK_{ff}^{\ell+1,m}=0,$ (81)

for the predominantly axial sector, and similarly

$\displaystyle(ab)=(TT)\rightarrow$ $\displaystyle~{}A^{\ell
m}_{00}+imJC^{\ell m}_{0}+JQ^{\ell
m}B_{0}^{\ell-1,m}-JQ^{\ell+1,m}B_{0}^{\ell+1,m}=0,$ (82)
$\displaystyle(\Theta\Theta)+(\Phi\Phi)\rightarrow$ $\displaystyle~{}A^{\ell
m}_{33}+imJC^{\ell m}_{3}+JQ^{\ell
m}B_{3}^{\ell-1,m}-JQ^{\ell+1,m}B_{3}^{\ell+1,m}=0,$ (83)
$\displaystyle(T\Theta)\rightarrow$ $\displaystyle~{}\ell(\ell+1)\alpha^{\ell
m}_{0}+imJ[(\ell-1)(\ell+2)\xi_{0}^{\ell m}-\tilde{\beta}^{\ell
m}_{0}-\zeta_{0}^{\ell m}]-JQ^{\ell
m}(\ell+1)[(\ell-2)(\ell-1)\chi_{0}^{\ell-1,m}$
$\displaystyle+(\ell-1)\tilde{\alpha}_{0}^{\ell-1,m}-\eta_{0}^{\ell-1,m}]-JQ^{\ell+1,m}\ell[(\ell+2)(\ell+3)\chi_{0}^{\ell+1,m}-(\ell+2)\tilde{\alpha}^{\ell+1,m}_{0}-\eta_{0}^{\ell+1,m}]=0,$
(84) $\displaystyle(\Theta\Phi)\rightarrow$
$\displaystyle~{}\ell(\ell+1)K_{s}^{\ell m}+imJK_{f}^{\ell m}-JQ^{\ell
m}(\ell+1)K_{g}^{\ell-1,m}+JQ^{\ell+1,m}lK_{g}^{\ell+1}=0,$ (85)

for predominantly polar sector. Here, $Q^{\ell
m}=\sqrt{(\ell^{2}-m^{2})/(4l^{2}-1)}$ and the other variables are functions
of the individual modes of axial and polar perturbations which help us to
organize the equations in a similar manner as in Ref. Kojima (1992); Pani
(2013). Note that among the terms proportional to $J$, each equation contains
functions that are proportional to $\ell$ and the neighbouring modes $\ell-1$,
$\ell+1$. Among these, the latter ones lead to a mixing of axial and polar
modes, as parity conservation prevents any coupling between axial and polar
modes with the same angular number. For example in Eq. (C), $\beta_{0}^{\ell
m}$, $\chi_{0}^{\ell m}$, $\tilde{\alpha}_{0}^{\ell m}$ and $\eta_{0}^{\ell
m}$ depend only on the axial perturbation functions, i.e., $h_{0/1/2}^{\ell
m}(r_{0})$ and $U^{\ell m}$; whereas $\xi_{0}^{\ell m}$,
$\tilde{\beta}_{0}^{\ell m}$ and $\zeta_{0}^{\ell m}$, which contribute as
neighbouring modes, depend only on the polar perturbation functions, i.e.,
$\xi_{r}^{\ell m}$, $V_{0}^{\ell m}$, $\delta p^{\ell m}$, $\delta\rho^{\ell
m}$, $H_{0/1/2}^{\ell m}(r_{0})$ and $K^{\ell m}(r_{0})$.

To leading order in spin, we can drop the mixing between axial and polar modes
without affecting the derived QNM frequencies. This is true both in the
equations of motions and in the junction conditions. To see this, we first
note that the equations are invariant under

$\displaystyle a^{\ell m}\rightarrow(-1)^{\ell+1}a^{\ell,-m},\quad p^{\ell
m}\rightarrow(-1)^{\ell}p^{\ell,-m},\quad J\rightarrow-J,\quad
m\rightarrow-m,$ (86)

where $a^{\ell m}$ collectively refers to all the linear functions of axial
perturbations in the above equations (e.g., $\beta_{0}^{\ell m}$) and $p^{\ell
m}$ collectively refers to all the linear functions of polar perturbations in
the above equations (e.g., $A_{00}^{\ell m}$). At the first order in spin, the
QNM frequencies can be expanded as
$\omega=\omega_{0}+m\chi\omega_{1}+\mathcal{O}(\chi^{2})$, where
$J=M^{2}\chi$. This implies that the linear-in-spin terms in the equations of
motion that do not have an extra factor of $m$ cannot affect $\omega_{1}$, and
may be dropped for their computation. This is also true for the perturbation
equations, as shown in Ref. Pani (2013). Thus, we can decouple the axial and
polar sectors and simplify the junction conditions to

$\displaystyle(ab)=(T\Phi)\rightarrow~{}$
$\displaystyle\ell(\ell+1)\beta^{\ell
m}_{0}+imJ[(\ell-1)(\ell+2)\chi_{0}^{\ell m}+\tilde{\alpha}^{\ell
m}_{0}+\eta_{0}^{\ell m}]=0,$ (87)
$\displaystyle(\Theta\Theta)-(\Phi\Phi)\rightarrow~{}$
$\displaystyle\ell(\ell+1)K_{00}^{\ell m}+imJK_{gg}^{\ell m}=0,$ (88)

for axial modes and

$\displaystyle(ab)=(TT)\rightarrow$ $\displaystyle~{}A^{\ell
m}_{00}+imJC^{\ell m}_{0}=0,$ (89)
$\displaystyle(\Theta\Theta)+(\Phi\Phi)\rightarrow$ $\displaystyle~{}A^{\ell
m}_{33}+imJC^{\ell m}_{3}=0,$ (90) $\displaystyle(T\Theta)\rightarrow$
$\displaystyle~{}\ell(\ell+1)\alpha^{\ell
m}_{0}+imJ[(\ell-1)(\ell+2)\xi_{0}^{\ell m}-\tilde{\beta}^{\ell
m}_{0}-\zeta_{0}^{\ell m}]=0,$ (91) $\displaystyle(\Theta\Phi)\rightarrow$
$\displaystyle~{}\ell(\ell+1)K_{s}^{\ell m}+imJK_{f}^{\ell m}=0,$ (92)

for polar modes to the first order in spin.

###  C.1 Axial boundary condition

Using Eqs. (87) and (88), we can eliminate the axial matter perturbation
($V^{\ell m}$) between these two equations to get the boundary condition for
the Regge-Wheeler function $\psi_{\text{RW}}(r)$ to linear order in spin to be

$\displaystyle\frac{1}{\psi_{\rm{RW}}(r)}\frac{d\psi_{\rm{RW}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}$
$\displaystyle=\frac{3M^{2}y^{2}\nu
V_{\text{RW}}(r_{0})-6iw^{3}+2iw^{2}y}{6Mw\nu(3w-y)}$
$\displaystyle\frac{m\chi}{6\ell(\ell+1)My^{5}(y-3w)^{2}\nu}[w^{3}(36w^{2}y(i(\ell^{2}+\ell+9)y+16\nu+\nu
y^{2})+6wy^{2}((\ell^{2}+\ell-7)\nu y^{2}$
$\displaystyle-2i(2\ell^{2}+2\ell+13)y-42\nu)+y^{3}(-3(\ell^{2}+\ell-4)\nu
y^{2}+4i(\ell^{2}+\ell+6)y+36\nu)-216iw^{3}$
$\displaystyle(y-2i\nu))-3M^{2}wy^{4}V_{\text{RW}}(r_{0})(4\nu w-6iwy+\nu
y^{3}+2iy^{2}-2\nu y)],$ (93)

where $w=M\omega$, $y=\omega r_{0}$, $\nu=16\pi\eta$ and

$\displaystyle
V_{\rm{RW}}(r_{0})=f(r)\Big{(}\frac{\ell(\ell+1)}{r^{2}}-\frac{6M}{r^{3}}+\frac{24mJ(-7M+3r_{0})}{\ell(\ell+1)r_{0}^{6}\omega}\Big{)}.$
(94)

The above boundary condition is consistent with the earlier results in Ref.
Maggio  _et al._ (2020) in the non spinning case for $\chi=0$. In the BH
limit, i.e., $\eta\rightarrow(16\pi)^{-1}$ and $r_{0}\rightarrow 2M$, we have

$\displaystyle\frac{1}{\psi_{\rm{RW}}(r)}\frac{d\psi_{\rm{RW}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}=-i\Big{(}\omega-\frac{m\chi}{4M}\Big{)}=-i(\omega-m\Omega_{H}),$
(95)

as expected.

###  C.2 Polar boundary condition

The polar boundary condition requires an extra ingredient, which is the
equation of state of the membrane fluid. Since there are four equations in the
polar sector and four matter perturbations, i.e., $\delta\rho^{\ell m}$,
$\delta p^{\ell m}$, $V^{\ell m}$ and $\xi_{r}^{\ell m}$, we do not have any
remaining constraint upon eliminating the matter variables to serve as the
boundary condition. Thus, we additionally impose that the perturbations to the
pressure and the density are related via the equilibrium sound-speed Kojima
(1992); Maggio  _et al._ (2020)

$\displaystyle c_{s}^{2}=\frac{\delta p^{\ell m}}{\delta\rho^{\ell
m}}=\frac{dp_{0}/dr}{d\rho^{0}/dr}=-\frac{3M^{2}-3Mr_{0}+r_{0}^{2}}{2(6M^{2}-5Mr_{0}+r_{0}^{2})}.$
(96)

With this extra constraint, we obtain the boundary condition to the linear
order in spin for the Zerilli function $\psi_{\text{Z}}(r)$, using the
relations in Appendix. B. We write it implicitly here as

$\displaystyle\frac{1}{\psi_{\text{Z}}(r)}\frac{d\psi_{\text{Z}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}=-16\pi\eta
i\omega+G(r_{0},\omega,\eta,\zeta)+m\chi H(r_{0},\omega,\eta,\zeta).\,,$ (97)

where $G(r_{0},\omega,\eta,\zeta)$ is given explicitly in Ref. Maggio  _et
al._ (2020). We provide the polar boundary condition in the supplemental
Mathematica document in git . Note that unlike the axial sector, there is
dependence on the bulk viscosity both in spinless case and at the linear order
in spin for the polar sector. In the BH limit, i.e.,
$\eta\rightarrow(16\pi)^{-1}$, $\zeta\rightarrow-(16\pi)^{-1}$ and
$r_{0}\rightarrow 2M$, we have

$\displaystyle\frac{1}{\psi_{\text{Z}}(r)}\frac{d\psi_{\text{Z}}(r)}{dr_{*}}\Big{|}_{r=r_{0}}=-i\Big{(}\omega-\frac{m\chi}{4M}\Big{)}=-i(\omega-m\Omega_{H}),$
(98)

as expected.

## References

  * Abbott  _et al._ (2019a) B. Abbott _et al._ (LIGO Scientific, Virgo), GWTC-1: A Gravitational-Wave Transient Catalog of Compact Binary Mergers Observed by LIGO and Virgo during the First and Second Observing Runs, Phys. Rev. X 9, 031040 (2019a), arXiv:1811.12907 [astro-ph.HE] .
  * Abbott  _et al._ (2021a) R. Abbott _et al._ (LIGO Scientific, Virgo), GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run, Phys. Rev. X 11, 021053 (2021a), arXiv:2010.14527 [gr-qc] .
  * Abbott  _et al._ (2024) R. Abbott _et al._ (LIGO Scientific, VIRGO), GWTC-2.1: Deep extended catalog of compact binary coalescences observed by LIGO and Virgo during the first half of the third observing run, Phys. Rev. D 109, 022001 (2024), arXiv:2108.01045 [gr-qc] .
  * Abbott  _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run, (2021b), arXiv:2111.03606 [gr-qc] .
  * Abbott  _et al._ (2016a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016a), arXiv:1602.03837 [gr-qc] .
  * Abbott  _et al._ (2017) B. P. Abbott _et al._ (LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc] .
  * Abbott  _et al._ (2020) R. Abbott _et al._ (LIGO Scientific, Virgo), GW190521: A Binary Black Hole Merger with a Total Mass of $150M_{\odot}$, Phys. Rev. Lett. 125, 101102 (2020), arXiv:2009.01075 [gr-qc] .
  * Li  _et al._ (2012) T. G. F. Li, W. Del Pozzo, S. Vitale, C. Van Den Broeck, M. Agathos, J. Veitch, K. Grover, T. Sidery, R. Sturani, and A. Vecchio, Towards a generic test of the strong field dynamics of general relativity using compact binary coalescence, Phys. Rev. D 85, 082003 (2012), arXiv:1110.0530 [gr-qc] .
  * Agathos  _et al._ (2014) M. Agathos, W. Del Pozzo, T. G. F. Li, C. Van Den Broeck, J. Veitch, and S. Vitale, TIGER: A data analysis pipeline for testing the strong-field dynamics of general relativity with gravitational wave signals from coalescing compact binaries, Phys. Rev. D 89, 082001 (2014), arXiv:1311.0420 [gr-qc] .
  * Ghosh  _et al._ (2016) A. Ghosh _et al._ , Testing general relativity using golden black-hole binaries, Phys. Rev. D 94, 021101 (2016), arXiv:1602.02453 [gr-qc] .
  * Ghosh  _et al._ (2018) A. Ghosh, N. K. Johnson-Mcdaniel, A. Ghosh, C. K. Mishra, P. Ajith, W. Del Pozzo, C. P. L. Berry, A. B. Nielsen, and L. London, Testing general relativity using gravitational wave signals from the inspiral, merger and ringdown of binary black holes, Class. Quant. Grav. 35, 014002 (2018), arXiv:1704.06784 [gr-qc] .
  * Brito  _et al._ (2018) R. Brito, A. Buonanno, and V. Raymond, Black-hole Spectroscopy by Making Full Use of Gravitational-Wave Modeling, Phys. Rev. D 98, 084038 (2018), arXiv:1805.00293 [gr-qc] .
  * Carullo  _et al._ (2019) G. Carullo, W. Del Pozzo, and J. Veitch, Observational Black Hole Spectroscopy: A time-domain multimode analysis of GW150914, Phys. Rev. D 99, 123029 (2019), [Erratum: Phys.Rev.D 100, 089903 (2019)], arXiv:1902.07527 [gr-qc] .
  * Ghosh  _et al._ (2021) A. Ghosh, R. Brito, and A. Buonanno, Constraints on quasinormal-mode frequencies with LIGO-Virgo binary–black-hole observations, Phys. Rev. D 103, 124041 (2021), arXiv:2104.01906 [gr-qc] .
  * Isi and Farr (2021) M. Isi and W. M. Farr, Analyzing black-hole ringdowns, (2021), arXiv:2107.05609 [gr-qc] .
  * Mehta  _et al._ (2023) A. K. Mehta, A. Buonanno, R. Cotesta, A. Ghosh, N. Sennett, and J. Steinhoff, Tests of general relativity with gravitational-wave observations using a flexible theory-independent method, Phys. Rev. D 107, 044020 (2023), arXiv:2203.13937 [gr-qc] .
  * Maggio  _et al._ (2023) E. Maggio, H. O. Silva, A. Buonanno, and A. Ghosh, Tests of general relativity in the nonlinear regime: A parametrized plunge-merger-ringdown gravitational waveform model, Phys. Rev. D 108, 024043 (2023), arXiv:2212.09655 [gr-qc] .
  * Gennari  _et al._ (2024) V. Gennari, G. Carullo, and W. Del Pozzo, Searching for ringdown higher modes with a numerical relativity-informed post-merger model, Eur. Phys. J. C 84, 233 (2024), arXiv:2312.12515 [gr-qc] .
  * Abbott  _et al._ (2016b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), Tests of general relativity with GW150914, Phys. Rev. Lett. 116, 221101 (2016b), [Erratum: Phys.Rev.Lett. 121, 129902 (2018)], arXiv:1602.03841 [gr-qc] .
  * Abbott  _et al._ (2019b) B. Abbott _et al._ (LIGO Scientific, Virgo), Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1, Phys. Rev. D 100, 104036 (2019b), arXiv:1903.04467 [gr-qc] .
  * Abbott  _et al._ (2021c) R. Abbott _et al._ (LIGO Scientific, Virgo), Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog, Phys. Rev. D 103, 122002 (2021c), arXiv:2010.14529 [gr-qc] .
  * Abbott  _et al._ (2021d) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), Tests of General Relativity with GWTC-3, (2021d), arXiv:2112.06861 [gr-qc] .
  * Hawking (1974) S. W. Hawking, Black hole explosions, Nature 248, 30 (1974).
  * Mathur (2009) S. D. Mathur, The Information paradox: A Pedagogical introduction,  _Strings, Supergravity and Gauge Theories. Proceedings, CERN Winter School, CERN, Geneva, Switzerland, February 9-13 2009_ , Class. Quant. Grav. 26, 224001 (2009), arXiv:0909.1038 [hep-th] .
  * Bekenstein (1973) J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  * Hawking (1976) S. W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D 13, 191 (1976).
  * Giudice  _et al._ (2016) G. F. Giudice, M. McCullough, and A. Urbano, Hunting for Dark Particles with Gravitational Waves, JCAP 1610 (10), 001, arXiv:1605.01209 [hep-ph] .
  * Cardoso and Pani (2017) V. Cardoso and P. Pani, Tests for the existence of black holes through gravitational wave echoes, Nat. Astron. 1, 586 (2017), arXiv:1709.01525 [gr-qc] .
  * Cardoso and Pani (2019) V. Cardoso and P. Pani, Testing the nature of dark compact objects: a status report, Living Rev. Rel. 22, 4 (2019), arXiv:1904.05363 [gr-qc] .
  * Maggio  _et al._ (2021a) E. Maggio, P. Pani, and G. Raposo, Testing the nature of dark compact objects with gravitational waves 10.1007/978-981-15-4702-7_29-1 (2021a), arXiv:2105.06410 [gr-qc] .
  * Mathur (2005) S. D. Mathur, The Fuzzball proposal for black holes: An Elementary review,  _The quantum structure of space-time and the geometric nature of fundamental interactions. Proceedings, 4th Meeting, RTN2004, Kolymbari, Crete, Greece, September 5-10, 2004_ , Fortsch. Phys. 53, 793 (2005), arXiv:hep-th/0502050 [hep-th] .
  * Bena and Warner (2008) I. Bena and N. P. Warner, Black holes, black rings and their microstates,  _Winter School on Attractor Mechanism (SAM 2006) Frascati, Italy, March 20-24, 2006_ , Lect. Notes Phys. 755, 1 (2008), arXiv:hep-th/0701216 [hep-th] .
  * Balasubramanian  _et al._ (2008) V. Balasubramanian, J. de Boer, S. El-Showk, and I. Messamah, Black Holes as Effective Geometries,  _Strings, supergravity and gauge theories. Proceedings, European RTN Winter School, CERN, Geneva, Switzerland, January 21-25, 2008_ , Class. Quant. Grav. 25, 214004 (2008), arXiv:0811.0263 [hep-th] .
  * Mazur and Mottola (2004) P. O. Mazur and E. Mottola, Gravitational vacuum condensate stars, Proc. Nat. Acad. Sci. 101, 9545 (2004), arXiv:gr-qc/0407075 [gr-qc] .
  * Buoninfante  _et al._ (2018) L. Buoninfante, A. S. Cornell, G. Harmsen, A. S. Koshelev, G. Lambiase, J. Marto, and A. Mazumdar, Towards nonsingular rotating compact object in ghost-free infinite derivative gravity, Phys. Rev. D 98, 084041 (2018), arXiv:1807.08896 [gr-qc] .
  * Buoninfante and Mazumdar (2019) L. Buoninfante and A. Mazumdar, Nonlocal star as a blackhole mimicker, Phys. Rev. D 100, 024031 (2019), arXiv:1903.01542 [gr-qc] .
  * Kaup (1968) D. J. Kaup, Klein-Gordon Geon, Phys. Rev. 172, 1331 (1968).
  * Ruffini and Bonazzola (1969) R. Ruffini and S. Bonazzola, Systems of selfgravitating particles in general relativity and the concept of an equation of state, Phys. Rev. 187, 1767 (1969).
  * Seidel and Suen (1991) E. Seidel and W. Suen, Oscillating soliton stars, Phys.Rev.Lett. 66, 1659 (1991).
  * Liebling and Palenzuela (2012) S. L. Liebling and C. Palenzuela, Dynamical Boson Stars, Living Rev. Rel. 15, 6 (2012), arXiv:1202.5809 [gr-qc] .
  * Brito  _et al._ (2016) R. Brito, V. Cardoso, C. A. R. Herdeiro, and E. Radu, Proca stars: Gravitating Bose-Einstein condensates of massive spin 1 particles, Phys. Lett. B752, 291 (2016), arXiv:1508.05395 [gr-qc] .
  * Morris and Thorne (1988) M. S. Morris and K. S. Thorne, Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56, 395 (1988).
  * Visser (1995) M. Visser,  _Lorentzian wormholes: From Einstein to Hawking_ (1995).
  * Damour and Solodukhin (2007) T. Damour and S. N. Solodukhin, Wormholes as black hole foils, Phys. Rev. D76, 024016 (2007), arXiv:0704.2667 [gr-qc] .
  * Datta  _et al._ (2020) S. Datta, R. Brito, S. Bose, P. Pani, and S. A. Hughes, Tidal heating as a discriminator for horizons in extreme mass ratio inspirals, Phys. Rev. D 101, 044004 (2020), arXiv:1910.07841 [gr-qc] .
  * Mukherjee  _et al._ (2022) S. Mukherjee, S. Datta, S. Tiwari, K. S. Phukon, and S. Bose, Toward establishing the presence or absence of horizons in coalescing binaries of compact objects by using their gravitational wave signals, Phys. Rev. D 106, 104032 (2022), arXiv:2202.08661 [gr-qc] .
  * Saketh  _et al._ (2023) M. V. S. Saketh, J. Steinhoff, J. Vines, and A. Buonanno, Modeling horizon absorption in spinning binary black holes using effective worldline theory, Phys. Rev. D 107, 084006 (2023), arXiv:2212.13095 [gr-qc] .
  * Cardoso  _et al._ (2019) V. Cardoso, A. del Rio, and M. Kimura, Distinguishing black holes from horizonless objects through the excitation of resonances during inspiral, Phys. Rev. D 100, 084046 (2019), [Erratum: Phys.Rev.D 101, 069902 (2020)], arXiv:1907.01561 [gr-qc] .
  * Asali  _et al._ (2020) Y. Asali, P. T. H. Pang, A. Samajdar, and C. Van Den Broeck, Probing resonant excitations in exotic compact objects via gravitational waves, Phys. Rev. D 102, 024016 (2020), arXiv:2004.05128 [gr-qc] .
  * Maggio  _et al._ (2021b) E. Maggio, M. van de Meent, and P. Pani, Extreme mass-ratio inspirals around a spinning horizonless compact object, Phys. Rev. D 104, 104026 (2021b), arXiv:2106.07195 [gr-qc] .
  * Pani (2015) P. Pani, I-Love-Q relations for gravastars and the approach to the black-hole limit, Phys. Rev. D92, 124030 (2015), arXiv:1506.06050 [gr-qc] .
  * Uchikata  _et al._ (2016) N. Uchikata, S. Yoshida, and P. Pani, Tidal deformability and I-Love-Q relations for gravastars with polytropic thin shells, Phys. Rev. D94, 064015 (2016), arXiv:1607.03593 [gr-qc] .
  * Cardoso  _et al._ (2017) V. Cardoso, E. Franzin, A. Maselli, P. Pani, and G. Raposo, Testing strong-field gravity with tidal Love numbers, Phys. Rev. D95, 084014 (2017), arXiv:1701.01116 [gr-qc] .
  * Sennett  _et al._ (2017) N. Sennett, T. Hinderer, J. Steinhoff, A. Buonanno, and S. Ossokine, Distinguishing Boson Stars from Black Holes and Neutron Stars from Tidal Interactions in Inspiraling Binary Systems, Phys. Rev. D96, 024002 (2017), arXiv:1704.08651 [gr-qc] .
  * Raposo  _et al._ (2019) G. Raposo, P. Pani, M. Bezares, C. Palenzuela, and V. Cardoso, Anisotropic stars as ultracompact objects in General Relativity, Phys. Rev. D99, 104072 (2019), arXiv:1811.07917 [gr-qc] .
  * Nair  _et al._ (2023) S. Nair, S. Chakraborty, and S. Sarkar, Dynamical Love numbers for area quantized black holes, Phys. Rev. D 107, 124041 (2023), arXiv:2208.06235 [gr-qc] .
  * Chakraborty  _et al._ (2023) S. Chakraborty, E. Maggio, M. Silvestrini, and P. Pani, Dynamical tidal Love numbers of Kerr-like compact objects, (2023), arXiv:2310.06023 [gr-qc] .
  * Cardoso  _et al._ (2016a) V. Cardoso, E. Franzin, and P. Pani, Is the gravitational-wave ringdown a probe of the event horizon?, Phys. Rev. Lett. 116, 171101 (2016a), arXiv:1602.07309 [gr-qc] .
  * Cardoso  _et al._ (2016b) V. Cardoso, S. Hopper, C. F. B. Macedo, C. Palenzuela, and P. Pani, Gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale, Phys. Rev. D94, 084031 (2016b), arXiv:1608.08637 [gr-qc] .
  * Abedi  _et al._ (2017) J. Abedi, H. Dykaar, and N. Afshordi, Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons, Phys. Rev. D96, 082004 (2017), arXiv:1612.00266 [gr-qc] .
  * Wang  _et al._ (2020) Q. Wang, N. Oshita, and N. Afshordi, Echoes from Quantum Black Holes, Phys. Rev. D 101, 024031 (2020), arXiv:1905.00446 [gr-qc] .
  * Mark  _et al._ (2017) Z. Mark, A. Zimmerman, S. M. Du, and Y. Chen, A recipe for echoes from exotic compact objects, Phys. Rev. D96, 084002 (2017), arXiv:1706.06155 [gr-qc] .
  * Testa and Pani (2018) A. Testa and P. Pani, Analytical template for gravitational-wave echoes: signal characterization and prospects of detection with current and future interferometers, Phys. Rev. D98, 044018 (2018), arXiv:1806.04253 [gr-qc] .
  * Maggio  _et al._ (2019a) E. Maggio, A. Testa, S. Bhagwat, and P. Pani, Analytical model for gravitational-wave echoes from spinning remnants, Phys. Rev. D 100, 064056 (2019a), arXiv:1907.03091 [gr-qc] .
  * Xin  _et al._ (2021) S. Xin, B. Chen, R. K. L. Lo, L. Sun, W.-B. Han, X. Zhong, M. Srivastava, S. Ma, Q. Wang, and Y. Chen, Gravitational-wave echoes from spinning exotic compact objects: Numerical waveforms from the Teukolsky equation, Phys. Rev. D 104, 104005 (2021), arXiv:2105.12313 [gr-qc] .
  * Ma  _et al._ (2022) S. Ma, Q. Wang, N. Deppe, F. Hébert, L. E. Kidder, J. Moxon, W. Throwe, N. L. Vu, M. A. Scheel, and Y. Chen, Gravitational-wave echoes from numerical-relativity waveforms via spacetime construction near merging compact objects, Phys. Rev. D 105, 104007 (2022), arXiv:2203.03174 [gr-qc] .
  * Vellucci  _et al._ (2023) V. Vellucci, E. Franzin, and S. Liberati, Echoes from backreacting exotic compact objects, Phys. Rev. D 107, 044027 (2023), arXiv:2205.14170 [gr-qc] .
  * Siemonsen (2024) N. Siemonsen, Nonlinear treatment of a black hole mimicker ringdown, (2024), arXiv:2404.14536 [gr-qc] .
  * Maggio  _et al._ (2020) E. Maggio, L. Buoninfante, A. Mazumdar, and P. Pani, How does a dark compact object ringdown?, Phys. Rev. D 102, 064053 (2020), arXiv:2006.14628 [gr-qc] .
  * Branchesi  _et al._ (2023) M. Branchesi _et al._ , Science with the Einstein Telescope: a comparison of different designs, JCAP 07, 068, arXiv:2303.15923 [gr-qc] .
  * Arun  _et al._ (2022) K. G. Arun _et al._ (LISA), New horizons for fundamental physics with LISA, Living Rev. Rel. 25, 4 (2022), arXiv:2205.01597 [gr-qc] .
  * Maselli  _et al._ (2018) A. Maselli, P. Pani, V. Cardoso, T. Abdelsalhin, L. Gualtieri, and V. Ferrari, Probing Planckian corrections at the horizon scale with LISA binaries, Phys. Rev. Lett. 120, 081101 (2018), arXiv:1703.10612 [gr-qc] .
  * Piovano  _et al._ (2023) G. A. Piovano, A. Maselli, and P. Pani, Constraining the tidal deformability of supermassive objects with extreme mass ratio inspirals and semianalytical frequency-domain waveforms, Phys. Rev. D 107, 024021 (2023), arXiv:2207.07452 [gr-qc] .
  * Barsanti  _et al._ (2023) S. Barsanti, A. Maselli, T. P. Sotiriou, and L. Gualtieri, Detecting Massive Scalar Fields with Extreme Mass-Ratio Inspirals, Phys. Rev. Lett. 131, 051401 (2023), arXiv:2212.03888 [gr-qc] .
  * Damour (1982) T. Damour, Surface Effects in Black-Hole Physics, in  _Marcel Grossmann Meeting: General Relativity_ (1982) p. 587.
  * Thorne  _et al._ (1986) K. S. Thorne, R. Price, and D. Macdonald, eds.,  _BLACK HOLES: THE MEMBRANE PARADIGM_ (1986).
  * Price and Thorne (1986) R. Price and K. Thorne, Membrane Viewpoint on Black Holes: Properties and Evolution of the Stretched Horizon, Phys. Rev. D 33, 915 (1986).
  * Darmois (1927) G. Darmois, Les équations de la gravitation einsteinienne, Mémorial de Sciences Mathématiques fascicule 25, 1 (1927).
  * Israel (1966) W. Israel, Singular hypersurfaces and thin shells in general relativity, Nuovo Cim. B44S10, 1 (1966), [Nuovo Cim.B44,1(1966)].
  * Visser (1996) M. Visser,  _Lorentzian wormholes: From Einstein to Hawking_ (AIP, Woodbury, USA, 1996) p. 412 p.
  * Abedi  _et al._ (2020) J. Abedi, N. Afshordi, N. Oshita, and Q. Wang, Quantum Black Holes in the Sky, Universe 6, 43 (2020), arXiv:2001.09553 [gr-qc] .
  * Berti  _et al._ (2015) E. Berti _et al._ , Testing General Relativity with Present and Future Astrophysical Observations, Class. Quant. Grav. 32, 243001 (2015), arXiv:1501.07274 [gr-qc] .
  * Kojima (1992) Y. Kojima, Equations governing the nonradial oscillations of a slowly rotating relativistic star, Phys. Rev. D 46, 4289 (1992).
  * Vines and Steinhoff (2018) J. Vines and J. Steinhoff, Spin-multipole effects in binary black holes and the test-body limit, Phys. Rev. D 97, 064010 (2018), arXiv:1606.08832 [gr-qc] .
  * Saketh and Vines (2022) M. V. S. Saketh and J. Vines, Scattering of gravitational waves off spinning compact objects with an effective worldline theory, Phys. Rev. D 106, 124026 (2022), arXiv:2208.03170 [gr-qc] .
  * Regge and Wheeler (1957) T. Regge and J. A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063 (1957).
  * Zerilli (1970) F. J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24, 737 (1970).
  * Pani (2013) P. Pani, Advanced Methods in Black-Hole Perturbation Theory, Int. J. Mod. Phys. A 28, 1340018 (2013), arXiv:1305.6759 [gr-qc] .
  * (89) https://github.com/elisamaggio/ThorGW.
  * Starobinsky (1973) A. A. Starobinsky, Amplification of waves reflected from a rotating ”black hole”., Sov. Phys. JETP 37, 28 (1973).
  * Starobinskij and Churilov (1973) A. A. Starobinskij and S. M. Churilov, Amplification of electromagnetic and gravitational waves scattered by a rotating black hole., Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki 65, 3 (1973).
  * Creci  _et al._ (2021) G. Creci, T. Hinderer, and J. Steinhoff, Tidal response from scattering and the role of analytic continuation, Phys. Rev. D 104, 124061 (2021).
  * Dolan (2008) S. R. Dolan, Scattering and Absorption of Gravitational Plane Waves by Rotating Black Holes, Class. Quant. Grav. 25, 235002 (2008), arXiv:0801.3805 [gr-qc] .
  * Saketh  _et al._ (2024) M. V. S. Saketh, Z. Zhou, and M. M. Ivanov, Dynamical tidal response of Kerr black holes from scattering amplitudes, Phys. Rev. D 109, 064058 (2024), arXiv:2307.10391 [hep-th] .
  * Penrose and Floyd (1971) R. Penrose and R. M. Floyd, Extraction of rotational energy from a black hole, Nature 229, 177 (1971).
  * Zel’Dovich (1971) Y. B. Zel’Dovich, Generation of Waves by a Rotating Body, Soviet Journal of Experimental and Theoretical Physics Letters 14, 180 (1971).
  * Blandford and Znajek (1977) R. D. Blandford and R. L. Znajek, Electromagnetic extractions of energy from Kerr black holes, Mon. Not. Roy. Astron. Soc. 179, 433 (1977).
  * Brito  _et al._ (2015) R. Brito, V. Cardoso, and P. Pani, Superradiance, Lect. Notes Phys. 906, pp.1 (2015), arXiv:1501.06570 [gr-qc] .
  * Pani  _et al._ (2012) P. Pani, V. Cardoso, L. Gualtieri, E. Berti, and A. Ishibashi, Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric, Phys. Rev. D 86, 104017 (2012), arXiv:1209.0773 [gr-qc] .
  * Maggio  _et al._ (2019b) E. Maggio, V. Cardoso, S. R. Dolan, and P. Pani, Ergoregion instability of exotic compact objects: electromagnetic and gravitational perturbations and the role of absorption, Phys. Rev. D99, 064007 (2019b), arXiv:1807.08840 [gr-qc] .
  * Leaver (1985) E. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A A402, 285 (1985).
  * Pani  _et al._ (2009) P. Pani, E. Berti, V. Cardoso, Y. Chen, and R. Norte, Gravitational wave signatures of the absence of an event horizon. I. Nonradial oscillations of a thin-shell gravastar, Phys. Rev. D80, 124047 (2009), arXiv:0909.0287 [gr-qc] .
  * Pierini and Gualtieri (2022) L. Pierini and L. Gualtieri, Quasinormal modes of rotating black holes in Einstein-dilaton Gauss-Bonnet gravity: The second order in rotation, Phys. Rev. D 106, 104009 (2022), arXiv:2207.11267 [gr-qc] .
  * Zhong  _et al._ (2023) Z. Zhong, V. Cardoso, and E. Maggio, Instability of ultracompact horizonless spacetimes, Phys. Rev. D 107, 044035 (2023), arXiv:2211.16526 [gr-qc] .