# Dynamical tidal Love numbers of Kerr-like compact objects

Sumanta Chakraborty  0000-0003-3343-3227<EMAIL_ADDRESS>School of Physical
Sciences, Indian Association for the Cultivation of Science, Kolkata-700032,
India  Elisa Maggio  0000-0002-1960-8185<EMAIL_ADDRESS>Max Planck
Institute for Gravitational Physics (Albert Einstein Institute), D-14476
Potsdam, Germany  Michela Silvestrini
<EMAIL_ADDRESS>Dipartimento di Fisica, “Sapienza”
Università di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma, Italy
Paolo Pani  0000-0003-4443-1761<EMAIL_ADDRESS>Dipartimento di Fisica,
“Sapienza” Università di Roma & Sezione INFN Roma1, P.A. Moro 5, 00185, Roma,
Italy

###### Abstract

We develop a framework to compute the tidal response of a Kerr-like compact
object in terms of its reflectivity, compactness, and spin, both in the static
and the frequency-dependent case. Here we focus on the low-frequency regime,
which can be solved fully analytically. We highlight some remarkable novel
features, in particular: i) Even in the zero-frequency limit, the tidal Love
numbers (TLNs) depend on the linear-in-frequency dependence of the object’s
reflectivity in a nontrivial way. ii) Intriguingly, the static limit of the
frequency-dependent TLNs is discontinuous, therefore the static TLNs differ
from the static limit of the (phenomenologically more interesting) frequency-
dependent TLNs. This shows that earlier findings regarding the static TLNs of
ultracompact objects correspond to a measure-zero region in the parameter
space, though the logarithmic behavior of the TLNs in the black hole limit is
retained. iii) In the non-rotating case, the TLNs _generically vanish_ in the
zero-frequency limit (just like for a black hole), except when the
reflectivity is ${\cal R}=1+\,{\cal O}(M\omega)$, in which case they vanish
with a model-dependent scaling, which is generically logarithmic, in the
black-hole limit. The TLNs initially grow with frequency, for any nonzero
reflectivity, and then display oscillations and resonances tied up with the
quasi-normal modes of the object. iv) For rotating compact objects, the TLNs
decrease when the reflectivity decreases or the rotation parameter increases.
Our results lay the theoretical groundwork to develop model-independent tests
of the nature of compact objects using tidal effects in gravitational-wave
signals.

##  I Introduction

According to the theory of general relativity (GR), a black hole (BH) has
vanishing tidal susceptibility. Namely, when immersed in a quasi-stationary
tidal field (for example the one produced by a binary companion at a large
distance) its induced multipole moments are all zero. This special property is
quantified by the vanishing of the static tidal Love numbers (TLNs) [1, 2, 3,
4, 5, 6, 7, 8, 9, 10] and can be related to some “hidden” symmetries of the
Kerr solution in GR [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. It is a quite
unique and fragile feature [21, 22]: it does not hold true for any other
object, even within GR [22, 23, 24, 25, 22, 26, 27, 28], for BHs in modified
gravity [22, 29], or in higher dimensions [30, 31, 32, 33, 34, 35, 36]. Its
violation can therefore be seen as a smoking gun for deviations from the
standard paradigm involving BHs in GR [22, 28], especially for supermassive
compact objects [37, 38, 39, 40] which, in the standard paradigm, can only be
BHs. So far the TLNs of compact objects other than Kerr BHs have been computed
on a case-by-case basis, including boson stars [22, 23, 24], gravastars [25,
22, 26], anisotropic stars [27], and other simple exotic compact objects
(ECOs) [41, 28] with stiff equation of state at the surface [22]. Furthermore,
they have been mostly computed for nonspinning objects and for static tidal
fields, with the notable exceptions of Refs. [9, 42, 43, 44, 45].

In the context of gravitational-wave tests of the nature of compact objects
based on their tidal response (see [28, 46] for some reviews), it would be
useful to have a framework to compute the TLNs from the general properties of
the object and not restricted to some specific underlying model. The scope of
this work is to provide such a general framework for the TLNs of a Kerr-like
compact object, by carefully taking into account the reflectivity of the
object, the role of the angular momentum, and the possible time dependence of
the tidal field. We exploit a better understanding of the role of the
reflectivity of ECOs, recently investigated in the context of their
quasinormal modes (QNMs) in the ringdown [47, 48]. While our framework is
valid for any frequency, here we focus on the low-frequency regime which is
amenable to analytical treatment using matching asymptotics.

In II we present our setup, which is based on considering a Kerr metric with
generic boundary conditions at a finite distance from the would-be horizon and
assuming GR (at least) in the exterior of the compact object. We use the
formalism developed in Ref. [9] (see also [43, 49]) to derive a master
equation for the TLNs, paying particular attention to the object’s
reflectivity, ${\cal R}$. We emphasize that the reflectivity of any compact
object is more naturally defined with respect to plane wave modes near the
horizon. It is therefore convenient to define the reflectivity in terms of a
master function describing linear perturbations that are wave-like near the
horizon. For this reason, we use the Detweiler function rather than the more
standard Teukolsky function, following closely Ref. [48].

Another important aspect of our analysis is the static limit, particularly for
the non-rotating case. As we will demonstrate, quite counter-intuitively, the
$\omega\rightarrow 0$ limit of the frequency-dependent TLNs for a non-rotating
compact object with $\mathcal{R}\neq 0$ is discontinuous111A similar
discontinuity was discussed in [50] in the context of the magnetic TLNs of
fluid stars. and does not coincide with the strictly static TLNs derived in
[22]. This happens because the solutions of the Teukolsky equation for an ECO,
admitting outgoing behavior near the horizon, with $\omega\neq 0$, do not tend
to the respective solutions in the zero-frequency limit222Note that in the BH
limit only the in-going solution to the Teukolsky equation matters, which has
a continuous zero-frequency limit. Thus, for BHs such a discontinuity does not
arise.. Hence the strictly static TLNs of a non-rotating compact object must
be determined in a separate manner and do not follow from a continuous
$\omega\rightarrow 0$ limit. Even more interesting is the result that the
zero-frequency limit of the frequency-dependent TLNs depends on the small-
frequency limit of the reflectivity, which we model as

${\cal R}(\omega)={\cal R}_{0}+iM\omega{\cal R}_{1}+{\cal
O}(M^{2}\omega^{2})\,,$ (1)

where $\omega$ is the frequency of the tidal perturbation, $M$ is the object’s
mass, and the (complex) coefficients ${\cal R}_{i}$ generically depend on the
object’s properties. One of our main findings is that, due to some
cancellations, the coefficient ${\cal R}_{1}$ of the ${\cal O}(M\omega)$ term
in the above equation enters the TLNs in the _static_ limit ($\omega\to 0$).
We, therefore, argue that modeling the tidal deformability of a compact object
requires knowledge of the frequency dependence of the reflectivity, even when
restricting to the _static_ TLNs. Moreover, we find that, in the static limit
and for nonrotating objects, the _TLNs vanish identically_ (as in the BH
case), except when ${\cal R}_{0}=1$, in which case the corresponding nonzero
value of the TLNs depends on ${\cal R}_{1}$ and exhibits a logarithmic
behavior as a function of the compactness parameter as the BH limit is
approached. A logarithmic dependence was also found in Ref. [22] by strictly
static perturbations of perfectly reflecting, non-spinning objects. However,
as we shall discuss, beside the logarithmic dependence the general expression
of the TLN in the zero-frequency limit differs from the one found in Ref. [22]
in the strictly static case. This show that a strictly static limit cannot be
obtained in a continuous manner from the frequency-dependent TLNs, as the
solutions of the Teukolsky equation with $\omega\neq 0$ do not have a smooth
limit to the solutions of the Teukolsky equation with $\omega=0$. Since the
process of binary coalescence is always dynamic, it seems more reasonable to
determine the ”dynamical TLNs” and then take the $\omega\to 0$ in order to
arrive at the static TLNs (see [50] for a related discussion). Intriguingly,
the logarithmic behavior as a function of the compactness parameter for the
TLNs of compact objects is a generic result, which holds in the strictly
static case as well as in the dynamical contexts, but in the dynamical case it
depends on the choice of $\mathcal{R}_{1}$ as well. In particular, it is
possible to choose $\mathcal{R}_{1}$, such that the TLNs do not have a
logarithmic dependence.

In III we present a collection of results for the TLNs of a compact object in
terms of its reflectivity, compactness, and spin, both in the static and in
the frequency-dependent case. For the case of a non-rotating compact object,
we discuss the static limit in two different manners — (a) substituting
$\omega=0$ in the Teukolsky equation and then determining the (strictly
static) TLNs from its solution, which coincides with the results of [22]; (b)
solving the dynamical case and thus determining the dynamical TLNs, and then
taking $\omega\rightarrow 0$ limit. As mentioned above, the static limit of
the TLNs associated with a non-rotating compact object depends on
$\mathcal{R}_{0}$ and $\mathcal{R}_{1}$, and the TLNs are either zero or
vanish in the BH limit with a scaling that depends on the compactness and on
the model-dependent choice of ${\cal R}_{1}$. Though the non-zero TLNs
generically depend on the compactness in a logarithmic fashion, only when
setting $\omega=0$ from the beginning, do we recover the results of the
previous work [22].

Finally, in the more general spinning and frequency-dependent case we
determine the TLNs in the small frequency limit. We present the dependence of
the TLNs of rotating compact objects on the model parameters, showing that the
induced multipole moments decrease as the rotation increases or as the
reflectivity decreases. Interestingly, the dynamical TLNs in the non-rotating
case depict certain oscillatory and resonant behaviors which are seemingly
characteristics of the low-frequency QNMs [51, 47, 48] associated with the
compact object.

We conclude in IV, with a discussion of our main findings and possible
extensions. Several detailed calculations have been presented in the
appendices for the benefit of the reader. Throughout this work, we use
geometrized $G=1=c$ units, unless otherwise mentioned. We also work with the
signature convention such that flat Minkowski metric in the Cartesian
coordinate system reads $\eta_{\mu\nu}=\textrm{diag.}(-1,+1,+1,+1)$.

##  II Setup

###  II.1 Exterior geometry of Kerr-like compact objects

In this work, we analyze the tidal effects on a spinning and horizonless
compact object whose exterior spacetime is described by the Kerr metric. This
is an approximation since, beyond spherical symmetry, the external spacetime
of a compact object can have arbitrary multipole moments, even in GR. At the
same time, regularity requires that any multipolar deviation from the Kerr
metric should vanish in the BH limit at least within GR [52].

It is useful to express the Kerr metric in terms of the ingoing null
coordinates $(v,r,\theta,\tilde{\phi})$ (their relation to Boyer-Lindquist
coordinates $(t,r,\theta,\phi)$ is given in A) such that the line element of
the exterior Kerr spacetime reads

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\left(\Delta-a^{2}\sin^{2}\theta\right)}{\Sigma}dv^{2}+2dvdr$
$\displaystyle-$ $\displaystyle 2a\sin^{2}\theta
drd\tilde{\phi}-\frac{4Mar\sin^{2}\theta}{\Sigma}dvd\tilde{\phi}$
$\displaystyle+$
$\displaystyle{\Sigma}d\theta^{2}+\frac{\left[(r^{2}+a^{2})^{2}-\Delta
a^{2}\sin^{2}\theta\right]}{\Sigma}\sin^{2}\theta d\tilde{\phi}^{2}\,,$

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\Delta=(r-r_{+})(r-r_{-})$, and
$r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ are the locations of the horizons. The
parameters $M$ and $J\equiv aM$ in the line element are the total mass and the
angular momentum of the compact object, respectively.

The radius $r_{0}$ of the compact object is assumed to be only slightly larger
than the event horizon, located at $r=r_{+}$, and smaller than the photon
sphere. Such a scenario can be motivated by models of microscopic corrections
at the horizon scale (see [28] for an overview). For the purpose of this work,
we consider the radius of the compact object to be

$r_{0}=r_{+}(1+\epsilon)\,,$ (3)

where $0<\epsilon\ll 1$. The properties of the object’s interior are
parametrized by a complex and frequency-dependent reflectivity
$\mathcal{R}(\omega)$ at the surface333Note that, within our formalism,
$r_{0}$ can be any effective radius in the exterior geometry, as long as the
reflectivity $\mathcal{R}(\omega)$ refers to the surface at $r=r_{0}$. For
example, if the object deviates from Kerr only near the would-horizon, one
could take $r_{0}$ to be sufficiently large so that the hypothesis of a Kerr
background for $r>r_{0}$ is satisfied. of the compact object, as will be
discussed in detail in Section II.3.

###  II.2 Tidal Love numbers (TLNs)

Here, we wish to provide the basic framework for computing the deformation of
a Kerr-like compact object when placed in an external tidal field. In
particular, we wish to study the TLNs, which encapsulate the linear response
of the compact object due to a perturbing tidal field. We first present the
Newtonian description of the tidal response and then generalize it to the
relativistic case.

####  II.2.1 Newtonian TLNs

The study of the tidal response of a self-gravitating object due to an
external tidal field was historically performed in the Newtonian framework,
where the total gravitational potential outside the object in the external
tidal field can be divided into two parts (see, e.g., [53]): (a) $U_{\rm
tidal}$, the gravitational potential due to the tidal field, and (b) $U_{\rm
body}$, the gravitational potential due to the compact object. The tidal
potential can be decomposed in the spherical harmonic basis as $U_{\rm
tidal}\sim\sum_{\ell m}\mathcal{E}_{\ell m}Y_{\ell m}r^{\ell}$, where
$\mathcal{E}_{\ell m}$ are the components of the tidal field associated with
the spherical harmonic decomposition, $\ell\geq 0$ is the angular number, and
$m$ is the azimuthal number (with $|m|\leq\ell$). In general,
$\mathcal{E}_{\ell m}$ will involve $\ell$ derivatives of the tidal potential.
Similarly, the Newtonian potential due to the central object can be expanded
in terms of its multipole moments, $I_{\ell m}$, as $U_{\rm
body}\sim\sum_{\ell m}I_{\ell m}Y_{\ell m}r^{-\ell-1}$. At the linear level,
the deviation $\delta I_{\ell m}$ from the background multipole moments (those
of a Kerr metric [54] in our case) are proportional to the tidal field
$\mathcal{E}_{\ell m}$ and its time derivatives,

$\displaystyle\delta I_{\ell m}\propto r_{0}^{2\ell+1}\left(2k_{\ell
m}\mathcal{E}_{\ell m}-\tau_{0}\nu_{\ell m}\dot{\mathcal{E}}_{\ell
m}+...\right)~{}.$ (4)

Here, $k_{\ell m}$ are the TLNs (related to the conservative response),
$\tau_{0}$ is a characteristic timescale associated with the tidal field,
$\nu_{\ell m}$ is associated with the tidal dissipation of the compact object
(sometimes called tidal heating in different contexts [55, 56, 57, 58, 59]),
and the dots denote higher-order derivatives with respect to the Boyer-
Lindquist time coordinate. Note that the proportionality factor due to the
linear relation between the multipole and tidal moments depends on the nature
of the compact object and its size, namely on $r_{0}$. Given the above
multipolar expansion, the total potential outside the compact object can be
expressed as

$\displaystyle U_{\rm total}$ $\displaystyle=U_{\rm body}+U_{\rm
tidal}=-\frac{M}{r}+{\rm intrinsic~{}moments}$ $\displaystyle+\int
d\omega\sum_{\ell m}\mathcal{E}_{\ell m}r^{\ell}\left[1+F_{\ell
m}(\omega)\left(\frac{r_{0}}{r}\right)^{2\ell+1}\right]Y_{\ell m}~{},$ (5)

where the frequency-dependent response function $F_{\ell m}(\omega)$ reads

$\displaystyle F_{\ell m}(\omega)=2k_{\ell m}(\omega)+i\omega\tau_{0}\nu_{\ell
m}(\omega)\,,$ (6)

and the (conservative) Newtonian TLNs are given by $k_{\ell
m}=(1/2)\textrm{Re}F_{\ell m}$.

####  II.2.2 Relativistic TLNs

A gauge-invariant, relativistic extension of the Newtonian TLNs can be
achieved with the use of Weyl scalars in the Newman-Penrose formalism [60].
The Weyl scalar which describes the behavior of the gravitational waves that
propagate towards asymptotic infinity is $\Psi_{4}$ and hence is the one we
will use in this work. In the Kerr-like background geometry, the Weyl scalar
$\Psi_{4}$ can be decomposed as [61]

$\rho^{4}\Psi_{4}=\int d\omega\sum_{\ell m}R_{\ell m}(r)~{}_{-2}S_{\ell
m}(\theta)e^{i(m\phi-\omega v)}\,,$ (7)

where $\rho=-(r-ia\cos\theta)^{-1}$, $R_{\ell m}(r)$ is the radial Teukolsky
function, and ${}_{-2}S_{\ell m}(\theta)e^{im\phi}$ is the spin-weighted
spheroidal harmonic with spin $s=-2$.

The fact that $\Psi_{4}$ is able to describe the tidal effects can be seen by
computing it in the Newtonian limit. In this case, the only nontrivial metric
functions read $g_{00}=-(1+2c^{-2}U_{\rm total})$ and $g_{ij}=(1+2c^{-2}U_{\rm
total})\delta_{ij}$, where $U_{\rm total}$ is the Newtonian potential
including the tidal effects and we have restored the speed of light for the
moment. In the Newtonian limit, obtained as $c\to\infty$, the computation of
$\Psi_{4}$ reveals the following behavior [9]:

$\displaystyle\lim_{c\rightarrow\infty}c^{2}\Psi_{4}\propto$
$\displaystyle\int d\omega\sum_{\ell m}\mathcal{E}_{\ell
m}r^{\ell-2}\left[1+F_{\ell
m}(\omega)\left(\frac{r_{0}}{r}\right)^{2\ell+1}\right]~{}_{-2}Y_{\ell m}\,,$
(8)

where ${}_{-2}Y_{\ell m}$ are the spin-weighted spherical harmonics. The above
behavior is valid in the intermediate region $r_{0}\ll r\ll 1/\omega$, i.e.
sufficiently far away both from the central object and the tidal source. As
evident, the radial part of the above equation is identical to the one derived
from the Newtonian potential in 5, with $F_{\ell m}$ being the tidal response
of the compact object to the external tidal field, modulo the external power
of the radial coordinate $r$. In 5, the external radial dependence scales as
$r^{\ell}$, while in II.2.2 the radial dependence behaves as $r^{\ell-2}$.
This difference can be accounted for by noting that in the Newtonian limit
$\Psi_{4}\sim\bar{m}^{i}\bar{m}^{j}\nabla_{i}\nabla_{j}U_{\rm total}$, where
$\bar{m}^{i}$ is one of the complex vector in the Newman-Penrose formalism,
$\nabla_{i}$ is the three-dimensional covariant derivative, and $U_{\rm
total}$ has been defined earlier. The null vector $\bar{m}^{i}$ has a radial
dependence $\sim(1/r)$ and is the origin of the $(1/r^{2})$ difference between
the Newtonian potential and the Weyl scalar.

The radial dependence in II.2.2 is valid also in the relativistic case in the
intermediate region $r_{0}\ll r\ll 1/\omega$. In this region, we can extract
the tidal response function $F_{\ell m}(\omega)$ from $\Psi_{4}$ and define
the relativistic TLNs in a gauge invariant manner as

$\displaystyle k_{\ell m}=\frac{1}{2}\text{Re}F_{\ell m}~{}.$ (9)

In general, the determination of the TLNs from $\Psi_{4}$ is achieved through
the following procedure: (i) find $\Psi_{4}$ in the near-horizon region and
apply the appropriate boundary conditions at $r_{0}$ (which will be discussed
in Section II.3), (ii) determine $\Psi_{4}$ in an intermediate region, which
is far away from both the compact object and the source of the tidal field,
(iii) for each $(\ell,m)$, determine the tidal response as the coefficient of
$r^{-\ell-3}$ in II.2.2, (iv) determine the real part of the tidal response,
which provides the corresponding TLN through 9. In the following, we perform
each of these steps and provide the basic set-up for obtaining the TLNs. For
brevity, we shall omit the $(\ell,m)$ subscripts in the radial and angular
functions in the Weyl scalar.

The radial function $R(r)$ satisfies the (source-free) radial Teukolsky
equation [61, 62, 63], which in the Kinnersley null tetrad [64] and in the
ingoing null coordinate system, reads

$\displaystyle\frac{d^{2}R(r)}{dr^{2}}+\left(\frac{2iP_{+}-1}{r-r_{+}}-\frac{2iP_{-}+1}{r-r_{-}}-2i\omega\right)\frac{dR(r)}{dr}$
$\displaystyle+\left(\frac{4iP_{-}}{(r-r_{-})^{2}}-\frac{4iP_{+}}{(r-r_{+})^{2}}+\frac{A_{-}+iB_{-}}{(r-r_{-})(r_{+}-r_{-})}\right.$
$\displaystyle\qquad-\left.\frac{A_{+}+iB_{+}}{(r-r_{+})(r_{+}-r_{-})}\right)R(r)=0\,,$
(10)

where the quantities $P_{\pm}$, $A_{\pm}$ and $B_{\pm}$ are defined as

$\displaystyle P_{\pm}$ $\displaystyle=$
$\displaystyle\frac{am-2r_{\pm}M\omega}{r_{+}-r_{-}}\,,$ (11) $\displaystyle
A_{\pm}$ $\displaystyle=$ $\displaystyle E_{\ell
m}-2-2(r_{+}-r_{-})P_{\pm}\omega$ (12)
$\displaystyle-(r_{\pm}+2M)r_{\pm}\omega^{2}\,,$ $\displaystyle B_{\pm}$
$\displaystyle=$ $\displaystyle 2r_{\pm}\omega\,,$ (13)

and $E_{\ell m}=\ell(\ell+1)+\mathcal{O}(a\omega)$. We would like to emphasize
that the structure of the above differential equation is different in the
static limit of the non-rotating case, i.e., for $a=0=\omega$. This will have
important implications for the static limit and for the dependence of the TLNs
on the compactness parameter $\epsilon$, which we will discuss in a later
section.

An analytical expression for the TLNs can be derived from the small-frequency
approximation of the radial Teukolsky equation in 10. The details of the
small-frequency expansion can be found in B. The near horizon limit of the
solution yields (see B for a derivation),

$\displaystyle
R(r)\sim\frac{\mathcal{A}}{(r_{+}-r_{-})^{4}}\Delta^{2}+\mathcal{B}e^{2i\bar{\omega}r_{*}}~{},\quad
r\to r_{+}~{},$ (14)

where, $\bar{\omega}=\omega-m\Omega_{\rm H}$, $\Omega_{\rm H}$ being the
angular velocity of the horizon, and $r_{*}$ is the tortoise coordinate
defined as $dr_{*}/dr=\left(r^{2}+a^{2}\right)/\Delta$. The unknown functions
$\mathcal{A}$ and $\mathcal{B}$ depend on the frequency and are to be
determined from the boundary conditions, which will be discussed in Section
II.3. After imposing these boundary conditions, the Teukolsky equation can be
solved up to the intermediate region, far from both the compact object and the
source of the tidal field, yielding the response function of the compact
object to the external tidal field. Finally, the TLN of the $(\ell,m)$ mode
reads in the small-frequency approximation (see B for a derivation),

$\displaystyle k_{\ell
m}=\textrm{Re}\Bigg{[}-i\frac{P_{+}}{2}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)$
$\displaystyle\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)$
$\displaystyle\qquad\times\left\\{\frac{1-\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right\\}$
$\displaystyle\Bigg{]}~{},$ (15)

where,

$\displaystyle\Gamma_{1}=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(3+2iP_{+}\right)_{\ell-2}}{\left(-1-2iP_{+}\right)_{\ell+2}}~{}.$
(16)

In the expression for $\Gamma_{1}$, we have introduced the Pochhammer symbol,
$z_{n}\equiv z(z+1)\times\cdots\times(z+n-1)$. A similar expression (related
to the imaginary part of $F_{\ell m}$) gives the dissipative response
$\nu_{\ell m}$.

As expected, the tidal response depends on the ratio ${\cal B}/{\cal A}$ and,
for $\mathcal{B}=0$ (i.e., in the BH limit), the response function becomes
imaginary, leading to vanishing TLNs. However, for compact objects other than
BHs, $\mathcal{B}$ is non-zero, and hence it is uniquely determined by the
boundary conditions at $r=r_{0}$, which we now discuss.

###  II.3 Reflectivity of compact objects

As shown in Section II.2, the TLNs of Kerr-like compact objects computed in
II.2.2 depend on the unknown constants of integration, $\mathcal{A}$ and
$\mathcal{B}$, whose ratio is fixed by the boundary conditions at $r=r_{0}$.
The radial Teukolsky function in the near horizon regime has two contributions
[63]: (a) an ingoing part, which behaves as $\Delta^{2}$ and (b) an outgoing
part, with radial dependence $e^{2i\bar{\omega}r_{*}}$. Thus, from 14, one may
identify the constant $\mathcal{A}$ to be associated with the ingoing mode at
the horizon, while the constant $\mathcal{B}$ is associated with the outgoing
mode at the horizon. For a BH, regularity of the perturbation at $r\sim r_{+}$
imposes no outgoing modes at the horizon since the BH absorbs everything and
hence one should set $\mathcal{B}=0$.

For non-BH objects, be it a neutron star, an ECO [41, 28, 46] or a quantum
corrected BH [65, 66, 67], there would be nontrivial reflection by the
object’s interior, and hence there would be outgoing modes for which
$\mathcal{B}\neq 0$. Overall, the properties of the compact object are
embedded into a single quantity, defined as the reflectivity of the object.
BHs have zero reflectivity, ordinary neutron stars have almost perfect
reflectivity [47] (due to viscosity they have tiny interaction with
gravitational waves, but see [68] for a recent analysis of the out-of-
equilibrium tidal dynamics), while non-BH compact objects will in general have
nonzero and frequency-dependent reflectivity. We parametrize the latter by a
complex number ${\cal R}(\omega)$, which might generically introduce a phase
shift in the reflected wave relative to the incident one.

Loosely speaking, the reflectivity is the ratio of the amplitudes of the
outgoing and the ingoing waves and one may consider the ratio
$(\mathcal{B}/\mathcal{A})$ as the reflectivity. However, this would not be a
correct assessment, since the outgoing and ingoing modes in terms of the
radial Teukolsky function in the $(v,r,\theta,\tilde{\phi})$ coordinate are
_not_ plane waves and hence the energy carried by ingoing and outgoing modes
is not $|{\cal A}|^{2}$ and $|{\cal B}|^{2}$, but rather a more complicated
function of the frequency [63]. The way around this difficulty is to express
the radial Teukolsky equation as a Schrödinger-like equation with a real
potential, which is constant in the near horizon regime, in which case the
mode functions near the horizon are exact plane waves. This is achieved by the
Detweiler function, $X$, which is defined as a linear combination of the
radial Teukolsky function $R^{(t)}$ in the Boyer-Lindquist coordinates and its
derivative as [69, 47]

$\displaystyle X=\frac{\sqrt{r^{2}+a^{2}}}{\Delta}\left[\alpha
R^{(t)}+\frac{\beta}{\Delta}\frac{dR^{(t)}}{dr}\right]\,,$ (17)

where the quantities $\alpha$ and $\beta$ are functions of $r$ and $\omega$,
their explicit dependence can be found in [69, 47]. With this transformation,
it turns out that the Detweiler function satisfies the following Schrödinger-
like equation,

$\displaystyle\frac{d^{2}X}{dr_{*}^{2}}-V(r,\omega)X=0~{},$ (18)

where the potential is purely real and has the following asymptotics: at the
horizon $V(r\to r_{+},\omega)\to-\bar{\omega}^{2}$, and at infinity
$V(r\to\infty,\omega)\to-\omega^{2}$. Hence, it is possible to define purely
outgoing and ingoing wave modes $\sim e^{\pm i\bar{\omega}r_{*}}$ at the
horizon and $\sim e^{\pm i\omega r_{*}}$ at infinity. Using these the
Detweiler function near the boundary of the compact object reads,

$X\sim
e^{-i\bar{\omega}(r_{*}-r^{0}_{*})}+\mathcal{R}(\omega)e^{i\bar{\omega}(r_{*}-r_{*}^{0})}~{},\quad
r_{*}\sim r_{*}^{0}\,,$ (19)

where $r_{*}^{0}$ is the location of the surface of the object in the tortoise
coordinate. Thus, we can define the reflectivity $\mathcal{R}(\omega)$ of the
compact object in a straightforward manner as

$\displaystyle\mathcal{R}(\omega)=\left[\frac{1-\frac{i}{\bar{\omega}}\left(\frac{1}{X}\frac{dX}{dr_{*}}\right)}{1+\frac{i}{\bar{\omega}}\left(\frac{1}{X}\frac{dX}{dr_{*}}\right)}\right]_{r_{*}^{0}}~{},$
(20)

which is in terms of the Detweiler function and its first radial derivative
near the surface of the compact object. To connect the ratio
$(\mathcal{B}/\mathcal{A})$, appearing in II.2.2, to the reflectivity derived
from the Detweiler function we first have to express the radial Teukolsky
function in the $(v,r,\theta,\tilde{\phi})$ coordinates, in terms of the
Boyer-Lindquist coordinates $(t,r,\theta,\phi)$. This has been performed
explicitly in A. Besides, one can use the transformation between the Teukolsky
function and the Detweiler function, in order to determine the connection
between the Detweiler reflectivity $\mathcal{R}$, with the Teukolsky
reflectivity $(\mathcal{B}/\mathcal{A})$ (for a detailed expression, see A).
Finally, the above relation can be inverted, and the ratio
$\mathcal{B}/\mathcal{A}$ can be expressed in terms of the Detweiler
reflectivity $\mathcal{R}$, which when substituted back into II.2.2, will
provide the TLNs in terms of the Detweiler reflectivity. In what follows we
will consider some generic form of the reflectivity, focusing on its small-
frequency expansion, as in 1, and shall derive the dependence of the TLNs on
the model parameters and frequency. In particular, our interest will be on the
scaling of the TLN with the parameter $\epsilon$, providing the departure of
the surface of the compact object from the BH horizon.

##  III TLNs of compact objects

Having set up the basic equations and boundary conditions, we compute the TLNs
for (generically rotating) compact objects, both in the static and frequency-
dependent case. As we shall show, our general framework provides results for
static perturbations and non-spinning objects, which does _not_ coincide with
the strictly static limit, more details will be discussed in Section III.4.
Here we will elucidate some general properties of the dynamical TLNs for
rotating and non-rotating compact objects.

###  III.1 The BH limit

To set the stage, we start from the BH limit of our results, which corresponds
to ${\cal R}=0$ and $\epsilon\to 0$. We discuss the TLNs of BHs in the linear-
frequency approximation and comment on the behavior of the TLNs when terms of
$\mathcal{O}(\omega^{2})$ are included in the analysis.

At linear order in the frequency, one can directly use the analytical results
derived in Sec. II.2. It is evident from II.2.2 that the tidal response
function of a rotating BH reads

$\displaystyle F^{\rm rot,BH}_{\ell m}$
$\displaystyle=-i\left(\frac{am-2r_{+}M\omega}{r_{+}-r_{-}}\right)\frac{(\ell-2)!(\ell+2)!}{(2\ell)!(2\ell+1)!}$
$\displaystyle\prod_{j=1}^{\ell}\left[j^{2}+4\left(\frac{am-2r_{+}M\omega}{r_{+}-r_{-}}\right)^{2}\right]\,,$
(21)

which agrees with previous results [9, 43] (see also [49]). For a
Schwarzschild BH, the tidal response function becomes

$F^{\rm
nonrot,BH}_{\ell}=2iM\omega\frac{(\ell-2)!(\ell+2)!}{(2\ell)!(2\ell+1)!}\prod_{j=1}^{\ell}\left[j^{2}+16M^{2}\omega^{2}\right]\,.$
(22)

Note that in the non-rotating case, the tidal response function does not
depend on the azimuthal number $m$, but only on the angular number $\ell$. For
$\ell=2$, the quadrupolar response function for the Schwarzschild BH takes the
form:

$F^{\rm nonrot,BH}_{2}=\frac{i}{15}M\omega+{\cal O}(\omega^{2}M^{2})\,.$ (23)

As evident in all of these cases, the response function in the small-frequency
regime is a purely imaginary quantity and hence the TLNs, which are the
conservative part of the response function and hence are given by the real
part of the response function, identically vanishes. Our analysis is
consistent with the findings of Ref. [9] and shows that for BHs the TLNs
identically vanish in the small frequency limit.

In the above expressions we have kept terms up to linear order in $\omega M$
and hence it is interesting to address what happens to the TLNs of BHs when
terms $\mathcal{O}(\omega^{2}M^{2})$ are taken into account. The starting
point is the radial Teukolsky equation where we keep all the terms up to
$\mathcal{O}(\omega^{2}M^{2})$ in the near-horizon approximation (for a
derivation, see C),

$\displaystyle\frac{d^{2}R}{dz^{2}}$
$\displaystyle+\left[\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega[2M+(r_{+}-r_{-})]}{(1+z)}\right]\frac{dR}{dz}+\Bigg{[}-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega[4M-(r_{+}-r_{-})]}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}a^{2}(1+E_{2})}{z(1+z)}\Bigg{]}R=0~{},$ (24)

where the rescaled radial coordinate $z$ is defined in 81, and $E_{1}$ and
$E_{2}$ are the coefficients of $\mathcal{O}(a\omega)$ and
$\mathcal{O}(a^{2}\omega^{2})$ terms in the expansion of the angular
eigenvalue in the angular Teukolsky equation, whose explicit forms can be
obtained from [70]. The first derivative term, namely $dR/dz$, does not depend
on $\omega^{2}$, while the coefficient of the radial perturbation $R$ does
only through the combination $a^{2}\omega^{2}$. Thus, for a Schwarzschild BH,
any correction of $\mathcal{O}(\omega^{2}M^{2})$ is absent and the TLNs of a
Schwarzschild BH vanish as

$\displaystyle k_{2}^{\textrm{nonrot, BH}}=0+\mathcal{O}(M^{3}\omega^{3})~{}.$
(25)

The same holds for slowly rotating BHs to linear order in the spin since any
possible correction to the TLNs is proportional to
$\mathcal{O}(a^{2}\omega^{2})$. For a generically rotating BH, there can be
non-trivial effects due to the $\mathcal{O}(a^{2}\omega^{2})$ term in the
Teukolsky equation. In other words, while it is guaranteed that a BH has
vanishing static TLNs, it could have non-trivial dynamical TLNs. We wish to
explore this interesting problem in a future work.

###  III.2 Non-spinning case

We present here the TLN of a non-rotating ultracompact object, which can be
obtained by taking the limit $a\to 0$ in II.2.2. The resulting expression
depends on the frequency, the mass of the compact object, the parameter
$\epsilon$, and the reflectivity. In this limit $P_{+}=-2M\omega$, thus the
expression for the TLN associated with the $\ell=2$ mode simplifies to,

$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{[}\left(\frac{iM\omega}{30}\right)\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)$
$\displaystyle\qquad\qquad\times\left\\{\frac{1-\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right\\}\Bigg{]}~{},$
(26)

where (see B for a derivation)

$\displaystyle\Gamma_{1}=\frac{3i(1-2iM\omega)}{M\omega(1+4M^{2}\omega^{2})(1+16M^{2}\omega^{2})}~{}.$
(27)

As we will demonstrate later, at frequencies satisfying the condition
$|1+(\mathcal{B}/\mathcal{A})\Gamma_{1}|\simeq 0$, there are resonances in the
TLNs. To see this explicitly, one must use the relation between the Teukolsky
reflectivity $(\mathcal{B}/\mathcal{A})$ with the Detweiler reflectivity
$\mathcal{R}(\omega)$, which in the non-rotating case reads (see B for
details),

$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\left(\frac{2M\omega}{3}\right)\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{2-iM\omega}$
$\displaystyle\qquad\times\mathcal{R(\omega)}e^{8\pi
M\omega-2i\omega\left(r_{*}^{0}-2M\right)}~{}.$ (28)

In the next sections, we will discuss the zero-frequency limit of the above
expressions and hence determine the static TLNs from the dynamical ones.

####  III.2.1 Zero-frequency limit

In this section, we discuss the zero-frequency limit of the TLNs of static and
spherically symmetric compact objects. Before proceeding, it is worth
emphasizing that we need to set the rotation parameter $a\to 0$ and then the
frequency $\omega\to 0$ to determine the static TLN for a non-rotating compact
object. As we will show in detail, the zero-frequency limit of the frequency-
dependent TLN differs from the strictly static TLN, as the branch of the
solution to the Teukolsky equation with outgoing behavior close to the horizon
becomes ill-defined in this limit. Therefore, we do not expect the TLN in the
zero frequency limit to match with the results derived in Ref. [22], but
nonetheless the zero-frequency limit seems more natural to consider as the
coalescence of binary compact objects is frequency-dependent.

Given the expression for $\Gamma_{1}$ in 27 and the relation between the
Teukolsky and Detweiler reflectivities in III.2, we can easily obtain the
zero-frequency limit of both of these expressions, by keeping the leading
order terms in $M\omega$. This yields
$\Gamma_{1}=(3i/M\omega)+6+\mathcal{O}(M\omega)$, while the Teukolsky
reflectivity reduces to

$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\frac{iM\omega\mathcal{R}}{3}+\frac{M^{2}\omega^{2}\mathcal{R}}{6}\left(3+16i\pi+8\epsilon+8\ln\epsilon\right)$
$\displaystyle+\mathcal{O}(M^{3}\omega^{3})~{}.$ (29)

Thus, keeping terms up to linear order in $M\omega$, we obtain

$\displaystyle\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma_{1}=-\mathcal{R}+\frac{iM\omega\mathcal{R}}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\,,$
(30)

and hence the TLN in the zero frequency limit becomes,

$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{\\{}\left(\frac{iM\omega}{30}\right)$
$\displaystyle\times\frac{1+\mathcal{R(\omega)}\left[1-\frac{iM\omega}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\right]}{1-\mathcal{R}(\omega)\left[1-\frac{iM\omega}{2}\left(7+16i\pi+8\epsilon+8\ln\epsilon\right)\right]}\Bigg{\\}}~{}.$
(31)

The above equation shows one of our main results: except for the single case
where the above expression has a pole (to be discussed below), the TLN
identically vanishes in the zero-frequency limit for any other reflectivity,
just as in the BH case. This includes all cases of partial reflection444It
might come as a surprise that the TLNs are zero if ${\cal R}\neq 0$ in the
zero-frequency limit. For example one could argue that a perfect-fluid star
has nonzero TLNs and viscosity (which effectively corresponds to $|{\cal
R(\omega)}|^{2}<1$) cannot drastically change the TLNs. However, note that
viscosity would affect the reflectivity starting at ${\cal O}(M\omega)$,
leaving ${\cal R}_{0}$ unaffected. Thus, our formalism also accounts for the
TLNs of a neutron star, wherein ${\cal R}_{1}$ depends on the equation of
state., i.e., $|{\cal R}(\omega)|^{2}<1$. In the exceptional case where
$\mathcal{R}(\omega)=1+iM\omega\mathcal{R}_{1}$, III.2.1 has a pole and the
TLN for the $\ell=2$ mode, in the zero-frequency limit, becomes

$\displaystyle
k_{2}=\frac{2}{15}\textrm{Re}\Bigg{[}\frac{1}{-2\mathcal{R}_{1}+\left\\{7+16i\pi+8(\epsilon+\ln\epsilon)\right\\}}\Bigg{]}~{}.$
(32)

The above result does not correspond to the strictly static case, as we must
keep terms $\mathcal{O}(M\omega)$ in the combination
$(\mathcal{B}/\mathcal{A})\Gamma_{1}$ as well as the reflectivity
$\mathcal{R}$ must be frequency-dependent (i.e., $\mathcal{R}_{1}\neq 0$) in
order to have a non-zero TLN. Note that, the magnitude of the reflectivity
reads
$|\mathcal{R}(\omega)|=1-M\omega\,\textrm{Im}\mathcal{R}_{1}+\mathcal{O}(\omega^{2})$.
Thus, if $\mathcal{R}_{1}$ is real, the object is perfectly reflecting, i.e.,
$|\mathcal{R}(\omega)|=1$. In general, if $\mathcal{R}_{1}$ is complex with
$\textrm{Im}\mathcal{R}_{1}>0$, the reflectivity is smaller than unity. Note
that $\textrm{Im}\mathcal{R}_{1}<0$ is not physical since it corresponds to a
reflectivity larger than unity.

Thus, the TLN associated with the $\ell=2$ mode depends on the frequency-
dependent part of the reflectivity and in general displays a logarithmic
behavior with $\epsilon$. Whenever $|\ln\epsilon|\gg|{\cal R}_{1}|/4$ in the
$\epsilon\to 0$ limit, the TLN vanishes as

$k_{2}=\frac{1}{60\ln\epsilon}\,.$ (33)

Although in this case we obtain a logarithmic scaling as found in [22] for
static TLNs of perfectly reflecting ECOs, the numerical coefficients differ
from those obtained in the zero-frequency limit of dynamical TLNs. Thus, the
static limit of the dynamical TLN does not coincide with the strictly static
case. This is because the solution of the Teukolsky equation in the zero-
frequency case differs from the one with linear-in-frequency terms. We will
discuss these aspects in detail in Sec. III.4.

The variation of the dynamical TLN in the zero-frequency limit with $\epsilon$
is presented in 1, for $\mathcal{R}_{0}=1$, and three different choices of
$\mathcal{R}_{1}$. The plot clearly demonstrates the dependence of the static
TLN on $\log\epsilon$, for constant values of $\mathcal{R}_{1}$. Also, 1 shows
that the magnitude of the static TLN depends heavily on $\epsilon$ and also
mildly on $\mathcal{R}_{1}$, as suggested by 32.

Figure 1: Absolute value of the quadrupolar TLN, $|k_{2}|$, the zero-frequency
limit and as a function of the compactness parameter $\epsilon$, in a
logarithmic scale, and for $\mathcal{R}_{0}=1$ and various choices of the
linear-in-frequency reflectivity $\mathcal{R}_{1}$. For larger and positive
values of $\mathcal{R}_{1}$ the absolute value of the TLN decreases, while for
negative choices of $\mathcal{R}_{1}$ the TLN increases. Overall, as
$\epsilon$ becomes larger, the TLN also increases. The plot shows the generic
logarithmic dependence of the static TLN on the compactness parameter
$\epsilon$.

####  III.2.2 Frequency dependence of the TLNs

Figure 2: Quadrupolar TLN as a function of the frequency, for three cases —
(a) BHs, with $\mathcal{R}=0$, (b) for a compact object, with
$\mathcal{R}_{0}=1$ and (c) another compact object with $\mathcal{R}_{0}=-1$.
In the latter two cases, involving non-rotating compact objects we have taken
$\epsilon=10^{-40}$ and $\mathcal{R}_{1}=1$. As evident, only for the choice
$\mathcal{R}_{0}=1$, the TLN goes to a finite and negative value in the zero-
frequency limit, while even for $\mathcal{R}_{0}=-1$ (which still corresponds
to a perfectly reflecting object), the TLN vanishes in the zero-frequency
limit. In both of these cases, the TLN increases with an increase in
frequency, and in general, the frequency-dependent TLN does not vanish.

Having discussed the static limit of the dynamical TLNs for non-rotating
compact objects, in this section, we will discuss the properties of the
dynamical TLN associated with the $\ell=2$ mode. By combining 27 and III.2,
the expression for the TLN associated with the $\ell=2$ mode of a non-rotating
compact object reads,

$\displaystyle k_{2}$
$\displaystyle=\textrm{Re}\Bigg{[}\left(\frac{iM\omega}{30}\right)\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)$
$\displaystyle\qquad\qquad\qquad\times\left(\frac{1+\mathcal{R}(\omega)G(\omega)}{1-\mathcal{R}(\omega)G(\omega)}\right)\Bigg{]}~{},$
(34)

where we have introduced the frequency-dependent quantity $G(\omega)$,

$\displaystyle G(\omega)$ $\displaystyle\equiv 2\frac{\exp\left[8\pi
M\omega-4iM\omega\left(\epsilon+\ln\epsilon\right)\right]}{2-iM\omega}$
$\displaystyle\qquad\times\left(\frac{1-2iM\omega+16M^{2}\omega^{2}-32iM^{3}\omega^{3}}{1+2iM\omega+16M^{2}\omega^{2}+32iM^{3}\omega^{3}}\right)~{}.$
(35)

Note that $G(\omega)\to 1$ in the zero-frequency limit, so one recovers
III.2.1. The frequency dependence of the TLN for a non-rotating compact
object, both in the perfectly reflecting and partially absorbing cases have
been depicted in 2 and 3, respectively. As evident, except for the case
$\mathcal{R}_{0}=1$, the TLN identically vanishes in the limit
$\omega\rightarrow 0$, which is consistent with III.2.1. Moreover, the TLN
associated with the $\ell=2$ mode in the zero-frequency limit is negative for
$\mathcal{R}_{0}=1$ and a positive choice of $\mathcal{R}_{1}$ and small
$\epsilon$, which is consistent with III.2.1. We will now discuss the features
associated with the frequency-dependent TLNs.

In all the cases involving either a perfectly reflecting compact object with
$\mathcal{R}_{0}=\pm 1$ (see 2), or a partially reflecting one with
$\mathcal{R}_{0}<1$ (see 3), the TLN associated with the $\ell=2$ mode
increases with frequency, and we observe oscillations. 3 also suggests that
the $\ell=2$ TLN decreases at higher frequencies, but our low-frequency
analysis, presented here, eventually breaks down, and thus the high-frequency
results should be taken with a grain of salt.

Figure 3: The TLN of a compact object as a function of the frequency for a
non-spinning and partially as well as perfectly reflecting compact object with
$\epsilon=10^{-40}$ and $\ell=2$. As evident, except for the case
$\mathcal{R}_{0}=1$, the TLN in the zero-frequency limit identically vanishes.
On the other hand, for $\mathcal{R}_{0}=1$, there are oscillations at non-zero
frequencies, while for $\mathcal{R}_{0}<1$, there are also resonances, which
drift to higher frequencies as the reflectivity decreases. See the main text
for more discussion.

We will now depict the one-to-one correspondence between the oscillation
frequencies as well as the resonances with the real part of the QNMs of the
compact object. Following Ref. [47], the QNMs in the small-$\epsilon$ limit
can be written as,

$\omega_{R}\sim-\frac{\pi(q+1)}{2|r_{*}^{0}|}+m\Omega_{\rm H}\,,$ (36)

where $r_{*}^{0}\sim M[1+(1-\chi^{2})^{-1/2}]\ln\epsilon$, $\chi=a/M$ is the
dimensionless spin, $q=2n-1\ (2n)$ for perfectly reflecting objects with
$\mathcal{R}_{0}=1$ ($\mathcal{R}_{0}=-1$), and $n\geq 1$ is the overtone
number. As discussed in Refs. [47, 51, 71], the above analytical approximation
for the real part of the QNMs is also valid for partially absorbing objects,
with $q=2n-1\ (2n)$ for compact objects with $\mathcal{R}_{0}>0$
($\mathcal{R}_{0}<0$). In the case of a non-spinning compact object with
$\mathcal{R}_{0}>0$ and $\epsilon=10^{-40}$, the analytical approximation for
the real part of the QNMs in 36 yields the following frequencies
$M\omega_{R}=0.017\ (n=1),0.034\ (n=2),0.051\ (n=3),0.068\ (n=4),0.085\
(n=5)$, etc. Intriguingly, at precisely these frequencies the oscillations in
the dynamical TLN occur, as shown in 3. The amplitudes of the oscillations and
resonances are then related to the absolute value of $\mathcal{R}_{0}$ and the
factor $G(\omega)$. As evident from III.2.2, for partially reflective compact
objects, close to the point $\mathcal{R}(\omega)G(\omega)\simeq 1$, the TLN
for the $\ell=2$ mode displays resonances. This can be verified from 3 as
well, which shows the appearance of resonances at two different frequencies
for the choices of reflectivities considered here. One can easily verify that
the locations of these resonances are exactly at the solutions of the equation
$\mathcal{R}(\omega)G(\omega)\simeq 1$, as predicted by the analytical
results. In particular, for smaller values of the reflectivity, these
resonances appear at larger values of $M\omega$, and hence, as the
reflectivity becomes very small, the resonance frequencies will become so
large that the small-frequency approximation will break down. We also observe
that the amplitudes of the resonances become larger for smaller
reflectivities, since the overall multiplication factor involving $M\omega$ in
the TLN becomes larger.

The above summarizes the behavior of the TLNs for non-rotating compact objects
both in the dynamical context, as well as in the zero-frequency limit. We now
turn our attention to the case of spinning compact objects.

###  III.3 Spinning case

Having discussed the features of the TLN in the case of a nonspinning compact
object, in this section, we will concentrate on determining the TLN in the
spinning case. The general expression has been provided in II.2.2, but we will
quote some results associated with the $\ell=2$ mode since that is the most
relevant one. For this case, the tidal response function can be determined
using 88, 91 and 92 in B, which reads,

$\displaystyle k_{2m}$
$\displaystyle=\textrm{Re}\Bigg{[}-\frac{iP_{+}}{60}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)$
$\displaystyle\qquad\qquad\times\left\\{\frac{1+\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{6i\left(1+iP_{+}\right)}{P_{+}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)}}{1-\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{6i\left(1+iP_{+}\right)}{P_{+}\left(1+4P_{+}^{2}\right)\left(1+P_{+}^{2}\right)}}\right\\}\Bigg{]}~{},$
(37)

where $P_{+}=\\{(am-2M\omega r_{+})/(r_{+}-r_{-})\\}$ as defined in 11. Note
that, in the non-rotating case, it was possible to derive a relation between
the Teukolsky reflectivity $(\mathcal{B}/\mathcal{A})$ and the Detweiler
reflectivity $\mathcal{R}$ in a closed form, see III.2. However, in the
rotating case, the analytical expression for such a relation is cumbersome and
we will not report it. Following this procedure, we express the TLN in III.3
in terms of the more physical Detweiler reflectivity. Note that, unlike the
non-rotating case, the TLN for a rotating compact object depends on the
azimuthal number $m$ through $P_{+}$. Moreover, even when the
$\omega\rightarrow 0$ limit is taken, due to the non-zero rotation parameter,
$P_{+}$ remains non-zero and hence the TLN may not vanish. However, the exact
zero-frequency limit depends on the relation between the Teukolsky and the
Detweiler function in the appropriate limits. In what follows, we will depict
the dependence of the TLN with the frequency as well as with $\epsilon$,
capturing the departure of the surface of the compact object from the would-be
horizon, for various choices of the reflectivity and rotation parameter.

The dependence of the TLN on $\epsilon$, the parameter which determines the
distance between the reflective surface and the would-be horizon, for a
rotating compact object, at zero-frequency is presented in 4 and 5,
respectively. In 4, we have plotted the absolute value of the TLN against
$\epsilon$ for a perfectly reflecting compact object, with
$\mathcal{R}_{0}=1=\mathcal{R}_{1}$, and for various choices of the
dimensionless rotation parameter. Interestingly, the zero-frequency TLN
decreases with increasing rotation, i.e., faster-rotating compact objects are
more difficult to deform than the slowly rotating ones. Furthermore, the TLN
decreases as the reflectivity of the compact object decreases, as expected
(cf. 5). This is because, the $\mathcal{R}_{0}\rightarrow 0$ limit corresponds
to the Kerr BH, having zero TLN at least for small frequencies, and hence it
is expected that a smaller zero-frequency reflectivity leads to a smaller TLN.
We would like to point out that the oscillatory behavior of the TLN with
$\epsilon$ originates from the term involving
$\exp[-2i\omega(r_{0}^{*}-r_{+})]$, appearing in the connection between the
Teukolsky reflectivity $(\mathcal{B}/\mathcal{A})$ and the Detweiler
reflectivity $\mathcal{R}$. Further, as evident from 5, the oscillation length
scale does not depend on the reflectivity, i.e., all the curves in the figure
are in phase with each other. However, the amplitude depends on the
reflectivity and decreases for smaller reflectivity. Note that both of these
plots are in the zero-frequency limit, and hence it is clear that, unlike the
case of a non-rotating ECO where the TLN generically vanishes in the zero-
frequency limit, for a rotating ECO the zero-frequency TLN is in general non-
zero.

Figure 4: The zero-frequency limit of the frequency-dependent TLN as a
function of $\epsilon$, which determines the distance between the surface of
the compact object and the would-be horizon for a spinning and perfectly
reflecting compact object ($\mathcal{R}_{0}=1=\mathcal{R}_{1}$) with $\ell=2$
and $m=2$. We considered three possible choices of the dimensionless rotation
parameter $\chi$. It is clear that, increasing the rotation, the zero-
frequency TLN decreases. Figure 5: Zero-frequency limit of the TLN as a
function of $\epsilon$ for a spinning compact object with $\chi=0.5$,
$\ell=2$, $m=2$ and for three different choices of the reflectivity. The plot
shows that a decrease in the zero-frequency reflectivity $\mathcal{R}_{0}$
decreases the TLN, while the oscillation pattern remains the same.

Besides the zero-frequency limit, we have also studied the dependence of the
$\ell=2$ TLN on the frequency. In 6, we show this dependence for different
choices of the spin, for a perfectly reflecting compact object with
$\mathcal{R}_{0}=1=\mathcal{R}_{1}$ and $\epsilon=10^{-10}$. The plot
explicitly demonstrates the presence of resonances in the structure of the
TLN, and these precisely appear where the denominator of III.3 vanishes. The
resonance frequencies associated with the dynamical TLNs also correspond to
the QNM frequencies of the rotating object, as discussedi n the previous
section. In the dynamical context, for higher frequencies, the behavior of the
TLN is very similar to that of a non-rotating compact object, involving
oscillations and resonances at specific frequencies. Thus, one can safely
argue that the frequency-dependent TLN of a compact object has similar
behavior in both the non-rotating and rotating case. In particular, rapidly
rotating compact objects have larger TLNs at higher frequencies, but very
small TLNs for low-frequency tidal fields. Therefore, at lower frequencies
faster-rotating objects are difficult to be tidally deformed, while at
slightly higher frequencies faster-rotating objects are the easiest to be
tidally deformed. On a similar note, 7 depicts the variation of the $\ell=2$
TLN with the frequency for various partially reflecting rotating compact
objects. In this case, the TLN also decreases with reflectivity, implying that
objects with smaller reflectivity are difficult to deform. This result is in
complete agreement with the fact that BHs, which are objects with no
reflectivity, cannot be tidally deformed. Thus, we have provided all the
details regarding the TLNs of a rotating compact object in the dynamical
context, as well as in the zero frequency limit, depicting non-trivial
behavior.

Figure 6: TLN of a spinning and perfectly reflecting
$(\mathcal{R}_{0}=1=\mathcal{R}_{1})$ compact object as a function of the
frequency with $\epsilon=10^{-10}$, $\ell=2$ and $m=2$. We observe resonances
in the spectrum of the TLN as in the case of non-rotating compact objects.
Interestingly, with an increase in rotation, the TLN decreases significantly,
validating our claim that faster-rotating objects are difficult to deform
tidally at lower frequencies. Figure 7: The absolute value of the TLN as a
function of the frequency for spinning compact objects with three different
choices of the reflectivity and the following choices of the physical
parameters: $\chi=0.5$, $\epsilon=10^{-10}$, $\ell=2$ and $m=2$. Following our
expectations, a decrease in the reflectivity results in a decrease in the TLN,
ultimately vanishing in the BH limit.

###  III.4 Static TLNs for a non-rotating compact object

As discussed in the previous sections, the strictly static TLN for a non-
rotating compact object needs to be derived from scratch. One of the prime
reasons is the fact that the outgoing solution of the Teukolsky equation
derived in B is ill-behaved in the limit $P_{+}\to 0$ (as we are interested in
non-rotating objects with $a=0$, and the static limit correspond to $\omega\to
0$). Thus, the static limit for a non-rotating compact object needs to be
studied by taking the zero-rotation and zero-frequency limit of the Teukolsky
equation, presented in 10, which reads,

$\displaystyle\dfrac{d^{2}R_{\ell}}{dr^{2}}$
$\displaystyle+\left(-\frac{1}{r-r_{+}}-\frac{1}{r-r_{-}}\right)\dfrac{dR_{\ell}}{dr}$
$\displaystyle+\left(-\frac{1}{r-r_{+}}+\frac{1}{r-r_{-}}\right)\frac{\gamma_{\ell}}{(r_{+}-r_{-})}R_{\ell}=0~{},$
(38)

where, we have defined $\gamma_{\ell}\equiv(\ell+2)(\ell-1)$. Introducing, the
variable $z$ as in B, in the present scenario we obtain $z=\frac{r}{2M}-1$, in
terms of which the above differential equation reduces to,

$\displaystyle\dfrac{d^{2}R_{\ell}}{dz^{2}}-\left(\frac{1}{z}+\frac{1}{1+z}\right)\dfrac{dR_{\ell}}{dz}+\left(-\frac{1}{z}+\frac{1}{1+z}\right)\gamma_{\ell}R_{\ell}=0~{}.$
(39)

This equation has the following explicit solution,

$\displaystyle
R_{\ell}=z(1+z)\left[c_{1}P_{\ell}^{2}(1+2z)+c_{2}Q_{\ell}^{2}(1+2z)\right]~{},$
(40)

where, $P_{\ell}^{2}(x)$ and $Q_{\ell}^{2}(x)$ are the associated Legendre
polynomials. To see explicitly how these solutions differ from the zero-
frequency and zero-rotation limit of B, we use the relations between the
associated Legendre polynomials and the hypergeometric functions,

$\displaystyle
P_{\ell}^{2}(1+2z)=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z(1+z)\,_{2}F_{1}(3+\ell,2-\ell;3;-z)~{}.$
(41)

As evident, the above hypergeometric function follows from the first solution
in B, under the limit $P_{+}\to 0$. While, for the other associated Legendre
polynomial,

$\displaystyle Q_{\ell}^{2}(1+2z)$
$\displaystyle=\frac{\sqrt{\pi}}{2^{\ell-1}}\frac{\Gamma(\ell+3)}{2\Gamma(\ell+\frac{3}{2})}\frac{z(1+z)}{(1+2z)^{\ell+3}}$
$\displaystyle\times\,_{2}F_{1}\left(2+\frac{\ell}{2},\frac{3}{2}+\frac{\ell}{2};\ell+\frac{3}{2};\frac{1}{(1+2z)^{2}}\right)~{},$
(42)

which has no analog in the static limit of B for a non-rotating system. This
is why the solution to the Teukolsky equation in the strictly static situation
differs from the static limit of the dynamical solution for the perturbation
of a reflecting compact object.

The above solution for the radial part of the Teukolsky function
$r^{4}\Psi_{4}$ holds for any values of $z$, and hence we can start by
considering the near-horizon, i.e., $z\to 0$ limit, as well as the far zone,
which corresponds to the large-$z$ limit of it. Let us first explore the far-
zone region. In this case, the associated Legendre polynomial
$P_{\ell}^{2}(1+2z)$ can be expanded as,

$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}\simeq\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z^{2}$
$\displaystyle\Big{[}\frac{\Gamma(3)\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell)}z^{-3-\ell}$
$\displaystyle+\frac{\Gamma(3)\Gamma(1+2\ell)}{\Gamma(\ell+3)\Gamma(1+\ell)}z^{\ell-2}\Big{]}~{}.$
(43)

Even though the ratio $\\{\Gamma(-1-2\ell)/\Gamma(-\ell)\\}$ has a finite
limit for integer values of $\ell$, the term $\Gamma(2-\ell)$ diverges for any
$\ell\geq 2$, and hence, the coefficient of the term $z^{-3-\ell}$ identically
vanishes. Therefore, for $P_{\ell}^{2}$, we obtain the following result in the
far-zone limit,

$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}=\frac{\Gamma(1+2\ell)}{\Gamma(\ell-1)\Gamma(1+\ell)}\left(\frac{r}{2M}\right)^{\ell}~{},$
(44)

such that for $\ell=2$, we obtain,

$\displaystyle P_{2}^{2}(1+2z)\Big{|}_{\rm
far}=3\left(\frac{r}{M}\right)^{2}~{},$ (45)

which matches exactly with the expressions derived in Ref. [72]. On a similar
footing, the other associated Legendre polynomial $Q_{\ell}^{2}$ in the far-
zone reads,

$\displaystyle Q_{\ell}^{2}(1+2z)\Big{|}_{\rm
far}=\frac{\sqrt{\pi}\Gamma(3+\ell)}{2^{2\ell+2}\Gamma(\ell+\frac{3}{2})}z^{-\ell-1}~{}.$
(46)

Thus, among the two independent solutions of the Teukolsky equation,
$P_{\ell}^{2}$ provides the tidal field, while $Q_{\ell}^{2}$ yields the
response of the non-rotating compact object to the tidal field. Note that, for
$\ell=2$, the far-zone limit of $Q_{2}^{2}(1+2z)$ reduces to $(8/5)(M/r)^{3}$,
which is again consistent with [72]. Therefore, the radial part of the
Teukolsky function, from 40, can be expressed as,

$\displaystyle R_{\ell}\Big{|}_{\rm far}$
$\displaystyle=c_{1}\frac{\Gamma(1+2\ell)}{\Gamma(\ell-1)\Gamma(1+\ell)}\left(\frac{r}{2M}\right)^{\ell+2}$
$\displaystyle\qquad\times\left[1+\mathcal{F}_{\ell}\left(\frac{r}{2M}\right)^{-2\ell-1}\right]~{}.$
(47)

Here, $\mathcal{F}_{\ell}$ is the tidal response function, which is purely
real, and thus the TLN becomes,

$\displaystyle
k_{\ell}=\frac{c_{2}}{c_{1}}\frac{\sqrt{\pi}\Gamma(3+\ell)}{2^{2\ell+3}\Gamma(\ell+\frac{3}{2})}\frac{\Gamma(\ell-1)\Gamma(1+\ell)}{\Gamma(1+2\ell)}~{}.$
(48)

Therefore the determination of the TLN boils down to determining the ratio
$(c_{2}/c_{1})$. For BHs, one can show that at the horizon, the associated
Legendre polynomials $P_{\ell}^{2}$ read

$\displaystyle P_{\ell}^{2}(1+2z)\Big{|}_{\rm near}$
$\displaystyle=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}z(1+z)$
$\displaystyle=\frac{\Gamma(\ell+3)}{2\Gamma(\ell-1)}\left(\frac{r}{2M}\right)^{2}\left(1-\frac{2M}{r}\right)~{},$
(49)

which is well-behaved at $r=2M$. While the other branch of the associated
Legendre polynomial becomes,

$\displaystyle Q_{\ell}^{2}(1+2z)\Big{|}_{\rm near}$
$\displaystyle=\frac{\sqrt{\pi}}{2^{\ell-1}}\frac{\Gamma(\ell+3)}{2\Gamma(\ell+\frac{3}{2})}\frac{z(1+z)}{(1+2z)^{\ell+3}}$
$\displaystyle\times\frac{\Gamma(\ell+\frac{3}{2})\Gamma(-2)}{\Gamma(\frac{\ell}{2}-\frac{1}{2})\Gamma(\frac{\ell}{2})}~{},$
(50)

which is ill-behaved due to the appearance of $\Gamma(-2)$ term. Thus,
regularity at the horizon demands $c_{2}=0$ in 40 and hence the BH TLNs
identically vanish. This is consistent with the static limit of the dynamical
TLNs associated with the BH spacetimes, as the term which is ill-behaved in
the static limit is also ill-behaved on the horizon, and hence does not appear
in the BH spacetime. For compact objects other than BHs, on the other hand,
one has to impose appropriate boundary conditions, e.g., in the perfectly
reflecting case one may impose the condition that the Zerilli and the Regge-
Wheeler function should vanish on the surface. Thus, to compare the above
result derived solely from the Teukolsky equation in the zero-frequency and
zero-rotation limit with those in the literature, we must connect the
Teukolsky function with the metric perturbations in the axial and polar
sectors. This is what we do next.

In the non-rotating case, the radial part of the Teukolsky function can be
decomposed into axial and polar parts, $R_{\ell}=R_{\ell}^{\rm
axial}+R_{\ell}^{\rm polar}$, each of which can be expressed in terms of the
Regge-Wheeler and the Zerilli functions respectively, such that in the zero-
frequency limit we obtain (see D for details),

$\displaystyle\frac{R_{\ell}^{\textrm{axial}}}{\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}$
$\displaystyle=\frac{r^{3}}{8}\Bigg{[}V^{\rm axial}\Psi^{\rm RW}_{\ell}+W^{\rm
axial}\left(\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}\right)\Bigg{]}~{},$ (51)
$\displaystyle\frac{R_{\ell}^{\textrm{polar}}}{\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}$
$\displaystyle=\frac{r^{3}}{8}\left[V^{\rm polar}\Psi^{\rm Z}_{\ell}+W^{\rm
polar}\left(\dfrac{d\Psi^{\rm Z}_{\ell}}{dr_{*}}\right)\right]~{},$ (52)

where, we have introduced

$\displaystyle V^{\rm axial}$ $\displaystyle=V_{\rm
RW}=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r^{4}}~{},$ (53)
$\displaystyle W^{\rm axial}$ $\displaystyle=\frac{2(r-3M)}{r^{2}}~{},$ (54)
$\displaystyle V^{\rm polar}$ $\displaystyle=V_{\rm
Z}=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\Big{[}\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}$
$\displaystyle+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\Big{]}~{},$
(55) $\displaystyle W^{\rm polar}$
$\displaystyle=\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}~{}.$
(56)

Here, $V_{\rm RW}$ is the Regge-Wheeler and $V_{\rm Z}$ is the Zerilli
potential. The above relations connect the radial part of the Teukolsky
function with the Regge-Wheeler and the Zerilli functions in the zero-
frequency limit. For our purpose, we wish to use the relations connecting the
Regge-Wheeler and the Zerilli functions with the axial metric perturbation
$h_{0}$ and the polar metric perturbation $H_{0}$, respectively, so that the
Teukolsky function can be directly related to the metric perturbations in the
zero-frequency case. We start with the polar sector, involving the functions
$V^{\rm polar}$ and $W^{\rm polar}$, which can be simplified further as

$\displaystyle V^{\rm polar}\Psi^{\rm Z}_{\ell}$ $\displaystyle+W^{\rm
polar}\left(\dfrac{d\Psi^{\rm
Z}_{\ell}}{dr_{*}}\right)=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\left[\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\right]\Psi^{\rm
Z}_{\ell}$
$\displaystyle+\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}\left(\frac{r-2M}{r}\right)\frac{d\Psi^{\rm
Z}_{\ell}}{dr}$
$\displaystyle=\frac{2(r-2M)}{r^{2}}\Bigg{[}\frac{\gamma_{\ell}r^{2}-3\gamma_{\ell}Mr-6M^{2}}{r(\gamma_{\ell}r+6M)}\frac{d\Psi^{\rm
Z}_{\ell}}{dr}+\left(\frac{\gamma_{\ell}^{2}(\frac{\gamma_{\ell}}{2}+1)r^{2}+3M\gamma_{\ell}^{2}r+18M^{2}\gamma_{\ell}+36\frac{M^{3}}{r}}{r(\gamma_{\ell}r+6M)^{2}}\right)\Psi^{\rm
Z}_{\ell}\Bigg{]}$ $\displaystyle=\frac{2(r-2M)}{r^{2}}H_{0}~{},$ (57)

where, we have used the relation between the polar metric perturbation $H_{0}$
and the Zerilli function $\Psi^{\rm Z}_{\ell}$ [22]. Therefore, the radial
part of the Teukolsky equation associated with the polar perturbation becomes,

$\displaystyle R_{\ell}^{\rm
polar}=\frac{r(r-2M)}{4}\sqrt{\gamma_{\ell}(\gamma_{\ell}+2)}H_{0}~{}.$ (58)

In the $z$ coordinate system, the above relation simply becomes $R_{\ell}^{\rm
polar}\propto z(1+z)H_{0}$. Thus, from 40, we can express the polar metric
perturbation $H_{0}$ in terms of the associated Legendre polynomials
$P_{\ell}^{2}$, $Q_{\ell}^{2}$, and the arbitrary constants linked with them.
Thus, imposing for example Dirichlet boundary conditions on the Zerilli
function at the surface of the compact object (note that this is the same
boundary condition imposed in [22] for a perfectly-reflective mirror), i.e.,
setting $\Psi_{\ell}^{\rm Z}(r_{*}^{0})=0$, we can determine the ratio of the
arbitrary constants, which in turn will yield the TLNs. For the $\ell=2$ mode,
we obtain

$\displaystyle k_{2}^{\rm polar}=\frac{8}{5\left(7+3\ln\epsilon\right)}~{},$
(59)

which exactly matches with [22]. Thus, the TLNs in the strictly zero-frequency
case can indeed be obtained by solving the Teukolsky equation in the non-
rotating scenario. Therefore, the formalism developed here can be reconciled
with earlier results in the literature [22].

For the axial case, we note that in the zero frequency limit the metric
perturbation $h_{0}$ can be expressed as,

$\displaystyle h_{0}$ $\displaystyle=\dfrac{d}{dr_{*}}\left(r\Psi_{\ell}^{\rm
RW}\right)=r\dfrac{d\Psi_{\ell}^{\rm
RW}}{dr_{*}}+\left(\frac{r-2M}{r}\right)\Psi_{\rm RW}~{},$ (60) $\displaystyle
h_{0}^{\prime}$ $\displaystyle=\left(\frac{r^{2}}{r-2M}\right)V_{\rm
RW}\Psi_{\rm RW}+2\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}$
$\displaystyle\qquad+\Psi_{\ell}^{\rm RW}\left(1+\frac{2M}{r^{2}}\right)~{},$
(61)

where the prime denotes the derivative with respect to the radial coordinate
$r$. Given the above expressions for the metric perturbation $h_{0}$ and its
derivative $h_{0}^{\prime}$ in terms of the Regge-Wheeler function and its
first derivative, we can consider the following combination
$h_{0}^{\prime}-(2h_{0}/r)$, which does not involve any derivatives of the
Regge-Wheeler function, such that,

$\displaystyle\Psi_{\ell}^{\rm
RW}=\frac{r^{3}}{\gamma_{\ell}}\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)~{}.$
(62)

Having expressed the Regge-Wheeler function in terms of $h_{0}$ and its
derivative, we can use 60 to write down the derivative of the Regge-Wheeler
function as well in a similar manner,

$\displaystyle\dfrac{d\Psi_{\ell}^{\rm RW}}{dr_{*}}$
$\displaystyle=\frac{h_{0}}{r}-\left(\frac{r-2M}{r^{2}}\right)\Psi_{\rm RW}$
$\displaystyle=\frac{h_{0}}{r}-\left(\frac{r(r-2M)}{\gamma_{\ell}}\right)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)~{}.$
(63)

Such that, we obtain the following combination of the Regge-Wheeler function
and its derivative, to read

$\displaystyle V^{\rm axial}\Psi^{\rm RW}_{\ell}$ $\displaystyle+W^{\rm
axial}\left(\dfrac{d\Psi_{\ell}^{\rm
RW}}{dr_{*}}\right)=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r\gamma_{\ell}}\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)$
$\displaystyle\qquad\qquad+\frac{2(r-3M)}{r^{2}}\left[\frac{h_{0}}{r}-\left(\frac{r(r-2M)}{\gamma_{\ell}}\right)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)\right]$
$\displaystyle=(r-2M)\dfrac{d}{dr}\left(\frac{h_{0}}{r^{2}}\right)+\frac{2(r-3M)}{r^{3}}h_{0}=\frac{(r-2M)}{r^{2}}\dfrac{dh_{0}}{dr}-\frac{2M}{r^{3}}h_{0}~{}.$
(64)

Therefore, the axial part of the Teukolsky radial function can be expressed in
terms of the metric perturbation $h_{0}$ as,

$\displaystyle R_{\ell}^{\textrm{axial}}$
$\displaystyle=\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}\left[\frac{r(r-2M)}{8}\dfrac{dh_{0}}{dr}-\frac{M}{4}h_{0}\right]~{}.$
(65)

The situation for generic choices of $\ell$ is more involved, as the axial
metric perturbation $h_{0}$ depends on the MeijerG function, with complicated
analytical properties. Thus, we will demonstrate the validity of the above
equation for the $\ell=2$ case, which will also be consistent with the results
in earlier literature. For the $\ell=2$ mode, the solution of the Teukolsky
equation reads

$\displaystyle R_{\ell}^{\textrm{axial}}$
$\displaystyle=c_{1}\left(\frac{3r^{2}}{M^{2}}\right)\left(r-2M\right)^{2}$
$\displaystyle+c_{2}\left(\frac{r-2M}{M}\right)\left(3r^{2}-6Mr-2M^{2}\right)$
$\displaystyle+c_{2}\left(\frac{3r^{2}}{2M^{2}}\right)\left(r-2M\right)^{2}\ln\left(1-\frac{2M}{r}\right)~{}.$
(66)

When substituted in 65, the above solution for the Teukolsky radial function
yields the following expression for the axial metric perturbation $h_{0}$,

$\displaystyle h_{0}$
$\displaystyle=c_{1}r^{2}(r-2M)+\frac{c_{2}}{24M^{5}r}\Big{[}4M^{4}+4M^{3}r+6M^{2}r^{2}$
$\displaystyle+6Mr^{3}\ln\left(1-\frac{2M}{r}\right)-3r^{4}\ln\left(1-\frac{2M}{r}\right)\Big{]}~{}.$
(67)

One can check that the above expression for the axial metric perturbation
$h_{0}$ satisfies the perturbed Einstein’s equations in the static limit.
Again, given this solution, one can impose Dirichlet boundary condition on the
Regge-Wheeler function at the surface of the compact object, by setting
$\Psi^{\rm RW}_{2}(r_{*}^{0})=0$. This in turn will relate the two unknown
coefficients $c_{1}$ and $c_{2}$ and one arrives at the following expression
for the TLN in terms of the compactness parameter $\epsilon$,

$\displaystyle k_{2}^{\rm
axial}=\frac{32}{5\left(25+12\ln\epsilon\right)}~{}.$ (68)

Again this coincides with the corresponding expression presented in [22] and
has the distinctive $\log\epsilon$ structure. Thus, the static limit of the
Teukolsky equation and the corresponding solution indeed provide the TLNs with
logarithmic dependence, consistent with earlier literature. In summary, the
TLNs in the $\omega=0$ case differ from the dynamical TLNs in the $\omega\to
0$ limit in numerical coefficients, though in both cases the logarithmic
dependence of the TLNs on $\epsilon$ remains. Thus, the strictly zero
frequency case is a set of measure zero, as it corresponds to a single
disjoint point in the frequency plane.

##  IV Discussion

TLNs provide a unique opportunity to test the nature of compact objects, as
well as the underlying gravitational theory, since only for BHs in GR, the
TLNs identically vanish. Earlier results regarding the TLNs of ECOs depicted
an astounding behavior, namely the TLNs are non-zero and behave as
$\log\epsilon$ in the BH limit, where $\epsilon$ is the difference between the
surface of the ECO and the location of the would-be horizon. This suggested
that even if $\epsilon$ is small, maybe of the Planckian order, the effect on
the TLNs are significant. However, such a result was derived in the context of
a static tidal field for non-rotating ECOs, while in practice all the
deformations are dynamic and all compact objects are rotating. Thus, it is
important to see if such a logarithmic behavior persists even in the dynamical
context for rotating ECOs. The inclusion of rotation and dynamics
automatically urges one to employ the Teukolsky formalism and it turns out
that the Weyl scalars entering the Teukolsky formalism are closely related to
the Newtonian potential in the appropriate limits. Thus, the Weyl scalars
provide the desired quantity that mimics the role of Newtonian potential in
the relativistic context and the Teukolsky equation governs the response
function of a compact object under an external tidal field. In this work, we
have solved the Teukolsky equation with a generic reflective boundary
condition at the surface of the ECO. This in turn leads to a modified tidal
response function, and hence on a TLN depending on the reflectivity of the
ECO. Since the mode functions are not plane waves in the Teukolsky formalism,
we converted the reflectivity in the Teukolsky formalism to the Detweiler
formalism using appropriate transformations, and presented the TLNs in terms
of the parameters of the ECO and the Detweiler reflectivity, ${\cal R}$. To
our surprise, it turns out that for generic reflectivity the zero-frequency
TLNs of a nonspinning compact object identically vanish and non-zero TLNs can
be obtained if and only if $\mathcal{R}(\omega\to 0)=1$. Intriguingly, the
$\log\epsilon$ behavior of the TLNs persists even in the zero-frequency limit
of the dynamical TLNs, but it differs in numerical coefficients from the
strictly static case and depends on the frequency-dependent part of the
reflectivity. Moreover, as the rotation of the compact object increases the
TLNs decrease sharply, implying that fast-rotating objects are difficult to
deform by applying external tidal forces. On the other hand, as the
reflectivity increases, the TLNs also increase, implying that the more
reflective an object is, the higher is its deformation under an external tidal
field. Further, we observe certain oscillation patterns in the TLNs, depending
on the parameter $\epsilon$. Also, both in the non-rotating and the rotating
case, there can be resonances in the TLNs, which happens at higher frequencies
for smaller reflectivities. Hence the dynamics of the TLNs are highly non-
trivial, as they depend on the parameter $\epsilon$ in an oscillatory manner,
while the amplitude of the TLNs depend crucially on the reflectivity and the
rotation of the compact object.

The situation in the zero-frequency limit needs an elaborate discussion.
Naively one would expect to arrive at the same result obtained in [22] for
static perturbations also from the zero-frequency limit of the dynamical TLNs.
However, this turns out not to be the case, because the solutions of the
Teukolsky equation in the frequency-dependent case become singular in the
zero-frequency limit for non-rotating compact objects. Therefore, the
$\omega=0$ is an isolated point, which cannot be arrived at by taking
$\omega\to 0$, suggesting that the strictly static behavior of the TLNs is a
set of measure zero in the phase space. In particular, the TLNs generically
vanish in the zero-frequency limit, except for the following reflectivity of
the compact object: $\mathcal{R}=1+i\omega
M\mathcal{R}_{1}+\mathcal{O}(M^{2}\omega^{2})$. Thus, the reflectivity must
have a frequency dependence, for the zero-frequency TLNs to be non-zero.
Moreover, the parameter $\mathcal{R}_{1}$ explicitly appears in the
reflectivity, and can potentially modify the dependence on $\epsilon$. The
$\log\epsilon$ behavior, though generic, can be eliminated by an appropriate
choice of $\mathcal{R}_{1}$. Thus, we arrive at the following conclusion:
static TLNs correspond to an idealized situation since the inspiral of a
binary system is _always_ dynamical. Moreover, the static limit of the
dynamical TLNs does _not_ coincide with the TLNs derived in the strictly
static case. Finally, when they are non-zero the TLNs vanish in the BH limit
with a typical logarithmic behavior as a function of the distance between the
surface of the compact object and the would-be horizon. This makes the
detection of non-zero TLNs for ECOs a tantalizing possibility for future
generations of GW detectors [22, 57].

There are several future directions of exploration. For example, whether such
behavior for the dynamical TLNs is observed also in the effective field theory
description, remains to be seen, see, e.g., two recent papers [73, 45]. The
first one claims the existence of logarithmic divergences in the field theory,
which must be renormalized to arrive at a finite TLN [73], while the second
one arrives at a $\log(r/2M)$ term in the TLN [45]. Both of these logarithmic
features are very different from the one discussed here in the context of
static TLNs. A detailed comparison with our results deserves further study.
Also, the decomposition of the TLNs in the axial and polar sectors is not very
clear in the context of Weyl scalars, governed by the Teukolsky equation.
Furthermore, it could be interesting to reassess the detectability of the TLNs
from binary coalescence signals, as the prospects are optimistic owing to the
logarithmic scaling [22, 57]. From the theoretical standpoint, our analysis
could be extended to background geometries with fewer symmetries than Kerr
[27], although that would require going beyond the Teukolsky formalism.
Finally, it is possible to map our general framework for TLNs into the
membrane paradigm [74, 75, 76], which can be used to describe ECOs as recently
developed in Ref. [48] (see also [77, 66] for recent work on the subject).
Work along this direction is underway [78].

###### Acknowledgements.

Research of S.C. is funded by the INSPIRE Faculty fellowship from the DST,
Government of India (Reg. No. DST/INSPIRE/04/2018/000893). E.M. acknowledges
funding from the Deutsche Forschungsgemeinschaft (DFG) - project number:
386119226. P.P. acknowledges the financial support provided under the European
Union’s H2020 ERC, Starting Grant agreement no. DarkGRA–757480, under MIUR
PRIN (Grant 2020KR4KN2 “String Theory as a bridge between Gauge Theories and
Quantum Gravity”) and FARE (GW-NEXT, CUP: B84I20000100001, 2020KR4KN2)
programs, and support from the Amaldi Research Center funded by the MIUR
program “Dipartimento di Eccellenza” (CUP: B81I18001170001).

##  Appendix A Connecting the near-horizon behavior of the Teukolsky function
to the Detweiler variable

The first step of connecting the near-horizon behavior of the radial Teukolsky
function in the $(v,r,\theta,\tilde{\phi})$ coordinate with the Detweiler
function, is to transform the radial Teukolsky function in the Boyer-Lindquist
coordinate system. For that the radial function in the Boyer-Lindquist
coordinate will have an additional factor of $e^{-i\omega r_{*}}$ (as,
$v=t+r_{*}$) and also a factor of $e^{im\bar{r}_{*}}$ (as,
$\tilde{\phi}=\phi+\bar{r}_{*}$) in the near horizon limit. Therefore the
radial Teukolsky function in the $(t,r,\theta,\phi)$ coordinate system
becomes,

$\,{}_{-2}R_{\ell
m}^{(t)}=\mathcal{A}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\omega
r_{*}}e^{im\bar{r}_{*}}+\mathcal{B}e^{-i\omega
r_{*}}e^{im\bar{r}_{*}}e^{2i\bar{\omega}r_{*}}$ (69)

where the tortoise coordinate $r_{*}$ and the azimuthal tortoise coordinate
$\bar{r}_{*}$ are given by,

$\displaystyle\frac{dr_{*}}{dr}=\frac{r^{2}+a^{2}}{\Delta}~{};\qquad\frac{d\bar{r}_{*}}{dr}=\frac{a}{\Delta}~{}.$
(70)

In the near horizon limit, both of these tortoise coordinates get related to
each other, such that,

$\displaystyle
d\bar{r}_{*}=\frac{a}{\Delta}dr=\frac{a}{r_{+}^{2}+a^{2}}\frac{r_{+}^{2}+a^{2}}{\Delta}dr=\frac{a}{r_{+}^{2}+a^{2}}dr_{*}~{}.$
(71)

Hence the radial Teukolsky function in the Boyer-Lindquist coordinate system
can expressed solely in terms of the tortoise coordinate $r_{*}$ and hence the
radial function becomes,

$\,{}_{-2}R_{\ell
m}^{(t)}=\mathcal{A}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\mathcal{B}e^{i\bar{\omega}r_{*}}~{},$
(72)

where,

$\displaystyle\bar{\omega}=\omega-\frac{am}{r_{+}^{2}+a^{2}}=\omega-m\Omega_{\rm
H}~{}.$ (73)

Here, $\Omega_{\rm H}$ is the angular velocity of the horizon. The above
expression for the radial Teukolsky function in the Boyer-Lindquist coordinate
has been used in the main text.

The next task is to transform the near-horizon radial Teukolsky function in
the Boyer-Lindquist coordinate to Detweiler function, through 17. Since the
transformation is singular on the horizon, the near horizon limit must be
taken with care. For this purpose, we expand the quantities $\mathcal{A}$ and
$\mathcal{B}$ in the near horizon regime as, $\mathcal{A}=A_{0}$ and
$\mathcal{B}=B_{0}+B_{1}\Delta+B_{2}\Delta^{2}$, where $A_{0}$, $B_{0}$,
$B_{1}$ and $B_{2}$ are constants, independent of the radial coordinate. Thus,
the radial Teukolsky function in the Boyer-Lindquist coordinate becomes

$\displaystyle\,_{-2}R_{\ell
m}^{(t)}=A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}~{}.$
(74)

Of course, the quantities $A_{0}$, $B_{0}$, $B_{1}$ and $B_{2}$ cannot be
arbitrary, rather they will be fixed by the Teukolsky equation. One
substitutes the above expression for the radial perturbation in the radial
Teukolsky equation and demands that the radial Teukolsky equation will be
satisfied at all orders in $\Delta$, thereby uniquely fixing the above
quantities. In what follows, we will choose these quantities to have precisely
those values, which are consistent with the radial Teukolsky equation.

Moving forward, the determination of the Detweiler function requires
determining the derivative of the radial Teukolsky function with respect to
$r$ in the Boyer-Lindquist coordinate, which yields,

$\displaystyle\dfrac{d}{dr}\left({}_{-2}R^{(t)}_{\ell m}\right)$
$\displaystyle=A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}\left(-\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+A_{0}\frac{2\Delta}{(r_{+}-r_{-})^{3}}e^{-i\bar{\omega}r_{*}}$
$\displaystyle+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+\left\\{B_{1}(r_{+}-r_{-})+2B_{2}\Delta(r_{+}-r_{-})\right\\}e^{i\bar{\omega}r_{*}}~{}.$
(75)

Substituting the above expression for the radial derivative of the radial
Teukolsky function, along with the radial Teukolsky function itself in 17, we
obtain the following expression for the Detweiler function,

$\,{}_{-2}X_{\ell m}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\left[\frac{\alpha}{\Delta}R_{-2}^{(t)}+\frac{\beta}{\Delta^{2}}\dfrac{dR_{-2}^{(t)}}{dr}\right]$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}\frac{\alpha}{\Delta}\left\\{A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\right\\}$
$\displaystyle\quad+\frac{\beta}{\Delta^{2}}\Bigg{\\{}A_{0}\frac{\Delta^{2}}{(r_{+}-r_{-})^{4}}e^{-i\bar{\omega}r_{*}}\left(-\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+A_{0}\frac{2\Delta}{(r_{+}-r_{-})^{3}}e^{-i\bar{\omega}r_{*}}$
$\displaystyle\quad+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)e^{i\bar{\omega}r_{*}}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta}\right)+\left\\{B_{1}(r_{+}-r_{-})+2B_{2}\Delta(r_{+}-r_{-})\right\\}e^{i\bar{\omega}r_{*}}\Bigg{\\}}\Bigg{]}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\left[\frac{\alpha\Delta}{(r_{+}-r_{-})^{4}}+\frac{2\beta}{\Delta(r_{+}-r_{-})^{3}}-\frac{\beta}{\Delta}\left(\frac{i\bar{\omega}(r_{+}^{2}+a^{2})}{(r_{+}-r_{-})^{4}}\right)\right]A_{0}e^{-i\bar{\omega}r_{*}}$
$\displaystyle\quad+\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}B_{0}\frac{\alpha}{\Delta}+B_{1}\alpha+B_{2}\alpha\Delta+B_{1}(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+2B_{2}(r_{+}-r_{-})\frac{\beta}{\Delta}$
$\displaystyle\quad+\left(B_{0}+B_{1}\Delta+B_{2}\Delta^{2}\right)\left(\frac{i\beta\bar{\omega}(r_{+}^{2}+a^{2})}{\Delta^{3}}\right)\Bigg{]}e^{i\bar{\omega}r_{*}}$
$\displaystyle\equiv\mathcal{A}_{X}e^{-i\bar{\omega}r_{*}}+\mathcal{B}_{X}e^{i\bar{\omega}r_{*}}~{},$
(76)

where the two radial coordinate dependent quantities, namely $\mathcal{A}_{X}$
and $\mathcal{B}_{X}$, sitting in front of the ingoing and outgoing solution
near the horizon can be expressed as,

$\displaystyle\mathcal{A}_{X}$
$\displaystyle=\frac{\sqrt{r_{+}^{2}+a^{2}}}{(r_{+}-r_{-})^{4}}\left[\alpha\Delta+\left\\{2(r_{+}-r_{-})-i\bar{\omega}(r_{+}^{2}+a^{2})\right\\}\frac{\beta}{\Delta}\right]A_{0}$
(77) $\displaystyle\mathcal{B}_{X}$
$\displaystyle=\sqrt{r_{+}^{2}+a^{2}}\Bigg{[}B_{0}\left\\{\frac{\alpha}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{3}}\right\\}+B_{1}\left\\{\alpha+(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{2}}\right\\}$
$\displaystyle\qquad+B_{2}\left\\{\alpha\Delta+2(r_{+}-r_{-})\frac{\beta}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta}\right\\}\Bigg{]}~{}.$
(78)

Thus, the reflectivity of the compact object, whose surface is very close to
the BH horizon, can be determined by taking the ratio
$(\mathcal{B}_{X}/\mathcal{A}_{X})$, which reads,

$\displaystyle\mathcal{R}$ $\displaystyle=\lim_{r\rightarrow
r_{0}}(r_{+}-r_{-})^{4}\left[\alpha\Delta+\left\\{2(r_{+}-r_{-})-i\bar{\omega}(r_{+}^{2}+a^{2})\right\\}\frac{\beta}{\Delta}\right]^{-1}\Bigg{\\{}\Bigg{[}\frac{B_{0}}{A_{0}}\left\\{\frac{\alpha}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{3}}\right\\}$
$\displaystyle\qquad+\frac{B_{1}}{A_{0}}\left\\{\alpha+(r_{+}-r_{-})\frac{\beta}{\Delta^{2}}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta^{2}}\right\\}+\frac{B_{2}}{A_{0}}\left\\{\alpha\Delta+2(r_{+}-r_{-})\frac{\beta}{\Delta}+i\bar{\omega}(r_{+}^{2}+a^{2})\frac{\beta}{\Delta}\right\\}\Bigg{]}\Bigg{\\}}~{}.$
(79)

Note that the coefficients $A_{0}$, $A_{1}$, $B_{0}$, $B_{1}$ and $B_{2}$,
appearing in the above expression for the Detweiler reflectivity, are to be
fixed by demanding that the radial function with the above coefficients, as in
74, follows the Teukolsky equation. Moreover, the reflectivity depends on the
quantities $\alpha$ and $\beta$ appearing in the transformation from the
Teukolsky to the Detweiler functions and have very specific behavior near the
horizon, which must also be taken into account. All in all, the above limit
needs to be taken carefully, since there can be various divergent
contributions canceling each other, yielding a finite value for the
reflectivity $\mathcal{R}$ as a function of the object parameters, frequency,
the ratio $\mathcal{B}/\mathcal{A}$ and of course $\epsilon$. This is the
result we have used in the main text.

##  Appendix B Analytical expression for the TLN in the small-frequency limit

In this appendix, we provide the derivation of the analytical expression for
the TLN for rotating spacetimes in the small-frequency regime, i.e.,
$M\omega\ll 1$. The starting point is the radial Teukolsky equation in the
$(v,r,\theta,\phi)$ coordinate, as presented in 10. As emphasized in the main
text, besides the small frequency approximation, we also require the near-
horizon limit in order to arrive at an analytical expression for the
frequency-dependent TLN. The near-horizon expansion of 10 at the leading order
in the frequency reads [9]

$\displaystyle\dfrac{d^{2}\,_{-2}R^{(v)}_{\ell
m}}{dz^{2}}+\left[\frac{2iP_{+}-1}{z}-\frac{2iP_{+}+1}{z+1}\right]\dfrac{d\,_{-2}R^{(v)}_{\ell
m}}{dz}+\left[\frac{4iP_{+}}{(1+z)^{2}}-\frac{4iP_{+}}{z^{2}}-\frac{\ell(\ell+1)-2}{z(1+z)}\right]\,_{-2}R^{(v)}_{\ell
m}=0~{},$ (80)

where we have defined the quantity $z$ as,

$\displaystyle z\equiv\frac{r-r_{+}}{r_{+}-r_{-}}~{};\qquad
P_{+}=\frac{am-2M\omega
r_{+}}{r_{+}-r_{-}}=-\frac{2Mr_{+}\bar{\omega}}{r_{+}-r_{-}}~{},$ (81)

and the expression for $P_{+}$ follows from 11. Further, we have introduced
the following definition for rescaled frequency $\bar{\omega}$ as,
$\bar{\omega}\equiv\omega-m\Omega_{\rm H}$, where $\Omega_{\rm H}$ is the
angular velocity of the horizon. The above equation can be explicitly solved
in terms of the Hypergeometric functions, yielding,

$\displaystyle\,_{-2}R^{(v)}_{\ell m}$
$\displaystyle=\mathcal{A}z^{2}(1+z)^{2}~\,{}_{2}F_{1}\left(2-\ell,3+\ell,3+2iP_{+};-z\right)$
$\displaystyle\qquad\qquad+\mathcal{B}z^{-2iP_{+}}(1+z)^{2}~\,{}_{2}F_{1}\left(-\ell-2iP_{+},1+\ell-2iP_{+},-1-2iP_{+};-z\right)~{},$
(82)

where, $\mathcal{A}$ and $\mathcal{B}$ are the arbitrary constants of
integration. Therefore, close to the horizon we can use the property that
$\lim_{x\rightarrow 0}\,{}_{2}F_{1}(a,b;c;x)=1$, in order to obtain the
following expression for the radial Teukolsky function:

$\,{}_{-2}R^{(v)}_{\ell m}\Big{|}_{\rm near-horizon}$
$\displaystyle=\mathcal{A}z^{2}+\mathcal{B}z^{-2iP_{+}}$
$\displaystyle=\frac{\mathcal{A}}{(r_{+}-r_{-})^{4}}\Delta^{2}+\mathcal{B}\exp\left[2i\left(\frac{r_{+}^{2}+a^{2}}{r_{+}-r_{-}}\right)\bar{\omega}\ln
z\right]~{}.$ (83)

In order to arrive at the first term in the second line, we have used the
relation between $z$ and the radial coordinate $r$, while the second term has
been obtained by relating the tortoise coordinate $r_{*}$ to the near-horizon
coordinate $z$. The determination of the arbitrary constants $\mathcal{A}$ and
$\mathcal{B}$ are to be performed by imposing appropriate boundary conditions
on the radial Teukolsky wave function close to the horizon.

The asymptotic limit of the hypergeometric functions, on the other hand, can
be used to determine the radial part of the rescaled and perturbed Weyl scalar
$\rho^{4}\Psi_{4}$ in the intermediate regime. This region is far from the
surface of the compact object ($r\gg r_{+}$) but is much closer to the compact
object than the distance of the compact object from the source of the tidal
field ($r\ll b$, where $b$ is the distance between the compact object and the
source of the tidal field). In this region, the radial part of the perturbed
and rescaled Weyl scalar, obtained in B, becomes,

$\,{}_{-2}R^{(v)}_{\ell m}\Big{|}_{\rm far}$
$\displaystyle=\mathcal{A}z^{4}\left\\{\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}z^{\ell-2}+\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}z^{-\ell-3}\right\\}$
$\displaystyle\qquad\qquad+\mathcal{B}z^{2-2iP_{+}}\left\\{\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}z^{\ell+2iP_{+}}+\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}z^{2iP_{+}-1-\ell}\right\\}$
$\displaystyle=z^{4}\Bigg{[}\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}\right\\}z^{\ell-2}$
$\displaystyle\qquad\qquad+\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}\right\\}z^{-\ell-3}\Bigg{]}$
$\displaystyle=z^{\ell+2}\left\\{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}\right\\}\Bigg{[}1+\mathcal{F}_{\ell}z^{-2\ell-1}\Bigg{]}~{}.$
(84)

A comparison of $\,{}_{-2}R^{(v)}_{\ell m}$, derived above, depicting the
radial part of the rescaled and perturbed Weyl scalar $\rho^{4}\Psi_{4}$ in
the intermediate zone, with II.2.2 reveals that one can identify the tidal
response function $\mathcal{F}_{\ell}$ (note that the response function being
independent of the azimuthal number $m$, we have omitted it from the
subscript) to have the following expression (see also [67])

$\displaystyle\mathcal{F}_{\ell}$
$\displaystyle=\frac{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(-\ell-2iP_{+})\Gamma(-2-\ell)}}{\mathcal{A}\frac{\Gamma(3+2iP_{+})\Gamma(1+2\ell)}{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}+\mathcal{B}\frac{\Gamma(-1-2iP_{+})\Gamma(1+2\ell)}{\Gamma(1+\ell-2iP_{+})\Gamma(\ell-1)}}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+2\ell)}\left[\frac{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{2}}{1+\frac{\mathcal{B}}{\mathcal{A}}\Gamma_{1}}\right]~{},$
(85)

where, we have introduced two quantities $\Gamma_{1}$ and $\Gamma_{2}$,
dependent on the frequency and the angular number $\ell$ as,

$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})}{\Gamma(1+\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(\ell-1)}~{},$
(86) $\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(-2-\ell)}~{}.$
(87)

The real part of the tidal response $\mathcal{F}_{\ell}$ provides the TLN.
Note that the above expression for the tidal response function can be further
simplified, along with the quantities $\Gamma_{1}$ and $\Gamma_{2}$, using
various identities involving Gamma functions with imaginary arguments. In
particular, we can express the term outside the square bracket in B as

$\displaystyle\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+2\ell)}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(1+\ell+2iP_{+})\Gamma(1+\ell-2iP_{+})\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(-\ell+2iP_{+})\Gamma(1+\ell-2iP_{+})\Gamma(1+2\ell)}$
$\displaystyle=\frac{\Gamma(3+\ell)\Gamma(-1-2\ell)}{\Gamma(2-\ell)\Gamma(1+2\ell)}\left[\frac{2\pi
P_{+}}{\sinh(2\pi
P_{+})}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]\left[\frac{\sin\left(-\pi\ell+2i\pi
P_{+}\right)}{\pi}\right]$
$\displaystyle=\left(-1\right)^{\ell}\frac{\Gamma(3+\ell)\Gamma(1+\ell)\Gamma(-1-2\ell)\Gamma(2+2\ell)}{\Gamma(1+\ell)\Gamma(-\ell)(\ell-1)\ell\Gamma(1+2\ell)\Gamma(2+2\ell)}\left[2iP_{+}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]$
$\displaystyle=\left(-1\right)^{\ell}\frac{\pi}{\sin(2\pi+2\pi\ell)}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)\frac{\sin(\pi+\pi\ell)}{\pi}\left[2iP_{+}\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)\right]$
$\displaystyle=-iP_{+}\left(\frac{(\ell+2)!(\ell-2)!}{(2\ell)!(1+2\ell)!}\right)\prod_{j=1}^{\ell}\left(j^{2}+4P_{+}^{2}\right)$
(88)

where, along with the results — (a) $\sin(ix)=i\sinh(x)$, (b)
$\sin(2x)=2\sin(x)\cos(x)$, and (c) $\Gamma(1+n)=n!$, we have also used the
following identities involving $\Gamma$ functions:

$\displaystyle\Gamma(z)\Gamma(1-z)$ $\displaystyle=\frac{\pi}{\sin(\pi
z)}~{},$ (89) $\displaystyle|\Gamma(1+n+ib)|^{2}$ $\displaystyle=\frac{\pi
b}{\sinh(\pi b)}\prod_{j=1}^{n}\left(j^{2}+b^{2}\right)~{};\qquad
j\in\mathbb{N}\,.$ (90)

The functions $\Gamma_{1}$ and $\Gamma_{2}$ can also be expressed as,

$\displaystyle\Gamma_{1}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\Gamma(-1-2iP_{+})\Gamma(1+\ell+2iP_{+})}{\Gamma(1+\ell-2iP_{+})\Gamma(3+2iP_{+})}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(\ell+2iP_{+}\right)\left(\ell-1+2iP_{+}\right)\times\cdots\times\left(3+2iP_{+}\right)}{\left(\ell-2iP_{+}\right)\left(\ell-1-2iP_{+}\right)\times\cdots\times\left(-2iP_{+}\right)\left(-1-2iP_{+}\right)}\,,$
(91) $\displaystyle\Gamma_{2}$
$\displaystyle=\frac{\Gamma(-1-2iP_{+})\Gamma(2-\ell)\Gamma(\ell-1)\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})\Gamma(-2-\ell)\Gamma(3+\ell)}\frac{\Gamma(3+\ell)}{\Gamma(\ell-1)}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\pi}{\sin\left[\pi(\ell-1)\right]}\frac{\sin\left[\pi(\ell+3)\right]}{\pi}\frac{\Gamma(-1-2iP_{+})\Gamma(-\ell+2iP_{+})}{\Gamma(-\ell-2iP_{+})\Gamma(3+2iP_{+})}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{\left(-2-2iP_{+}\right)\left(-3-2iP_{+}\right)\times\cdots\times\left(-\ell-2iP_{+}\right)}{\left(2+2iP_{+}\right)\left(1+2iP_{+}\right)\left(2iP_{+}\right)\times\cdots\times\left(-\ell+2iP_{+}\right)}$
$\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\frac{(-1)^{\ell-1}\left(2+2iP_{+}\right)\left(3+2iP_{+}\right)\times\cdots\times\left(\ell+2iP_{+}\right)}{(-1)^{\ell+2}\left(2+2iP_{+}\right)\left(-1-2iP_{+}\right)\left(-2iP_{+}\right)\times\cdots\times\left(\ell-2iP_{+}\right)}=-\Gamma_{1}\,,$
(92)

where, we have used the result,
$\sin[\pi(3+\ell)]=\sin[\pi(\ell-1)]\cos(4\pi)+\sin[4\pi]\cos[\pi(\ell-1)]=\sin[\pi(\ell-1)]$.
Therefore, the tidal response function depends on $\Gamma_{1}$ alone. This
result, along with the expression for $\Gamma_{1}$ has been presented in the
main text.

Having discussed the most general situation with arbitrary choices for $\ell$
and $a$, let us briefly touch upon the interesting limit, with $\ell=2$ and
$a=0$. In this case, it follows that $P_{+}=-2M\omega$ and the expression for
$\Gamma_{1}$ simplifies to,

$\displaystyle\Gamma_{1}$
$\displaystyle=24\frac{\Gamma(-1+4iM\omega)}{\Gamma(3+4iM\omega)}=\frac{24}{(2+4iM\omega)(1+4iM\omega)4iM\omega(-1+4iM\omega)}\simeq\frac{3i}{M\omega}~{}.$
(93)

Moreover, we also obtain,

$\displaystyle\Gamma_{2}=24\frac{\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}\simeq-\frac{3i}{M\omega}~{}.$
(94)

Thus, in the non-rotating case, with $\ell=2$, we obtain the following
expression for the response function

$\displaystyle F_{2}$
$\displaystyle=\frac{8iM\omega}{5!}\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\frac{1+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}}{1+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\frac{\Gamma(-1+4iM\omega)}{\Gamma(3+4iM\omega)}}$
$\displaystyle=\frac{8iM\omega}{5!}\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)}$
$\displaystyle\qquad\times\Bigg{[}\frac{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(3+4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)}\Bigg{]}~{}.$
(95)

Further simplification can be achieved by using the following identity,

$\displaystyle\Gamma(3-4iM\omega)$
$\displaystyle=(2-4iM\omega)(1-4iM\omega)(-4iM\omega)(-1-4iM\omega)(-2-4iM\omega)\Gamma(-2-4iM\omega)$
$\displaystyle=-16iM\omega\left(1+16M^{2}\omega^{2}\right)\left(1+4M^{2}\omega^{2}\right)\Gamma(-2-4iM\omega)~{},$
(96)

so that the response function becomes,

$\displaystyle F_{2}$
$\displaystyle=-\frac{1}{240}\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(-2-4iM\omega)}$
$\displaystyle\qquad\qquad\qquad\times\Bigg{[}\frac{\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{\Gamma(3+4iM\omega)+24\left(\frac{\mathcal{B}}{\mathcal{A}}\right)\Gamma(-1+4iM\omega)}\Bigg{]}~{}.$
(97)

Note that the above expression is in terms of the Teukolsky reflectivity
$(\mathcal{B}/\mathcal{A})$, which needs to be converted to the Detweiler
reflectivity $\mathcal{R}$. The steps involved in performing this modification
will be detailed in A, while here we simply quote the final expression for the
response function of the $\ell=2$ mode, in the non-rotating limit, in terms of
the Detweiler reflectivity,

$\displaystyle F_{2}$
$\displaystyle=-\frac{1}{240}\frac{\Gamma(3+4iM\omega)}{\Gamma(-2+4iM\omega)\Gamma(-2-4iM\omega)}$
$\displaystyle\qquad\qquad\qquad\times\Bigg{[}\frac{e^{4iM\omega(\epsilon+\ln\epsilon)}(2-iM\omega)\Gamma(-2+4iM\omega)\Gamma(3-4iM\omega)+\widetilde{\mathcal{R}}\Gamma(-1+4iM\omega)\Gamma(-2-4iM\omega)}{e^{4iM\omega(\epsilon+\ln\epsilon)}\Gamma(2-iM\omega)\Gamma(3+4iM\omega)+\widetilde{\mathcal{R}}\Gamma(-1+4iM\omega)}\Bigg{]}~{},$
(98)

where the frequency dependent quantity $\widetilde{\mathcal{R}}$ is related to
the Detweiler reflectivity $\mathcal{R}$ through the following relation,

$\displaystyle\widetilde{\mathcal{R}}=16e^{8\pi
M\omega}M\omega(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})\mathcal{R}~{}.$
(99)

Therefore, a comparison between the response function in B, obtained by using
the Teukolsky function, and the one in B, determined by the Detweiler
function, reveals the following connection between the two reflectivities,

$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}$
$\displaystyle=\frac{2}{3}\frac{e^{8\pi
M\omega}M\omega\mathcal{R}(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{e^{4iM\epsilon\omega}\epsilon^{4iM\omega}(2-iM\omega)}$
$\displaystyle=e^{8\pi
M\omega-4iM\omega\left(\epsilon+\ln\epsilon\right)}\frac{M\omega}{3}\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{1-\frac{iM\omega}{2}}\mathcal{R}~{}.$
(100)

Note that, the tortoise coordinate $r^{*}_{0}$ at the location of the
reflective surface can be expressed in terms of the parameter $\epsilon$,
which depicts the shift of the reflective surface from the would-be horizon as

$\displaystyle
r^{*}_{0}=2M\left(1+\epsilon\right)+2M\ln\epsilon=2M\left(1+\epsilon+\ln\epsilon\right)~{},$
(101)

and hence the relation between the Teukolsky and the Detweiler reflectivity
takes the following form,

$\displaystyle\frac{\mathcal{B}}{\mathcal{A}}=e^{8\pi
M\omega-2i\omega\left(r^{*}_{0}-2M\right)}\frac{(i+2M\omega+16iM^{2}\omega^{2}+32M^{3}\omega^{3})}{1-\frac{iM\omega}{2}}\left(\frac{M\omega\mathcal{R}}{3}\right)~{}.$
(102)

In the limit of $\omega\rightarrow 0$, the Teukolsky and the Detweiler
reflectivity gets related in a simple form, which reads,
$(\mathcal{B}/\mathcal{A})=(iM\omega\mathcal{R}/3)$. We will use this
particular form in the main text while discussing the static limit of the TLN
for a non-rotating BH.

##  Appendix C Teukolsky equation with quadratic-in-frequency corrections

We start by simplifying 10 by keeping terms up to
$\mathcal{O}(M^{2}\omega^{2})$. For this purpose, we note the following
identities among the parameters of 10,

$\displaystyle P_{+}-P_{-}$ $\displaystyle=-2M\omega~{},\qquad
P_{-}=P_{+}+2M\omega~{},$ (103) $\displaystyle A_{+}-A_{-}$
$\displaystyle=-2(r_{+}-r_{-})(P_{+}-P_{-})\omega-(r_{+}+2M)r_{+}\omega^{2}+(r_{-}+2M)r_{-}\omega^{2}$
$\displaystyle=-\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)~{},$ (104)
$\displaystyle B_{+}-B_{-}$ $\displaystyle=2\omega(r_{+}-r_{-})~{}.$ (105)

Besides, note that $E_{\ell
m}=\ell(\ell+1)+E_{1}a\omega+E_{2}a^{2}\omega^{2}$, where $E_{1}$ and $E_{2}$
depends on the angular and azimuthal numbers $\ell$ and $m$, along with the
mass of the BH. Changing the radial coordinate $r$ to a new coordinate $z$,
defined in 81, and using the above identities, the coefficient of $(dR/dz)$
term in the radial Teukolsky equation reads,

$\displaystyle\textrm{Coefficient~{}of}~{}\frac{dR}{dz}$
$\displaystyle=\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega\\{2M+(r_{+}-r_{-})(1+z)\\}}{(1+z)}\,,$
(106)

which in the near-horizon approximation ($z\approx 0$), reduces to,

$\displaystyle\textrm{Coefficient~{}of}~{}\frac{dR}{dz}\Bigg{|}_{\rm
near~{}horizon}$
$\displaystyle=\frac{2iP_{+}-1}{z}-\frac{1+2iP_{+}+2i\omega\\{2M+(r_{+}-r_{-})\\}}{(1+z)}~{}.$
(107)

Note that there are no terms of $\mathcal{O}(M^{2}\omega^{2})$ in the
coefficient of $(dR/dz)$. Similarly, the coefficient of the radial Teukolsky
function $R$, in the radial Teukolsky equation, becomes,

$\displaystyle\textrm{Coefficient~{}of}~{}R$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+8iM\omega}{(1+z)^{2}}-\frac{A_{+}+iB_{+}}{z(1+z)}+\frac{(A_{-}-A_{+})+i(B_{-}-B_{+})}{(1+z)}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+8iM\omega}{(1+z)^{2}}-\frac{\ell(\ell+1)-2+(E_{1}-2m)a\omega+2i\omega
r_{+}+\omega^{2}(-r_{+}^{2}+2Mr_{+}+E_{2}a^{2})}{z(1+z)}$
$\displaystyle\qquad+\frac{\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)-2i\omega(r_{+}-r_{-})}{(1+z)}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega\\{4M-(1+z)(r_{+}-r_{-})\\}+\omega^{2}(r_{+}-r_{-})(r_{+}+r_{-}-2M)(1+z)}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}(-r_{+}^{2}+2Mr_{+}+E_{2}a^{2})}{z(1+z)}~{},$
(108)

which depends on terms involving $M^{2}\omega^{2}$. Using the result that
$r_{+}+r_{-}=2M$, as well as $r_{+}-2M=-r_{-}$ and $r_{+}r_{-}=a^{2}$, along
with the near-horizon approximation, the coefficient of the Teukolsky radial
function in the Teukolsky equation becomes,

$\displaystyle\textrm{Coefficient~{}of}~{}R\Bigg{|}_{\rm near~{}horizon}$
$\displaystyle=-\frac{4iP_{+}}{z^{2}}+\frac{4iP_{+}+2i\omega\\{4M-(r_{+}-r_{-})\\}}{(1+z)^{2}}$
$\displaystyle\qquad-\frac{\ell(\ell+1)-2}{z(1+z)}+\frac{2ma\omega}{z(1+z)}\left(1-\frac{E_{1}}{2m}\right)-\frac{2i\omega
r_{+}}{z(1+z)}-\frac{\omega^{2}a^{2}(1+E_{2})}{z(1+z)}~{}.$ (109)

These expressions for the coefficients of $(dR/dz)$ and $R$ term in the radial
Teukolsky equation have been used in the main text.

##  Appendix D Zero-rotation limit of the Detweiler function

In this appendix, we will discuss the perturbation of compact objects having
zero rotation. In this limit, we first write down the relation between the
Detweiler and the Teukolsky function with $s=-2$, which reads [79],

$\,{}_{-2}X^{0}_{\ell m}$
$\displaystyle=\left\\{\left(\frac{r}{r^{2}-2Mr}\right)\alpha_{-2}^{0}\right\\}\,_{-2}R_{\ell
m}^{0}+\left\\{\left(\frac{r}{\left(r^{2}-2Mr\right)^{2}}\right)\beta_{-2}^{0}\right\\}\dfrac{\,{}_{-2}R_{\ell
m}^{0}}{dr}$
$\displaystyle=\left\\{\left(\frac{1}{r-2M}\right)\alpha_{-2}^{0}\right\\}\,_{-2}R_{\ell
m}^{0}+\left\\{\left(\frac{1}{\left(r-2M\right)^{3}}\right)\beta_{-2}^{0}\right\\}\dfrac{\,{}_{-2}R_{\ell
m}^{0}}{dr_{*}}\,,$ (110)

where $\alpha_{-2}^{0}$ and $\beta_{-2}^{0}$ are the coefficients necessary to
make the potential for the Detweiler function real with zero rotation and
$r_{*}$ is the tortoise coordinate. For $s=-2$ and in the zero rotation limit,
the functions $\alpha_{-2}^{0}$ and $\beta_{-2}^{0}$ read as follows [79],

$\displaystyle\alpha_{-2}^{0}$
$\displaystyle=\frac{\kappa(A_{1}+iA_{2})+|\kappa|^{2}}{\sqrt{2}|\kappa|\sqrt{A_{1}+\textrm{Re}\kappa}}~{},$
(111) $\displaystyle\beta_{-2}^{0}$ $\displaystyle=\frac{i\kappa
B_{2}}{\sqrt{2}|\kappa|\sqrt{A_{1}+\textrm{Re}\kappa}}~{},$ (112)

where, we have introduced the quantities $\kappa$, $A_{1}$, $A_{2}$ and
$B_{2}$, defined as,

$\displaystyle\kappa$ $\displaystyle=4\gamma_{\ell}(\gamma_{\ell}+2)+48i\omega
M~{},\qquad\gamma_{\ell}=\left(\ell+2\right)\left(\ell-1\right)~{},$ (113)
$\displaystyle A_{1}$
$\displaystyle=4\left[\frac{8r^{8}\omega^{4}}{(r^{2}-2Mr)^{2}}+\frac{8r^{4}\omega^{2}}{(r^{2}-2Mr)}\left(\frac{M^{2}}{(r^{2}-2Mr)}-\gamma_{\ell}\right)-\frac{4r^{2}\omega^{2}}{(r^{2}-2Mr)}\left(3r^{2}+2Mr\right)+12r^{2}\omega^{2}+\gamma_{\ell}(\gamma_{\ell}+2)\right]~{},$
(114) $\displaystyle A_{2}$
$\displaystyle=4\left[-\frac{24r^{5}\omega^{3}}{(r^{2}-2Mr)}+\frac{4\gamma_{\ell}(r-M)r^{2}\omega}{(r^{2}-2Mr)}+4\omega
r\gamma_{\ell}+12\omega M\right]~{},$ (115) $\displaystyle B_{2}$
$\displaystyle=32r^{6}\omega^{3}+16r^{2}\omega(r^{2}-2Mr)\left\\{\frac{2M^{2}}{(r^{2}-2Mr)}-\gamma_{\ell}\right\\}-8M\omega
r(r^{2}-2Mr)~{}.$ (116)

Further, the Teukolsky function $\,{}_{-2}R_{\ell m}^{0}$ can be related to
the Regge-Wheeler and Zerilli function in the zero-rotation limit. The first
step in this direction is to decompose the Teukolsky function into polar and
axial parts,

$\,{}_{-2}R_{\ell m}^{0}=\,_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}+\,_{-2}R_{\ell m}^{0,~{}\textrm{polar}}~{}.$ (117)

Since the Detweiler function depends on the Teukolsky function in a linear
manner, it follows that such a decomposition will hold for the Detweiler
function as well,

$\,{}_{-2}X_{\ell m}^{0}=\,_{-2}X_{\ell
m}^{0,~{}\textrm{axial}}+\,_{-2}X_{\ell m}^{0,~{}\textrm{polar}}~{}.$ (118)

The axial and polar parts of the Teukolsky function can then be expressed in
terms of the Regge-Wheeler and Zerilli functions, respectively, as

$\,{}_{-2}R_{\ell m}^{0,~{}\textrm{axial}}$
$\displaystyle=\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[V^{\rm
axial}\Psi^{\rm RW}_{\ell m}+\left(W^{\rm
axial}+2i\omega\right)\left(\dfrac{d\Psi_{\ell m}^{\rm
RW}}{dr_{*}}+i\omega\Psi_{\ell m}^{\rm RW}\right)\right]$
$\displaystyle=\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi_{\ell m}^{\rm
RW}}{dr_{*}}\right)\right]~{},$ (119) $\,{}_{-2}R_{\ell
m}^{0,~{}\textrm{polar}}$
$\displaystyle=-\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left[\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}\Psi^{\rm Z}_{\ell
m}+\left(W^{\rm polar}+2i\omega\right)\left(\dfrac{d\Psi^{\rm Z}_{\ell
m}}{dr_{*}}\right)\right]~{},$ (120)

where we have introduced four functions of the radial coordinate $r$, namely
$V_{\rm axial}$, $V_{\rm polar}$, $W_{\rm axial}$ and $W_{\rm polar}$, each of
which can be defined as,

$\displaystyle V^{\rm axial}$
$\displaystyle=\frac{(r-2M)\left\\{(\gamma_{\ell}+2)r-6M\right\\}}{r^{4}}=V_{\rm
RW}~{},$ (121) $\displaystyle W^{\rm axial}$
$\displaystyle=\frac{2(r-3M)}{r^{2}}~{},$ (122) $\displaystyle V^{\rm polar}$
$\displaystyle=\frac{(r-2M)}{r^{4}(\gamma_{\ell}r+6M)^{2}}\left[\gamma_{\ell}^{2}(\gamma_{\ell}+2)r^{3}+6M\gamma_{\ell}^{2}r^{2}+36M^{2}\gamma_{\ell}r+72M^{3}\right]=V_{\rm
Z}~{},$ (123) $\displaystyle W^{\rm polar}$
$\displaystyle=\frac{2\gamma_{\ell}r^{2}-6\gamma_{\ell}Mr-12M^{2}}{r^{2}(\gamma_{\ell}r+6M)}~{}.$
(124)

Therefore, one can directly relate the axial and polar parts of the Detweiler
function to the Regge-Wheeler and the Zerilli function, respectively,

$\,{}_{-2}X_{\ell m}^{\rm axial}$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\,_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\dfrac{\,{}_{-2}R_{\ell
m}^{0,~{}\textrm{axial}}}{dr_{*}}$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi^{\rm RW}_{\ell
m}}{dr_{*}}\right)\right]\Bigg{]}$
$\displaystyle+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr_{*}}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left[\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Psi^{\rm RW}_{\ell
m}+\left(W^{\rm axial}+2i\omega\right)\left(\dfrac{d\Psi^{\rm RW}_{\ell
m}}{dr_{*}}\right)\right]\Bigg{]}$ $\displaystyle\equiv A_{\rm RW}(r)\Psi^{\rm
RW}_{\ell m}+B_{\rm RW}(r)\left(\frac{\Psi^{\rm RW}_{\ell m}}{dr}\right)~{},$
(125)

where, we have defined the two functions $A_{\rm RW}$ and $B_{\rm RW}$, in the
following expressions,

$\displaystyle A_{\rm RW}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}\Bigg{]}$
$\displaystyle\qquad+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\left(\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\right)\left(W^{\rm
axial}+2i\omega\right)V_{\rm RW}~{},$ (126) $\displaystyle B_{\rm RW}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left(W^{\rm
axial}+2i\omega\right)f(r)$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left\\{V^{\rm
axial}+i\omega\left(W^{\rm axial}+2i\omega\right)\right\\}$
$\displaystyle\qquad\qquad+\frac{d}{dr}\left\\{\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8\omega}\left(W^{\rm
axial}+2i\omega\right)\right\\}\Bigg{]}~{}.$ (127)

Hence, we can express, the ratio of the derivative of the axial part of the
Detweiler function with the Detweiler function itself in terms of the same
ratio for the Regge-Wheeler function as,

$\displaystyle\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
axial}}{dr}\right)}{\,{}_{-2}X_{\ell m}^{\rm axial}}=\frac{\left[A_{\rm
RW}^{\prime}+B_{\rm RW}\left(\frac{r^{2}V_{\rm
RW}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm RW}+B_{\rm RW}^{\prime}-B_{\rm
RW}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\Psi_{\rm
RW}^{\prime}}{\Psi_{\rm RW}}}{A_{\rm RW}+B_{\rm RW}\frac{\Psi_{\rm
RW}^{\prime}}{\Psi_{\rm RW}}}~{}.$ (128)

The above ratio can also be expressed in terms of the tortoise coordinate
$r_{*}$,

$\displaystyle\frac{1}{f}\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
axial}}{dr_{*}}\right)}{\,{}_{-2}X_{\ell m}^{\rm
axial}}=\frac{f(r)\left[A_{\rm RW}^{\prime}+B_{\rm RW}\left(\frac{r^{2}V_{\rm
RW}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm RW}+B_{\rm RW}^{\prime}-B_{\rm
RW}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\frac{d\Psi_{\rm
RW}}{dr_{*}}}{\Psi_{\rm RW}}}{f(r)A_{\rm RW}+B_{\rm RW}\frac{\frac{d\Psi_{\rm
RW}}{dr_{*}}}{\Psi_{\rm RW}}}~{}.$ (129)

Along identical lines, for the polar sector, we obtain,

$\,{}_{-2}X_{\ell m}^{\rm polar}=-A_{\rm Z}(r)\Psi^{\rm Z}_{\ell m}-B_{\rm
Z}(r)\left(\frac{\Psi^{\rm Z}_{\ell m}}{dr}\right)~{},$ (130)

where the functions $A_{\rm Z}$ and $B_{\rm Z}$ are defined as,

$\displaystyle A_{\rm Z}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\frac{d}{dr}\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}\Bigg{]}$
$\displaystyle\qquad+\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\left(\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\right)\left(W^{\rm
polar}+2i\omega\right)V_{\rm Z}~{},$ (131) $\displaystyle B_{\rm Z}(r)$
$\displaystyle=\left(\frac{\alpha_{-2}^{0}}{r-2M}\right)\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left(W^{\rm
polar}+2i\omega\right)f(r)$
$\displaystyle\qquad+f(r)\left(\frac{\beta_{-2}^{0}}{\left(r-2M\right)^{3}}\right)\Bigg{[}\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left\\{V^{\rm
polar}+i\omega\left(W^{\rm polar}+2i\omega\right)\right\\}$
$\displaystyle\qquad\qquad+\frac{d}{dr}\left\\{\frac{r^{3}\sqrt{\gamma_{\ell}(2+\gamma_{\ell})}}{8}\left(W^{\rm
polar}+2i\omega\right)\right\\}\Bigg{]}~{}.$ (132)

Therefore, the ratio of the derivative of the polar sector of the Detweiler
function with respect to the tortoise coordinate and the Detweiler function
itself becomes,

$\displaystyle\frac{\left(\frac{\,{}_{-2}X_{\ell m}^{\rm
polar}}{dr_{*}}\right)}{\,{}_{-2}X_{\ell m}^{\rm
polar}}=\frac{f(r)\left[A_{\rm Z}^{\prime}+B_{\rm Z}\left(\frac{r^{2}V_{\rm
Z}}{(r-2M)^{2}}\right)\right]+\left[A_{\rm Z}+B_{\rm Z}^{\prime}-B_{\rm
Z}\left(\frac{2M}{r(r-2M)}\right)\right]\frac{\frac{d\Psi_{\rm
Z}}{dr_{*}}}{\Psi_{\rm Z}}}{f(r)A_{\rm Z}+B_{\rm Z}\frac{\frac{d\Psi_{\rm
Z}}{dr_{*}}}{\Psi_{\rm Z}}}~{}.$ (133)

These results have been used in the main text, as well as while discussing the
static limit of the TLNs for non-rotating compact objects.

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