# Vanishing Love of Black Holes in General Relativity:
From Spacetime Conformal Symmetry of a Two-dimensional Reduced Geometry

Takuya Katagiri  Astronomical Institute, Graduate School of Science, Tohoku
University, Aoba, Sendai 980-8578, Japan  Niels Bohr International Academy,
Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark  Masashi Kimura
Department of Informatics and Electronics, Daiichi Institute of Technology,
Tokyo 110-0005, Japan  Department of Physics, Rikkyo University, Toshima,
Tokyo 171-8501, Japan  Hiroyuki Nakano  Faculty of Law, Ryukoku University,
Kyoto 612-8577, Japan  Kazuyuki Omukai  Astronomical Institute, Graduate
School of Science, Tohoku University, Aoba, Sendai 980-8578, Japan

###### Abstract

Linear response of a self-gravitating object to an external static
perturbation field is measured in terms of the Love numbers. Four-dimensional
black holes in vacuum in General Relativity have zero Love numbers for
spin-$s$ field perturbations. In this paper, we study the underlying structure
of the vanishing of spin-$s$-field Love numbers of Schwarzschild and Kerr
black holes in terms of spacetime conformal symmetry in a unified manner for
the static fields. The perturbations can be reduced with the harmonic
decomposition to a set of infinite static scalar fields in a two-dimensional
anti-de Sitter spacetime $({\rm AdS}_{2})$. In the reduced system, each scalar
field is associated with another, implying that all multipole modes of the
perturbation can be regarded as symmetric partners, which can be understood
from the property of the supersymmetry algebra. The generator of the
supersymmetric structure is constructed from a closed conformal Killing vector
field of ${\rm AdS}_{2}$. The associated conserved quantity allows one to show
no static response without explicitly solving the perturbation equations. The
symmetric structure corresponds to the “ladder” symmetry in Hui et al. [JCAP
01, no.01, 032 (2022)] although the geometrical origin is different. Our
ladder operator includes the generators of hidden symmetries in previous works
that discuss the connection between Love numbers and hidden symmetries from
geometrical or algebraic perspectives.

##  I Introduction

The black hole is one of the most surprising predictions of General
Relativity. Its only component is the concept of spacetime. Black holes have
remarkable properties in General Relativity, i.e., no hair PhysRevLett.26.331
; Bekenstein:1972ky  and uniqueness Hawking:1971vc ; Robinson:1975bv ;
PhysRevD.81.024033 . As a consequence, all the astrophysical black holes are
believed to be well described by the Kerr geometry, which is characterized
solely by the mass and spin parameters. On the other hand, General Relativity
might need modification in the vicinity of the event horizon. In modified
theories of gravity, black holes can possess a scalar hair
PhysRevLett.106.151104 ; PhysRevD.93.044047 , unlike the Kerr black holes. The
detection of the deviation from the Kerr geometry by future gravitational-wave
observations can be a smoking gun for such new physics.

Binary black hole systems are a strong candidate for gravitational-wave
sources LIGOScientific:2016aoc  and also an astrophysical laboratory to test
strong gravitational field Berti:2015itd ; Berti:2005ys . During the early
inspiral phase of a compact binary, the two bodies behave as point masses, as
their internal structure does not affect the orbital motion PhysRev.136.B1224
. As the orbital separation sufficiently decreases due to the gravitational-
wave emission, however, the tidal interaction between the bodies becomes
remarkable and higher-order post-Newtonian correction comes into effect. Its
correction appears at a $5$PN order in the gravitational waveform as a
function of a quantity called tidal deformability, which characterizes how the
bodies deform as a static response to the perturbative tidal field generated
by each other Vines:2011ud . The tidal deformability is encoded in a set of
tidal Love numbers, which are determined by the internal structure of the body
Hinderer:2007mb ; Binnington:2009bb ; Damour:2009vw ; poisson_will_2014 . This
means that a measurement of the tidal Love numbers from a gravitational-wave
signal provides information about the internal structure of the bodies as an
inverse problem Flanagan:2007ix . Indeed, the equation of state of a neutron
star has been constrained from GW170817 Flanagan:2007ix ;
LIGOScientific:2017vwq ; LIGOScientific:2017ync . In the same manner, it is
expected that a test of the strong-field gravity can be performed with tidal
Love numbers Cardoso:2017cfl .

An intriguing result regarding tidal Love numbers is their vanishing for
(four-dimensional) black holes in vacuum in General Relativity, i.e.,
Schwarzschild Binnington:2009bb ; Gralla:2017djj  and Kerr black holes
LeTiec:2020spy ; Chia:2020yla ; LeTiec:2020bos ; Charalambous:2021mea . The
Schwarzschild and Kerr black holes have no static response for spin-$0$ or
spin-$1$ field perturbations either Hui:2020xxx ; Charalambous:2021mea .
Alternatively, it is known that black holes in some modified theories of
gravity acquire nonzero tidal Love numbers Cardoso:2017cfl ; Cardoso:2018ptl .
One might then think that an additional field gives nonzero Love numbers. In
fact, the Love numbers of black holes can be nonzero with an anisotropic fluid
Cardoso:2019upw ; Cardoso:2021wlq  or with a bosonic scalar-field condensate
DeLuca:2021ite  within General Relativity. On the other hand, black holes have
zero Love numbers in some modified theories of gravity or in the presence of
some kind of “matter” field, e.g., the Schwarzschild black hole in the Brans-
Dicke theory and the Reissner-Nordström black hole in the Einstein-Maxwell
theory Cardoso:2017cfl ; Cardoso:2019upw . In addition, even in vacuum in
General Relativity, the Schwarzschild-Tangherlini spacetime has nonzero Love
numbers Hui:2020xxx . In other contexts, several literatures Porto:2016zng ;
Penna:2018gfx ; Hui:2021vcv  pointed out that from the viewpoint of an
effective field theory approach Goldberger:2004jt ; Goldberger:2005cd ;
Kol:2011vg , the no tidal response of the Schwarzschild black hole is puzzling
and seems to be a result of fine-tuning. These suggest that some non-trivial
underlying structure is prohibiting finite Love numbers.

Several authors independently argue that the vanishing of Love numbers may be
a result of “hidden” symmetries governing linear perturbations.111Here, the
terminology “hidden” symmetries means that they do not correspond to
isometries of a spacetime. Penna Penna:2018gfx  suggested in the context of
the black-hole membrane paradigm that the emergent local Carroll symmetry
Duval:2014lpa  in the near-horizon region may have a role in explaining the no
static response of the Schwarzschild black hole. Charalambous, Dubovsky, and
Ivanov Charalambous:2021kcz  argued that a long-wavelength spin-$s$ field
perturbation in the Kerr spacetime has a hidden ${\rm SL}(2,\mathbb{R})\times
U(1)$ symmetry dubbed Love symmetry and spin-$s$-field Love numbers of the
Kerr black hole vanish as a consequence of this symmetry based on the ${\rm
SL}(2,\mathbb{R})$ representation theory. Recently, Hui, Joyce, Penco,
Santoni, and Solomon Hui:2021vcv  showed the vanishing of scalar-field Love
numbers from the perspective of two types of hidden symmetries, one of which
originates from the presence of a Killing vector field in a three-dimensional
Euclidian anti-de Sitter space (see also Ref. BenAchour:2022uqo ). More
recently, Charalambous, Dubovsky, and Ivanov Charalambous:2022rre  conducted a
detailed study on the vanishing of spin-$s$-field Love numbers of Kerr-Newman
black holes in terms of ${\rm SL}(2,\mathbb{R})$ representations.

The recent arguments above on the connection between the static response and a
hidden symmetry are reminiscent of the Laplace-Runge-Lenz vector, a conserved
quantity associated with a dynamical symmetry in the Kepler problem
goldstein:mechanics . The Laplace-Runge-Lenz vector explains the absence of
the periapsis shift in the inverse-square central force. The Laplace-Runge-
Lenz vector also appears in the structure of the hydrogen atom and explains
the degeneracy of the energy levels with different orbital angular momenta
2003AmJPh..71..171V . Some hidden symmetries may also explain the vanishing of
the Love numbers of black holes based on the analogy of quantum mechanics and
the black-hole perturbation theory as in Refs. PhysRevLett.52.1361 ;
PhysRevD.30.295 ; PhysRevD.43.605 ; PhysRevD.101.024008 ; Cardoso:2017qmj ;
Cardoso:2017egd ; PhysRevD.106.044052 . However, we still have a fundamental
question: why does such a hidden symmetric structure exist?

It is natural to expect that a geometrical property such as spacetime symmetry
gives symmetry for perturbation fields in a given spacetime. In this paper,
for the above problem, we study the underlying symmetric structure of the
vanishing of the spin-$s$-field Love numbers of the Schwarzschild and Kerr
black holes in terms of spacetime conformal symmetry in a unified manner. In
the analysis of the Schwarzschild black hole case, we reduce a perturbation
with the harmonic decomposition into a set of infinite static scalar fields in
the two-dimensional anti-de Sitter spacetime $({\rm AdS}_{2})$. We then
discuss a symmetric structure generated by a ladder operator constructed from
a closed conformal Killing vector field of ${\rm AdS}_{2}$.222The hidden
symmetric structure corresponds to the “ladder” symmetry in Ref. Hui:2021vcv
although the geometrical origin is different. Hui et al. Hui:2021vcv  gave a
generator, i.e., ladder operator, arising from a Killing vector field of a
Euclidean ${\rm AdS}_{3}$ for a scalar field case $s=0$; they also found a
ladder operator for a tidal field case $s=2$ but its construction seems to be
heuristic. Our viewpoint gives a ladder operator for spin-$s$ fields from a
closed conformal Killing vector field in ${\rm AdS}_{2}$. Our operator
includes the ladder operators for $s=0$ and $2$ in Ref. Hui:2021vcv ; the
ladder operator in Ref. BenAchour:2022uqo  and the mass ladder operator in
Refs. Cardoso:2017qmj ; Cardoso:2017egd ; PhysRevD.106.044052 ;
Katagiri:2021scx ; Katagiri:2021xig  for a scalar field case. We further show
no static response of the Schwarzschild black hole by exploiting an associated
conserved quantity without explicitly solving the perturbation equations. To
our knowledge, this is the first to explain the vanishing of the
spin-$s$-field Love numbers of the Schwarzschild black hole in terms of the
symmetric structure arising from spacetime symmetry.333We mention here that
for a scalar field case, the generators of ${\rm SL}(2,\mathbb{R})$ symmetry
in Refs. Charalambous:2021kcz ; Charalambous:2022rre  are Killing vector
fields of ${\rm AdS}_{2}$. On the other hand, the geometrical interpretation
of the generators for spin-$s(\geq 1)$ fields is not clear. We also discuss
the Kerr black hole in a parallel manner.

The organization of the rest is as follows. Section II presents a brief review
of a static response of the Schwarzschild black hole to spin-$s$ field
perturbations. In Sec. III, we reduce the problem of the static spin-$s$ field
perturbation on the Schwarzschild black hole into that of static scalar fields
in ${\rm AdS}_{2}$, and further show a hidden symmetric structure governing
the perturbations from the perspective of spacetime conformal symmetry of the
reduced geometry. We then explain in Sec. IV no static response of the
Schwarzschild black hole in terms of the symmetric structure. In Sec. V, we
discuss no static response of Kerr black holes and a hidden symmetric
structure. Section VI is devoted to the summary and discussion of this paper.
Appendices give the supplemental materials; a weakly time-dependent
perturbation case; derivation of a ladder operator from spacetime conformal
symmetry and the application of the operator to perturbations to the
Schwarzschild-Tangherlini black hole Tangherlini:1963bw .

##  II Static response of Schwarzschild black holes

In this section, we briefly review a linear response of Schwarzschild black
holes to an external spin-$s$ field Binnington:2009bb . We assume that the
amplitude of the field is small, and its wavelength is much longer than the
horizon scale. The field is then described by a linear perturbation theory of
the Schwarzschild black hole in the static limit.

###  II.1 Linear perturbation theory of the Schwarzschild black hole

The Schwarzschild black hole spacetime is described by

$g_{\mu\nu}^{(0)}dx^{\mu}dx^{\nu}=-\frac{\Delta}{z^{2}}dt^{2}+\frac{z^{2}}{\Delta}dz^{2}+z^{2}d\Omega^{2},$
(1)

where

$\Delta=z(z-1),$ (2)

and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$ is the line element
of a two-dimensional unit sphere $S^{2}$. In the current coordinate system
$(t,z,\theta,\varphi)$, the radial coordinate $z$ is defined in the range
$z\in(1,\infty)$, where $z=1$ and $\infty$ correspond to the locations of the
event horizon and spatial infinity, respectively.

We first explain the linear theory of a tidal field perturbation, i.e., $s=2$.
On the Schwarzschild background (1), the linearly perturbed metric takes a
form,

$g_{\mu\nu}=g_{\mu\nu}^{(0)}+h_{\mu\nu},$ (3)

where $h_{\mu\nu}$ is a linear perturbation. Each independent component of
$h_{\mu\nu}$ is classified as either scalar or vector part on $S^{2}$, which
can be expanded in terms of scalar spherical harmonics or vector spherical
harmonics, respectively Higuchi:1986wu ; Kodama:2003jz . The scalar and vector
parts correspond to the even and odd parts, respectively, under the parity
transformation $(\theta,\varphi)\to(\pi-\theta,\varphi+\pi)$. The former and
latter are also called the polar-type and axial-type perturbations,
respectively. Thus, we have

$h_{\mu\nu}=h_{\mu\nu}^{\rm(polar)}+h_{\mu\nu}^{\rm(axial)},$ (4)

where $h_{\mu\nu}^{\rm(polar)}$ and $h_{\mu\nu}^{\rm(axial)}$ denote the
polar- and axial-type perturbations, respectively.

In the Regge-Wheeler gauge Regge:1957td , the linearized Einstein equation for
the axial- and polar-type perturbations is reduced to two independent master
equations. One for the axial-type perturbation $\Phi_{\ell}^{-}$ is the so-
called Regge-Wheeler equation Regge:1957td  in the static limit:

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell}^{-}}{dz}\right)-\frac{\Delta}{z^{2}}\left[\frac{\ell(\ell+1)}{z^{2}}-\frac{3}{z^{3}}\right]\Phi_{\ell}^{-}=0,$
(5)

where $\ell=2,3,\cdots$ is the index of multipoles. Note that the azimuthal
number $m$ does not appear because of the spherical symmetry of the background
spacetime. The other for the polar-type perturbation is the so-called Zerilli
equation PhysRevD.2.2141 . As shown in Appendix A, with the Chandrasekhar
transformation Chandrasekhar:1975zza , a solution of the static Zerilli
equation can be generated from that of the static Regge-Wheeler equation
(5).444The master variable of the static polar-type perturbation can be
defined well in the static limit as well Jhingan:2002kb .

We next give a perturbation equation for the spin-$s$ fields. Static scalar
field $(s=0)$, vector field $(s=1)$, and axial-type tidal field $(s=2)$
perturbations are uniformly described by Refs. Kodama:2003jz ; Berti:2009kk ;
Hatsuda:2021gtn

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell}^{(s)}}{dz}\right)-\frac{\Delta}{z^{2}}\left[\frac{\ell(\ell+1)}{z^{2}}-\frac{s^{2}-1}{z^{3}}\right]\Phi_{\ell}^{(s)}=0,$
(6)

where $\ell\geq s$. In this work, we regard Eq. (6) as a fundamental equation
governing static spin-$s$ field perturbations to the Schwarzschild black hole
in a unified manner because the polar-type tidal field perturbation can also
be described by the axial-type one with the Chandrasekhar transformation
Chandrasekhar:1975zza  as stated above.

###  II.2 Spin-$s$-field Love numbers

Here, we introduce the notion of the spin-$s$-field Love numbers. We consider
asymptotic behaviors of solutions of Eq. (6) at the horizon and large
distances. On the one hand, at the horizon $z=1$, we have two linearly
independent asymptotic solutions, i.e.,

$\Phi_{\ell}^{(s)}|_{z\to 1}\sim{\rm const.},~{}~{}\ln(1-1/z).$ (7)

Note that, at the horizon, the former and latter are regular and
logarithmically divergent, respectively. On the other hand, at large distances
$z\gg 1$, we have two linearly independent asymptotic solutions, i.e.,

$\Phi_{\ell}^{(s)}|_{z\gg 1}\sim z^{\ell+1},~{}~{}z^{-\ell}.$ (8)

A generic expectation might be that in the distant region $z\gg 1$, the
regular solution at the horizon, i.e., $\Phi_{\ell}^{(s)}|_{z\to 1}\sim{\rm
const.}$ would be written in the form of a linear combination of two linearly
independent solutions, i.e.,

$\Phi_{\ell}^{(s)}|_{z\gg 1}\sim z^{\ell+1}(1+2k_{\ell}^{(s)}z^{-2\ell-1}).$
(9)

Physically, the first and second terms are interpreted as the external
perturbation field and the static response of the black hole, respectively.
The coefficients $k_{\ell}^{(s)}$ are called the spin-$s$-field Love numbers
Cardoso:2017cfl ; DeLuca:2021ite  based on the Newtonian analogy. In the same
manner, the polar-type one for the gravitational field perturbation can also
be defined poisson_will_2014 ; Binnington:2009bb .

###  II.3 Example: Lowest multipole $(\ell=s)$ case

As an example, we show no static response to the lowest multipole mode of the
physical solution. The analytic expressions for $\Phi_{s}^{(s)}$ are given by

$\Phi_{0}^{(0)}=C_{1}^{(0)}z+C_{2}^{(0)}z\ln\left(1-1/z\right),$ (10)

for $s=0$, and

$\Phi_{1}^{(1)}=C_{1}^{(1)}z^{2}+C_{2}^{(1)}\left[1+2z+2z^{2}\ln\left(1-1/z\right)\right],$
(11)

for $s=1$, and

$\begin{split}\Phi_{2}^{(2)}=&C_{1}^{(2)}z^{3}+C_{2}^{(2)}z^{-1}\left[3+4z+6z^{2}+12z^{3}+12z^{4}\ln\left(1-1/z\right)\right],\end{split}$
(12)

for $s=2$, where $C_{1}^{(s)}$ and $C_{2}^{(s)}$ are constants. Note here that
the term of $C_{2}^{(s)}$ has a logarithmic contribution, which is divergent
in the horizon limit $z\to 1$. To ensure the regularity of the perturbations
at the horizon, we must set $C_{2}^{(s)}=0$. Therefore, the solution regular
at the horizon takes the form,

$\displaystyle\Phi_{s}^{(s)}=$ $\displaystyle C_{1}^{(s)}z^{s+1}.$ (13)

Note that this solution has no decaying term of $z$ at large distances $z\gg
1$. Thus, the spin-$s$-field Love number $k_{s}^{(s)}$ is precisely zero for
the Schwarzschild black hole. One can show that the higher-multipolar
spin-$s$-field Love numbers also vanish in the same manner. The tidal Love
numbers for the polar-type tidal field perturbation are also precisely zero as
shown in Appendix A.

##  III Hidden supersymmetric structure of static perturbations

In this section, we show that there exists a “supersymmetric” structure
governing static spin-$s$ field perturbations to the Schwarzschild black hole
from a geometrical point of view. We first demonstrate that the static
spin-$s$ field perturbation to the Schwarzschild black hole can be reduced to
a set of infinite static scalar fields in the two-dimensional anti-de Sitter
spacetime (${\rm AdS}_{2}$). We then construct a ladder operator that is a
generator of the symmetric structure, from a particular spacetime conformal
symmetry of ${\rm AdS}_{2}$, and further derive a conserved quantity
associated with the symmetric structure. In Appendix B, we show that the
following discussion is completely parallel also for a weakly time-dependent
perturbation.

###  III.1 Reduction to ${\rm AdS}_{2}$

The static spin-$s$ field perturbation satisfies

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell}}{dz}\right)-\frac{\Delta}{z^{2}}\left[\frac{\ell(\ell+1)}{z^{2}}-\frac{s^{2}-1}{z^{3}}\right]\Phi_{\ell}=0,$
(14)

which is the same as Eq. (6). Henceforth, we omit the superscript $s$ denoting
the spin of the field. We rewrite Eq. (14) into more convenient form for later
discussion:

$\displaystyle\frac{d}{dz}\left(\Delta\frac{d\phi_{\ell}}{dz}\right)-\left(\ell(\ell+1)-\frac{s^{2}}{z}\right)\phi_{\ell}=$
$\displaystyle 0,$ (15)

where we have defined

$\phi_{\ell}(z)=\frac{\Phi_{\ell}}{z}.$ (16)

In what follows, we shall see that Eq. (15) can be identified as the equation
of motion for a static scalar field in ${\rm AdS}_{2}$. Consider a line
element in the static coordinate $(t,z)$,

$g_{ab}^{\rm AdS_{2}}dx^{a}dx^{b}=-\Delta dt^{2}+\frac{1}{\Delta}dz^{2},$ (17)

where $a,b$ run $t$ and $z$. The scalar curvature of this metric is calculated
as

$R^{(2)}=-2.$ (18)

The spacetime described by Eq. (17) has a negative constant curvature, i.e.,
${\rm AdS}_{2}$. The d’Alembertian in ${\rm AdS}_{2}$ is given by

$\square_{{\rm
AdS}_{2}}:=-\frac{1}{\Delta}\partial_{t}^{2}+\partial_{z}\left(\Delta\partial_{z}\right).$
(19)

It should be noted that the first term of the left-hand side of Eq. (15)
corresponds to the d’Alembertian (19) for a static field, i.e.,

$\square_{{\rm AdS}_{2}}=\frac{d}{dz}\left(\Delta\frac{d}{dz}\right).$ (20)

This indicates that the variable $\phi_{\ell}$ in Eq. (15) can be identified
as a scalar field in ${\rm AdS}_{2}$ of Eq. (17) satisfying an equation of
motion,

$\displaystyle\left[\square_{{\rm
AdS}_{2}}-\left(\ell(\ell+1)-\frac{s^{2}}{z}\right)\right]\phi_{\ell}=$
$\displaystyle 0.$ (21)

Thus, the static spin-$s$ field perturbation to the Schwarzschild black hole
can be reduced to a set of infinite static scalar fields in ${\rm AdS}_{2}$ of
Eq. (17). Note that this reduction differs from the so-called near-horizon
limit for extremal black holes Bardeen:1999px  because the emergence of the
${\rm AdS}_{2}$ geometry is not inherited from the enhancement of the isometry
of the background spacetime itself Castro:2010fd ; Bertini:2011ga .

###  III.2 Ladder operator from spacetime conformal symmetry

We construct a generator of a hidden symmetric structure, i.e., ladder
operator, based on the ${\rm AdS}_{2}$ geometry,555For the case of $s=0$, the
operator (22) corresponds to a mass ladder operator which maps a solution of a
massive Klein-Gordon equation into another solution of that with different
mass squared Cardoso:2017qmj ; Cardoso:2017egd ; PhysRevD.106.044052 ;
Katagiri:2021scx ; Katagiri:2021xig .

$\mathcal{D}_{k_{\pm}}:=\mathcal{L}_{\zeta}-\frac{k_{\pm}}{2}\left(\nabla_{a}\zeta^{a}-\frac{s^{2}}{k_{\pm}^{2}}\right),$
(22)

where

$k_{+}=-\ell-1,~{}~{}k_{-}=\ell.$ (23)

Here, $\mathcal{L}_{\zeta}$ is the Lie derivative with respect to a vector
field $\zeta^{a}$ called a closed conformal Killing vector field in ${{\rm
AdS}_{2}}$ of Eq. (17),666Although there exist three independent closed
conformal Killing vector fields in ${\rm AdS}_{2}$, the time independence of
the potential term in Eq. (21), i.e., $s^{2}/z$, forces one to choose the
vector field (24) only (see also footnote 16).

$\zeta^{a}\frac{\partial}{\partial x^{a}}=\Delta\frac{\partial}{\partial z}.$
(24)

which satisfies a conformal Killing equation,

$\mathcal{L}_{\zeta}g_{ab}=\left(\nabla_{c}\zeta^{c}\right)g_{ab},$ (25)

with the closed condition,

$\nabla_{a}\zeta_{b}=\nabla_{b}\zeta_{a}.$ (26)

In Appendix C, we derive a ladder operator from spacetime conformal symmetry
of ${\rm AdS}_{2}$, in particular, show that the type of the system (21) is
unique in ${\rm AdS}_{2}$ of Eq. (17) admitting a ladder operator for a
generic time-dependent scalar field under the requirement of the regularity of
an effective potential, and further discuss the application of the operator to
perturbations to the Schwarzschild-Tangherlini black hole Tangherlini:1963bw .

The operator $\mathcal{D}_{k_{\pm}}$ in Eq. (22) can be written
as777Surprisingly, the ladder operator (27) with $s=0$ corresponds to that in
Eq. (2.4) in Ref. Hui:2021vcv  and that in Eq. (2.20) in Ref.
BenAchour:2022uqo  although their constructions are completely different. The
operator (27) with $s=2$ also corresponds to Eqs. (C.4) and (C.5) in Ref.
Hui:2021vcv , which are heuristically constructed; our result thus explains
the geometrical origin of them.

$\mathcal{D}_{k_{\pm}}=\Delta\frac{d}{dz}-\frac{k_{\pm}}{2}\left(2z-1-\frac{s^{2}}{k_{\pm}^{2}}\right).$
(27)

We here point out that the ladder operator satisfies a commutation relation,

$\left[\Box_{{\rm
AdS}_{2}}+\frac{s^{2}}{z},\mathcal{D}_{k_{\pm}}\right]=-2k_{\pm}\mathcal{D}_{k_{\pm}}+\left(\nabla_{a}\zeta^{a}\right)\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)-\frac{s^{2}}{z}\right)\right].$ (28)

###  III.3 Transformation of static scalar fields by ladder operators

Now, we act the commutation relation (28) with $k_{-}$ in Eq. (23) on a smooth
function $\phi_{\ell}(z)$, yielding

$\begin{split}&\left(\mathcal{D}_{\ell}+\nabla_{a}\zeta^{a}\right)\left[\square_{{\rm
AdS}_{2}}-\left(\ell(\ell+1)-\frac{s^{2}}{z}\right)\right]\phi_{\ell}=\left[\square_{{\rm
AdS}_{2}}-\left((\ell-1)\ell-\frac{s^{2}}{z}\right)\right]\mathcal{D}_{\ell}\phi_{\ell}.\end{split}$
(29)

When the function $\phi_{\ell}$ satisfies Eq. (21), the left-hand side of the
relation (LABEL:relation) vanishes; thus, obtaining

$\left[\square_{{\rm
AdS}_{2}}-\left((\ell-1)\ell-\frac{s^{2}}{z}\right)\right]\mathcal{D}_{\ell}\phi_{\ell}=0.$
(30)

This implies that the function $\mathcal{D}_{\ell}\phi_{\ell}$ generated by a
transformation $\phi_{\ell}\to\mathcal{D}_{\ell}\phi_{\ell}$ is a solution of
the equation of motion for $\phi_{\ell}$ in Eq. (21) with $\ell\to\ell-1$. In
other words, the action of the ladder operator $\mathcal{D}_{\ell}$ generates
a solution with the multipole index shifted as $\ell\to\ell-1$. In the same
manner, by acting the commutation relation (28) with $k_{+}$ in Eq. (23) on
$\phi_{\ell}$, one can show that the ladder operator generates a function
$\mathcal{D}_{-\ell-1}\phi_{\ell}$, which satisfies

$\left[\square_{{\rm
AdS}_{2}}-\left((\ell+1)(\ell+2)-\frac{s^{2}}{z}\right)\right]\mathcal{D}_{-\ell-1}\phi_{\ell}=0.$
(31)

The function $\mathcal{D}_{-\ell-1}\phi_{\ell}$ is a solution with the
multipole index shifted as $\ell\to\ell+1$.

Given the commutation relation (28), the multiple action of the ladder
operators further shifts the index $\ell$ of the solution as follows. When
considering $N$-th order multiple actions of the ladder operators on
$\phi_{\ell}$ in Eq. (21),

$\phi_{\ell}\to\mathcal{D}_{\ell-N+1}\cdots\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\phi_{\ell},$
(32)

we obtain

$\left[\square_{{\rm
AdS}_{2}}-\left((\ell-N)(\ell-N+1)-\frac{s^{2}}{z}\right)\right]\mathcal{D}_{\ell-N+1}\cdots\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\phi_{\ell}=0.$
(33)

This multiple action generates a solution with a shifted index as
$\ell\to\ell-N$. In particular, as will be seen later, the $(\ell-s)$-th order
action generates the lowest multipole mode from the higher multipole mode
indexed with $\ell$. In the same manner, $N$-th order multiple action
$\mathcal{D}_{-\ell-N}\cdots\mathcal{D}_{-\ell-2}\mathcal{D}_{-\ell-1}$ on
$\phi_{\ell}$ generates a solution with the index shifted as $\ell\to\ell+N$.

###  III.4 Supersymmetric structure

The ladder operator relates a static solution with a given $\ell$ to other
static solutions with a shifted multipole index. That means that the
transformation generated by the ladder operator keeps the “shape” of the
equation of motion. We discuss here the underlying structure of the shape
invariance, i.e., hidden supersymmetric structure.

Let us consider the system that consists of the pair of fields
$(\phi_{\ell},\phi_{\ell\pm 1})$ governed by the action,

$S=\frac{1}{2}\int
dz\left[-\left(\Delta\frac{d}{dz}\phi_{\ell}\right)^{2}-V_{\ell}\phi_{\ell}^{2}-\left(\Delta\frac{d}{dz}\phi_{\ell\pm
1}\right)^{2}-V_{\ell\pm 1}\phi_{\ell\pm 1}^{2}\right],$ (34)

where the potentials are given by888 In terms of $k_{\pm}$ in Eq. (23), the
potentials take the forms,
$\begin{split}V_{\ell}=\Delta\left[k_{\pm}\left(k_{\pm}+1\right)-\frac{s^{2}}{z}\right],~{}~{}V_{\ell\pm
1}=\Delta\left[\left(k_{\pm}-1\right)k_{\pm}-\frac{s^{2}}{z}\right].\end{split}$
(35)

$\begin{split}V_{\ell}=\Delta\left[\ell\left(\ell+1\right)-\frac{s^{2}}{z}\right],~{}~{}V_{\ell\pm
1}=\Delta\left[\left(\ell\pm 1\right)\left(\ell\pm
1+1\right)-\frac{s^{2}}{z}\right].\end{split}$ (36)

The action (34) is invariant under the continuous transformation,

$\phi_{\ell}\to\phi_{\ell}+\epsilon\mathcal{D}_{-k_{\pm}}\phi_{\ell\pm
1},~{}~{}\phi_{\ell\pm 1}\to\phi_{\ell\pm
1}+\epsilon\mathcal{D}_{k_{\pm}}\phi_{\ell},$ (37)

where $\epsilon$ is an infinitesimal parameter. We discuss the corresponding
conserved quantity in the next subsection. One can inductively generalize the
above symmetry of the pair $(\phi_{\ell},\phi_{\ell\pm 1})$ to that of
infinite pairs $(\phi_{\ell},\phi_{\ell\pm 1},\phi_{\ell\pm 2},\cdots)$. As
will be seen below, the transformation (37) generates a supersymmetric
structure.

We now discuss the detailed property of the above symmetry. By varying the
action (34) with respect to $\phi_{\ell}$ and $\phi_{\ell\pm 1}$, we obtain
the equations of motion for $\phi_{\ell}$ and $\phi_{\ell\pm 1}$, which are
consistent with Eq. (21),

$\mathcal{H}_{\ell}\phi_{\ell}=0,~{}~{}\mathcal{H}_{\ell\pm 1}\phi_{\ell\pm
1}=0,$ (38)

where we have defined the Hamiltonians,

$\mathcal{H}_{\ell}:=-\Delta\frac{d}{dz}\left(\Delta\frac{d}{dz}\right)+V_{\ell},~{}~{}\mathcal{H}_{\ell\pm
1}:=-\Delta\frac{d}{dz}\left(\Delta\frac{d}{dz}\right)+V_{\ell\pm 1}.$ (39)

Introducing a function,

$W_{k_{\pm}}:=\frac{k_{\pm}}{2}\left(2z-1-\frac{s^{2}}{k_{\pm}^{2}}\right),$
(40)

the potentials $V_{\ell}$ and $V_{\ell\pm 1}$ in Eq. (36) can be written as

$\begin{split}V_{\ell}=W_{k_{\pm}}^{2}+\Delta\frac{dW_{k_{\pm}}}{dz}+\beta_{k_{\pm}}^{2},~{}~{}V_{\ell\pm
1}=W_{k_{\pm}}^{2}-\Delta\frac{dW_{k_{\pm}}}{dz}+\beta_{k_{\pm}}^{2},\end{split}$
(41)

where we have defined

$\beta_{k_{\pm}}=\frac{k_{\pm}^{2}-s^{2}}{2k_{\pm}}.$ (42)

In terms of $W_{k_{\pm}}$ in Eq. (40), the ladder operator takes the form of
$\mathcal{D}_{k_{\pm}}=\Delta\dfrac{d}{dz}-W_{k_{\pm}}$. One can then show the
following relations:

$\begin{split}\mathcal{H}_{\ell}=\mathcal{D}_{-k_{\pm}}\mathcal{D}_{k_{\pm}}+\beta_{k_{\pm}}^{2},~{}~{}\mathcal{H}_{\ell\pm
1}=\mathcal{D}_{k_{\pm}}\mathcal{D}_{-k_{\pm}}+\beta_{k_{\pm}}^{2}.\end{split}$
(43)

This implies

$\begin{split}\mathcal{D}_{k_{\pm}}\mathcal{H}_{\ell}\phi_{\ell}&=\mathcal{D}_{k_{\pm}}\left(\mathcal{D}_{-k_{\pm}}\mathcal{D}_{k_{\pm}}+\beta_{k_{\pm}}^{2}\right)\phi_{\ell}=\mathcal{H}_{\ell\pm
1}\mathcal{D}_{k_{\pm}}\phi_{\ell}=0,\\\
\mathcal{D}_{-k_{\pm}}\mathcal{H}_{\ell\pm 1}\phi_{\ell\pm
1}&=\mathcal{D}_{-k_{\pm}}\left(\mathcal{D}_{k_{\pm}}\mathcal{D}_{-k_{\pm}}+\beta_{k_{\pm}}^{2}\right)\phi_{\ell\pm
1}=\mathcal{H}_{\ell}\mathcal{D}_{-k_{\pm}}\phi_{\ell\pm 1}=0.\end{split}$
(44)

It can be seen that given the solution $\phi_{\ell}$ of $\mathcal{H}_{\ell}$,
the function $\mathcal{D}_{k_{\pm}}\phi_{\ell}$ is that of
$\mathcal{H}_{\ell\pm 1}$; given the solution $\phi_{\ell\pm 1}$ of
$\mathcal{H}_{\ell\pm 1}$, the function $\mathcal{D}_{-k_{\pm}}\phi_{\ell\pm
1}$ is that of $\mathcal{H}_{\ell}$.999This is the emergence of the vertical
symmetry in Ref. Hui:2021vcv . In other words, $\phi_{\ell}$ and
$\phi_{\ell\pm 1}$ are associated with each other through the transformation
generated by the ladder operator. In this sense, $\phi_{\ell}$ and
$\phi_{\ell\pm 1}$ are a symmetric partner under the transformation by the
ladder operators. Note that for a given $\phi_{\ell}$, there exist infinite
symmetric partners with different multipoles generated by the multiple action
of the ladder operators. This feature corresponds to the supersymmetric
structure seen in supersymmetric quantum mechanics Cooper:1994eh .

The above symmetric structure can be understood from the property of the
following algebra. Defining matrices,

$\mathcal{H}=\begin{pmatrix}\mathcal{H}_{\ell}-\beta_{k_{\pm}}^{2}&0\\\
0&\mathcal{H}_{\ell\pm 1}-\beta_{k_{\pm}}^{2}\\\
\end{pmatrix},~{}~{}\mathcal{Q}=\begin{pmatrix}0&0\\\
\mathcal{D}_{k_{\pm}}&0\\\
\end{pmatrix},~{}~{}\bar{\mathcal{Q}}=\begin{pmatrix}0&\mathcal{D}_{-k_{\pm}}\\\
0&0\\\ \end{pmatrix},$ (45)

then, we have

$\left[\mathcal{H},\mathcal{Q}\right]=\left[\mathcal{H},\bar{\mathcal{Q}}\right]=O,~{}~{}\left\\{\mathcal{Q},\bar{\mathcal{Q}}\right\\}=\mathcal{H},~{}~{}\left\\{\mathcal{Q},{\mathcal{Q}}\right\\}=\left\\{\bar{\mathcal{Q}},\bar{{\mathcal{Q}}}\right\\}=O,$
(46)

where $O$ is the zero matrix; $[~{},~{}]$ and $\\{~{},~{}\\}$ denote the
commutator and the anti-commutator, respectively. This is known as the
supersymmetry algebra, where $\mathcal{D}_{k_{\pm}}$ and
$\mathcal{D}_{-k_{\pm}}$ correspond to an intertwiner, and $\mathcal{Q}$ and
$\bar{\mathcal{Q}}$ correspond to supercharges in supersymmetric quantum
mechanics Cooper:1994eh .

###  III.5 Conserved quantity from the supersymmetric structure

We derive a radially conserved quantity associated with the supersymmetric
structure. When performing the transformation (37), the invariance of the
action (34) leads to a conserved quantity,101010There exists another conserved
quantity, $\mathcal{\tilde{W}}_{\ell\pm
1}=(\mathcal{D}_{k_{\pm}}\phi_{\ell})\left(\Delta\dfrac{d\phi_{\ell\pm
1}}{dz}\right)-\left(\Delta\dfrac{d\mathcal{D}_{k_{\pm}}\phi_{\ell}}{dz}\right)\phi_{\ell\pm
1}$, which is equivalent to the quantity (47) for the fields satisfying the
equations of motion.

$\begin{split}\mathcal{W}_{\ell}:=\left(\mathcal{D}_{-k_{\pm}}\phi_{\ell\pm
1}\right)\left(\Delta\frac{d\phi_{\ell}}{dz}\right)-\left(\Delta\frac{d\mathcal{D}_{-k_{\pm}}\phi_{\ell\pm
1}}{dz}\right)\phi_{\ell}.\end{split}$ (47)

One can show the radial conservation of $\mathcal{W}_{\ell}$ for the fields
that satisfy the equations of motion, $\mathcal{H}_{\ell}\phi_{\ell}=0$ and
$\mathcal{H}_{\ell\pm 1}\phi_{\ell\pm 1}=0$.

We discuss the property of the conserved quantity in the radial direction. Let
us consider the pair of the lowest multipole mode $\phi_{s}$ for a given $s$
and an originally “unphysical” mode $\phi_{s-1}$. From the explicit forms of
the ladder operator, $\mathcal{D}_{\mp s}=z(z-1)\dfrac{d}{dz}\pm s(z-1)$, it
follows that $\mathcal{D}_{-s}\phi_{s-1}|_{z\to 1}=0$ and
$\mathcal{D}_{s}\phi_{s}|_{z\to 1}=0$ for $\phi_{s-1}$ and $\phi_{s}$ with the
regularity condition at $z=1$, while this is not the case for the
logarithmically divergent solutions. This is a unique property for the pair
with $\phi_{s-1}$, and is not the case for pairs without it, e.g.,
$(\phi_{s},\phi_{s+1})$. The radial conservation of $\mathcal{W}_{s}$ then
leads to

$\left(\mathcal{D}_{-s}\phi_{s-1}\right)\frac{d}{dz}\left[z^{s}\mathcal{D}_{s}\phi_{s}\right]=0.$
(48)

This means that with the general solution of $\phi_{s-1}$ prepared, the
conservation of $\mathcal{W}_{s}$ is equivalent to that of the following
quantity:

$Y_{s}:=z^{s}\mathcal{D}_{s}\phi_{s}.$ (49)

From the observations above, the radially conserved quantity $Y_{s}$
identically vanishes for $\phi_{s}$ with regularity at $z=1$, while that is
nonzero for the logarithmically divergent solution. This shows that
$\mathcal{D}_{s}\phi_{s}$ itself identically vanishes for $\phi_{s}$ with
regularity. The vanishing of $D_{s}\phi_{s}$ under the regularity condition is
similar to the phenomenon known as “unbroken supersymmetry” in the context of
supersymmetric quantum mechanics Cooper:1994eh .111111One can show from
another conserved quantity $\mathcal{\tilde{W}}_{s-1}$ that
$\mathcal{D}_{-s}\phi_{s-1}$ also identically vanishes; this is also similar
to unbroken supersymmetry Cooper:1994eh .

One can obtain a radially conserved quantity for general mulipoles. The ladder
operators map a given $\phi_{\ell}$ into the lowest multipole mode,
$\tilde{\phi}_{s}:=\mathcal{D}_{s+1}\cdots\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\phi_{\ell}$,
thereby yielding a conserved quantity in the radial direction,

$Y_{\ell}:=z^{s}\mathcal{D}_{s}\tilde{\phi_{s}}.$ (50)

##  IV Vanishing Love of Schwarzschild black hole from hidden symmetry

In this section, we discuss no static response of the Schwarzschild black hole
to the spin-$s$ field perturbation in terms of the supersymmetric structure.
In Appendix D, we show that for the case of the polar-type tidal field
perturbation.

###  IV.1 Lowest multipole mode case

First, we show the vanishing of the Love number for the lowest multipole mode
$\phi_{s}$ with a given $s$, by exploiting the radial conservation of
$Y_{\ell}$ in Eq. (49). For the general solution of $\phi_{s}$, we have121212
The radial conservation of ${Y}_{s}$ includes horizontal symmetry found by
direct inspection of the Regge-Wheeler equation in Ref. Hui:2021vcv .

$\begin{split}{Y}_{s}=&z^{s}\mathcal{D}_{s}\phi_{s}\\\
=&z^{2s}\Delta\frac{d}{dz}\left(z^{-s}\phi_{s}\right).\end{split}$ (51)

Without knowledge of the exact solutions of the perturbations, we instead use
the asymptotic solutions at the horizon and large distances, i.e.,

$\left.\phi_{s}\right|_{z\to 1}\sim{\rm const.},~{}~{}\ln\left(1-1/z\right),$
(52)

and

$\left.\phi_{s}\right|_{z\gg 1}\sim z^{s},~{}~{}z^{-s-1}.$ (53)

The translational conservation of $Y_{s}$ of Eq. (51) tells us the asymptotic
behavior of the solution at large distances under the requirement of the
regularity at the horizon as follows. The evaluation of $Y_{s}$ in Eq. (51)
for $\phi_{s}|_{z\to 1}\sim{\rm const}.$ yields $Y_{s}=0$ at the horizon, as
we have already known, for $\phi_{s}|_{z\gg 1}\sim z^{s}$ it also leads to
$Y_{s}=0$. On the other hand, the values of $Y_{s}$ are nonzero for
$\phi_{s}|_{z\to 1}\sim\ln(1-1/z)$ and $\phi_{s}|_{z\gg 1}\sim z^{-s-1}$.
Therefore, the solution regular (logarithmically divergent) at the horizon
must connect to a purely growing (decaying, respectively) solution of $z$ at
large distances. This shows that for $\phi_{s}$, the spin-$s$-field Love
number of the Schwarzschild black hole vanishes, i.e., $k_{s}^{(s)}=0$.

###  IV.2 General multipole case

Next, we extend the discussion above to the general multipole case by using
the ladder operators. As stated at the end of Sec. III, the ladder operators
map a given mode $\phi_{\ell}$ into the lowest multipole mode
$\tilde{\phi}_{s}=\mathcal{D}_{s+1}\cdots\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\phi_{\ell}$,
giving rise to a conserved quantity in the radial direction,

$\begin{split}{Y}_{\ell}=&z^{s}\mathcal{D}_{s}\tilde{\phi}_{s}\\\
=&z^{2s}\Delta\frac{d}{dz}\left(z^{-s}\tilde{\phi}_{s}\right),\end{split}$
(54)

which is the same form as $Y_{s}$ in Eq. (51).

Asymptotic solutions of $\phi_{\ell}$ at the horizon and large distances are,
respectively,

$\left.\phi_{\ell}\right|_{z\to 1}\sim{\rm
const.},~{}~{}\ln\left(1-1/z\right),$ (55)

and

$\left.\phi_{\ell}\right|_{z\gg 1}\sim z^{\ell},~{}~{}z^{-\ell-1}.$ (56)

Noting that the ladder operators have no logarithmic term, the solution
regular at the horizon, $\phi_{\ell}|_{z\to 1}\sim{\rm const}.$, should give
rise to $\tilde{\phi}_{s}|_{z\to 1}\sim{\rm const}.$, showing the vanishing of
the conserved quantity $Y_{\ell}$. On the other hand, the solution
logarithmically divergent at the horizon, i.e., $\phi_{\ell}|_{z\to
1}\sim\ln(1-1/z)$, leads to $\tilde{\phi}_{s}|_{z\to 1}\sim\ln(1-1/z)$, giving
the nonzero $Y_{\ell}$.

From the asymptotic form of the ladder operators at large distances,

$\mathcal{D}_{\ell}=z^{2}\left(1+\mathcal{O}\left(1/z\right)\right)\dfrac{d}{dz}-\ell
z\left(1+\mathcal{O}\left(1/z\right)\right)~{}~{}{\rm for}~{}~{}z\gg 1,$ (57)

it follows that the ladder operator $\mathcal{D}_{\ell}$ cancels out the
leading term of $\phi_{\ell}|_{z\gg 1}\sim z^{\ell}$. Hence, the leading
behavior of $\mathcal{D}_{\ell}\phi_{\ell}|_{z\gg 1}$ comes from the
subleading term of $\phi_{\ell}|_{z\gg 1}$, leading to
$\mathcal{D}_{\ell}\phi_{\ell}|_{z\gg 1}\sim z^{\ell-1}$. In the same manner,
one can show $\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\phi_{\ell}|_{z\gg 1}\sim
z^{\ell-2}$. This indicates that $(\ell-s)$-th order action of the ladder
operators on $\phi_{\ell}|_{z\gg 1}\sim z^{\ell}$ leads to
$\tilde{\phi}_{s}|_{z\gg 1}\sim z^{s}$, giving the vanishing of $Y_{\ell}$. On
the other hand, the ladder operator on $\phi_{\ell}|_{z\gg 1}\sim z^{-\ell-1}$
leads to $\mathcal{D}_{\ell}\phi_{\ell}|_{z\gg 1}\sim z^{-\ell}$. Hence,
$(\ell-s)$-th order action leads to $\tilde{\phi}_{s}|_{z\gg 1}\sim z^{-s-1}$,
giving nonzero $Y_{\ell}$.

To summarize the above analysis, the solution regular at the horizon, i.e.,
$\phi_{\ell}|_{z\to 1}\sim{\rm const}.$, and the solution purely growing at
large distances, i.e., $\phi_{\ell}|_{z\gg 1}\sim z^{\ell}$, give the
vanishing $Y_{\ell}$, while the solution divergent at the horizon, i.e.,
$\phi_{\ell}|_{z\to 1}\sim\ln(1-1/z)$, and the solution purely decaying at
large distances, i.e., $\phi_{\ell}|_{z\gg 1}\sim z^{-\ell-1}$, lead to the
nonzero $Y_{\ell}$. This shows that the solution regular at the horizon
connects to that purely growing at the distant region, and does not contain
the decaying term, i.e., the vanishing of spin-$s$-field Love numbers for all
$\ell$.

##  V Vanishing Love of Kerr black holes

We remark the vanishing of (conservative) Love numbers of Kerr black holes
LeTiec:2020spy ; Chia:2020yla ; LeTiec:2020bos ; Charalambous:2021mea . Our
viewpoint from the hidden symmetric structure based on the ${\rm AdS}_{2}$
geometry will provide an alternative insight into the vanishing of the Love
numbers of the Kerr black hole in the following manner.

Spin-$s$ linear field perturbations to the Kerr black hole are governed by the
so-called Teukolsky equation PhysRevLett.29.1114 . Our analysis on the
Schwarzschild black hole does not straightforwardly extend to the Kerr black
hole case because the Teukolsky equation is not reduced to either of the
Regge-Wheeler or Zerilli equations in the nonrotating limit. In Ref.
Hatsuda:2020iql , it is conjectured that the radial Teukolsky equation is
isospectral to the following equation, namely, having the same spectra as

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell
m}^{(s)}\left(z;\omega\right)}{dz}\right)+\left[\left(2M\omega\right)^{2}-V_{\ell
m}^{(s)}(z)\right]\Phi_{\ell m}^{(s)}\left(z;\omega\right)=0,$ (58)

with

$V_{\ell
m}^{(s)}(z)=\frac{\Delta}{z^{2}}\left[4a^{2}\omega^{2}+\frac{4a\omega\left(m-a\omega\right)}{z}+\frac{{}_{s}A_{\ell
m}+s(s+1)-a\omega\left(2m-a\omega\right)}{z^{2}}-\frac{s^{2}-1}{z^{3}}\right],$
(59)

where $M$ and $a$ are mass and spin parameter of the Kerr black hole, $\omega$
is a frequency of the linear fields, and ${}_{s}A_{\ell m}$ is a separation
constant of the spin-weighted spheroidal harmonics. Here, we have defined

$\begin{split}\Delta=&z\left(z-1\right),\\\
z=&\frac{r-r_{-}}{r_{+}-r_{-}},\end{split}$ (60)

where $r_{+}$ and $r_{-}$ are, respectively, radii of the outer and inner
horizons. In the current coordinate, the locations of the outer horizon and
infinity correspond to $z=1$ and $z=\infty$, respectively.

Equation (58) can be reduced to the static Regge-Wheeler equation (5) smoothly
in the static limit $\omega\to 0$.131313The valuable $\Phi^{(s)}_{\ell m}$
seems to be generated from the Newman-Penrose valuable by a highly-nontrivial
integral transformation Whiting:1988vc . For the validity of the
isospectrality including the static limit, the further investigation on the
integral transformation needs. The authors thank this point by Masato Nozawa.
By introducing a new variable defined by $\phi_{\ell}^{(s)}(z)=\Phi_{\ell
m}^{(s)}/z$ as in Eq. (16), one can show the existence of a hidden
supersymmetric structure in a completely parallel manner as the Schwarzschild
black hole case. The variable $\Phi_{\ell m}^{(s)}$ thus has no decaying terms
at large distances under the regularity condition at the horizon.

##  VI Summary and Discussion

We have investigated the underlying symmetric structure of the vanishing of
spin-$s$-field Love numbers of the Schwarzschild and Kerr black holes in terms
of spacetime symmetry in a unified manner for the static perturbations. This
is the first to explain the vanishing of spin-$s$-field Love numbers of the
Schwarzschild black hole in terms of the hidden symmetric structure arising
from spacetime symmetry. For the Kerr black hole case, completely parallel
argument can be made based on the conjecture Hatsuda:2020iql .

The key observation is that the static spin-$s$ field perturbation to the
Schwarzschild black hole can be reduced with the harmonic decomposition to a
set of infinite static scalar fields in ${\rm AdS}_{2}$. Note that the
emergence of the ${\rm AdS}_{2}$ geometry is not derived from the enhancement
of the isometry of the background spacetime itself. A weakly time-dependent
perturbation can also be reduced into a set of infinite time-dependent scalar
fields in ${\rm AdS}_{2}$ in a parallel manner.

The hidden supersymmetric structure exists for the static spin-$s$ field
perturbation. In the reduced system of scalar fields in ${\rm AdS}_{2}$, each
scalar field is associated with its pair, implying that all multipole modes of
the perturbation can be regarded as symmetric partners which can be understood
from the property of the supersymmetry algebra. The generator of the
supersymmetric structure is constructed from a closed conformal Killing vector
field of ${\rm AdS}_{2}$. Consequently, there exists a radially conserved
quantity. The vanishing of the spin-$s$-field Love numbers of the
Schwarzschild black hole can be understood from this conserved quantity
without explicitly solving the perturbation equations in a unified manner.

We comment on previous works that have independently studied the relation
between the vanishing of Love numbers and hidden symmetries. While the
generators of those hidden symmetries appear to come from geometrical
Hui:2021vcv  or algebraic properties Charalambous:2021kcz ; BenAchour:2022uqo
; Charalambous:2022rre , they would have deeper connection with the hidden
effective ${\rm AdS}_{2}$ geometry. We particularly mention that our ladder
operator includes 1) the generators of the ladder symmetry in Ref. Hui:2021vcv
, which arise from an isometry of the Euclidean ${\rm AdS}_{3}$ at least for
scalar fields and are heuristically constructed for spin-$2$ fields, and 2)
the generator in Ref. BenAchour:2022uqo , which is constructed in terms of the
${\rm SL}(2,\mathbb{R})$ symmetry. Our result supports the claim, “the Love
symmetry exactly reduces to the ${\rm SL}(2,\mathbb{R})$ isometry of the ${\rm
AdS}_{2}$ near horizon black hole geometry” in Ref. Charalambous:2022rre .

Our work has several extensions. First, it is important to confirm the
isospectrality between the radial Teukolsky equation and the alternative
equation (58) in Sec. V. Alternatively, for a straightforward extension of our
analysis to the Kerr black hole, Sasaki-Nakamura Sasaki:135698  or
Chandrasekhar-Detweiler Chandrasekhar:1976zz  formalism might be better than
the Teukolsky’s one because those are smoothly reduced to the Regge-Wheeler
and Zerilli equations in the non-rotating limit. Next, we expect that the
vanishing of Love numbers of black holes beyond General Relativity, e.g., the
Schwarzschild black hole in the Brans-Dicke theory, the Reissner-Nordström
black hole in the Einstein-Maxwell theory Cardoso:2017cfl ; Cardoso:2019upw
etc., can be understood in terms of another “hidden” symmetry. Based on the
hidden symmetry, our forthcoming paper suggests that more than a few black
holes with zero Love numbers exist either beyond General Relativity or in non-
vacuum. For testing theories of gravity, we need deeper theoretical
understanding of a system with or without non-zero Love numbers in advance of
future gravitational-wave observations.

###### Acknowledgements.

The authors wish to express their cordial gratitude to Prof. Takahiro Tanaka,
the Leader of Innovative Area Grants-in-Aid for Scientific Research
“Gravitational wave physics and astronomy: Genesis”, for his continuous
interest and encouragement. The authors also would like to thank Yasuyuki
Hatsuda, Kazumi Kashiyama, Koutarou Kyutoku, Masato Nozawa, and Shijun Yoshida
for fruitful discussions and useful comments. This research is supported by
Grants-in-Aid for Scientific Research (TK and KO: 17H06360, MK: 22K03626, HN:
21K03582, 21H01082, 17H06358, KO: 17H01102, 17H02869, 22H00149) from the Japan
Society for the Promotion of Science. TK is also supported by VILLUM FONDEN
(grant no. 37766), by the Danish Research Foundation, and under the European
Union’s H2020 ERC Advanced Grant “Black holes: gravitational engines of
discovery” grant agreement no. Gravitas–101052587. KO acknowledges support
from the Amaldi Research Center funded by the MIUR program “Dipartimento di
Eccellenza” (CUP:B81I18001170001).

##  Appendix A Polar-type static tidal-field perturbation to Schwarzschild
black holes: Construction and tidal Love numbers

Here, we show the vanishing of tidal Love numbers for the polar-type
perturbation with the aid of the Chandrasekhar transformation
Chandrasekhar:1975zza . Static polar-type perturbations $\Phi_{\ell}^{+}(z)$
are governed by the Zerilli equation in the static limit PhysRevD.2.2141 ;
Jhingan:2002kb ,

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell}^{+}}{dz}\right)-V_{\ell}^{+}(z)\Phi_{\ell}^{+}=0,$
(61)

where

$V_{\ell}^{+}(z):=\frac{\Delta}{z^{2}}\left(\frac{9+9\lambda
z+3\lambda^{2}z^{2}+\lambda^{2}(\lambda+2)z^{3}}{z^{3}(\lambda
z+3)^{2}}\right),$ (62)

with $\lambda=\ell^{2}+\ell-2$. Here, the index $\ell=2,3,\cdots$ is that of
multipole modes.

###  A.1 Construction of polar-type static tidal fields

We generate polar-type perturbations $\Phi_{\ell}^{+}$ from an axial-type
perturbation $\Phi_{\ell}^{-}(z)$ which satisfies the static Regge-Wheeler
equation,

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\Phi_{\ell}^{-}}{dz}\right)-V_{\ell}^{-}(z)\Phi_{\ell}^{-}=0,$
(63)

with

$V_{\ell}^{-}(z):=\frac{\Delta}{z^{2}}\left(\frac{\ell(\ell+1)}{z^{2}}-\frac{3}{z^{3}}\right),$
(64)

which is the same as Eq. (5). We perform the Chandrasekhar transformation
Chandrasekhar:1975zza ,

$\Phi_{\ell}^{-}\to\tilde{\Phi}_{\ell}^{+}=D_{+}\Phi_{\ell}^{-},$ (65)

where

$D_{+}:=\frac{\Delta}{z^{2}}\frac{d}{dz}+\frac{\Delta}{z^{3}\left(1+\lambda
z/3\right)}+\frac{\lambda(\lambda+2)}{6}.$ (66)

Then, one can show that $\tilde{\Phi}_{\ell}^{+}$ satisfies the following
equation,

$\frac{\Delta}{z^{2}}\frac{d}{dz}\left(\frac{\Delta}{z^{2}}\frac{d\tilde{\Phi}_{\ell}^{+}}{dz}\right)-V_{\ell}^{+}\tilde{\Phi}_{\ell}^{+}=0.$
(67)

This is nothing but the static Zerilli equation (61). Thus, the function
$\tilde{\Phi}_{\ell}^{+}$ describes a polar-type static tidal-field
perturbation.

###  A.2 Vanishing Love: Quadrupole case $\ell=2$

We show the vanishing of the quadrupole tidal Love number for the polar-type
perturbation from the axial-type one. The general solution of the axial-type
perturbation with $\ell=2$ is given in Eq. (12), i.e.,

$\begin{split}\Phi_{2}^{-}=&C_{1}z^{3}+C_{2}\frac{3+4z+6z^{2}+12z^{3}+12z^{4}\ln\left(1-1/z\right)}{12z},\end{split}$
(68)

where $C_{1}$, $C_{2}$ are constants. Acting the operator $D_{+}$ in Eq. (66),
we obtain

$\begin{split}\tilde{\Phi}_{2}^{+}=&C_{1}\frac{z\left(-3+6z^{2}+4z^{3}\right)}{3+4z}\\\
&+C_{2}\frac{z\left(13+24z+12z^{2}-12z^{3}-3\left(3-6z^{2}-4z^{3}\right)\ln\left(1-1/z\right)\right)}{9+12z}.\end{split}$
(69)

This is indeed a solution of the static Zerilli equation (61) with $\ell=2$.
Equation (69) shows that the term of $C_{2}$ is logarithmically divergent in
the horizon limit $z\to 1$; to eliminate that term, we impose $C_{2}=0$,
thereby obtaining

$\begin{split}\tilde{\Phi}_{2}^{+}=&C_{1}\frac{z\left(-3+6z^{2}+4z^{3}\right)}{3+4z}.\end{split}$
(70)

It follows that the solution $\tilde{\Phi}_{2}^{+}$ in Eq. (70) is regular at
the horizon $z=1$ and is purely growing at large distances $z\gg 1$, i.e.,
$\Phi_{2}^{+}|_{z\gg 1}\sim z^{3}$. Thus, the quadrupolar tidal Love number of
the polar-type perturbation vanishes. One can show that this is also the case
of higher multipoles in the same manner.

##  Appendix B Hidden supersymmetric structure of a weakly time-dependent
perturbation

We discuss a hidden supersymmetric structure of a weakly time-dependent
spin-$s$ field perturbation. Let us consider a time-dependent spin-$s$ field
perturbation that satisfies the Regge-Wheeler equation Regge:1957td :

$-\partial_{t}^{2}\Phi_{\ell}\left(t,z\right)+\frac{\Delta}{z^{2}}\partial_{z}\left(\frac{\Delta}{z^{2}}\partial_{z}\Phi_{\ell}\left(t,z\right)\right)-\frac{\Delta}{z^{2}}\left(\frac{\ell(\ell+1)}{z^{2}}-\frac{s^{2}-1}{z^{3}}\right)\Phi_{\ell}\left(t,z\right)=0.$
(71)

Now, we assume that the time dependence of the perturbation is weak: the time
scale of the change of the field is much longer than the black hole radius,
i.e., $|\partial_{t}\Phi_{\ell}|^{-1}\gg 1$. We further focus on a “near zone”
such that the time dependence of the field is still weak at a distant region
from the black hole horizon, i.e., $1<z\ll|\partial_{t}\Phi_{\ell}|^{-1}$. We
then have

$\begin{split}\frac{z^{4}}{\Delta}\partial_{t}^{2}\Phi_{\ell}=&\frac{1}{\Delta}\partial_{t}^{2}\Phi_{\ell}+z^{2}\partial_{t}^{2}\Phi_{\ell}+z\partial_{t}^{2}\Phi_{\ell}+\partial_{t}^{2}\Phi_{\ell}+\frac{1}{z}\partial_{t}^{2}\Phi_{\ell}\\\
\simeq&\frac{1}{\Delta}\partial_{t}^{2}\Phi_{\ell},\end{split}$ (72)

where $\simeq$ means an equality within the approximations
$z\ll|\partial_{t}\Phi_{\ell}|^{-1}$ and $|\partial_{t}\Phi_{\ell}|^{-1}\gg
1$.141414This corresponds to the near-zone approximation Castro:2010fd ;
Bertini:2011ga .  Using this property, we obtain an equation for the weakly
time-dependent spin-$s$ field perturbation in the near zone
$1<z\ll|\partial_{t}\Phi_{\ell}|^{-1}$ from Eq. (71):

$-\frac{1}{\Delta}\partial_{t}^{2}\Phi_{\ell}+z^{2}\partial_{z}\left(\frac{\Delta}{z^{2}}\partial_{z}\Phi_{\ell}\right)-\left(\ell(\ell+1)-\frac{s^{2}-1}{z}\right)\Phi_{\ell}=0.$
(73)

We redefine the valuable $\Phi_{\ell}$ in Eq. (73) as

$\phi_{\ell}(t,z)=\frac{\Phi_{\ell}}{z}.$ (74)

One can then obtain

$\left[\Box_{{\rm
AdS}_{2}}-\left(\ell(\ell+1)-\frac{s^{2}}{z}\right)\right]\phi_{\ell}=0.$ (75)

Here, the d’Alembertian $\Box_{{\rm AdS}_{2}}$ is given by

$\square_{{\rm
AdS}_{2}}=-\frac{1}{\Delta}\partial_{t}^{2}+\partial_{z}\left(\Delta\partial_{z}\right),$
(76)

which is the same as Eq. (19), and is based on the line element of ${\rm
AdS}_{2}$,

$g_{ab}^{{\rm AdS}_{2}}dx^{a}dx^{b}=-\Delta dt^{2}+\frac{1}{\Delta}dz^{2},$
(77)

which is the same as Eq. (17). Thus, the weakly time-dependent spin-$s$ field
perturbation can be reduced to a set of infinite time-dependent scalar fields
in ${\rm AdS}_{2}$. As can be seen from the fact that Eq. (75) is the same
form as Eq. (21), one can define the ladder operator (22) and can show a
hidden symmetric structure in a completely parallel manner as the static-field
case.

##  Appendix C Ladder operators from spacetime conformal symmetry and
application to perturbations to Schwarzschild-Tangherlini black holes

We here derive a ladder operator from spacetime conformal symmetry of ${\rm
AdS}_{2}$ described by

$g_{ab}^{{\rm AdS}_{2}}dx^{a}dx^{b}=-\Delta
dt^{2}+\frac{1}{\Delta}dz^{2},~{}~{}\Delta=z(z-1),$ (78)

which is the same as Eq. (17). In particular, we show that the type of the
system (21) is unique in ${\rm AdS}_{2}$ of Eq. (78) admitting a ladder
operator for a generic time-dependent scalar field under the requirement of
the regularity of an effective potential at $z=1$. We further discuss the
application to a weakly time-dependent scalar field perturbation to the
Schwarzschild-Tangherlini black holes Tangherlini:1963bw .

###  C.1 Generic time-dependent scalar field in ${\rm AdS}_{2}$

We consider a time-dependent scalar field in ${\rm AdS}_{2}$ of Eq. (78),
which satisfies

$\left[\Box_{{\rm
AdS}_{2}}-\left({\hat{\ell}}({\hat{\ell}}+1)+P(z)\right)\right]\Psi_{\hat{\ell}}(t,z)=0,$
(79)

where ${\hat{\ell}}$ is a nonnegative real-valued constant, $P$ is assumed to
be a time-independent potential because we are currently interested in a
reduced problem from linear perturbations to a static black hole. Note that
$\hat{\ell}$ is not necessarily an integer. Here, we have defined the
d’Alembertian with the line element (78),

$\Box_{{\rm
AdS}_{2}}=-\frac{1}{\Delta}\partial_{t}^{2}+\partial_{z}\left(\Delta\partial_{z}\right),$
(80)

which is the same as Eq. (19). For later convenience, we introduce a new real-
valued parameter $k$ such that

$k(k+1)={\hat{\ell}}({\hat{\ell}}+1).$ (81)

Solving this quadratic equation for $k$, we obtain

$k_{+}=-\hat{\ell}-1,~{}~{}k_{-}=\hat{\ell},$ (82)

where we have assigned $k_{\pm}$ on the solutions so that $k_{+}<k_{-}$. Note
that $k_{\pm}$ is an integer if and only if $\hat{\ell}$ is an integer. In
terms of $k_{\pm}$, Eq. (79) can be rewritten as

$\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)+P\right)\right]\Psi_{\hat{\ell}}=0.$ (83)

###  C.2 Conditions for commutation relations holding

We first require a commutation relation:

$\left[\Box_{{\rm
AdS_{2}}}-P,\mathcal{D}_{k_{\pm}}\right]=-2k_{\pm}\mathcal{D}_{k_{\pm}}+2Q\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)+P\right)\right],$ (84)

where $\mathcal{D}_{k_{\pm}}$ is a derivative operator, and $Q$ is a function
in ${\rm AdS}_{2}$. Acting this commutation relation on a smooth function
$\Psi_{\hat{\ell}}$ leads to

$\left[\Box_{{\rm
AdS}_{2}}-\left(\left(k_{\pm}-1\right)k_{\pm}+P\right)\right]\mathcal{D}_{k_{\pm}}\Psi_{\hat{\ell}}=\left(\mathcal{D}_{k_{\pm}}+2Q\right)\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)+P\right)\right]\Psi_{\hat{\ell}}.$ (85)

One can see that when $\Psi_{\hat{\ell}}$ is a solution of Eq. (83), the
right-hand side vanishes, obtaining

$\left[\Box_{{\rm
AdS}_{2}}-\left(\left(k_{\pm}-1\right)k_{\pm}+P\right)\right]\mathcal{D}_{k_{\pm}}\Psi_{\hat{\ell}}=0.$
(86)

Noting that $(k_{+}-1)k_{+}=(\hat{\ell}+1)(\hat{\ell}+2)$ and
$(k_{-}-1)k_{-}=(\hat{\ell}-1)\hat{\ell}$, this is an equation of motion for a
scalar field $\mathcal{D}_{k_{\pm}}\Psi_{\hat{\ell}}$ with a shifted parameter
$\hat{\ell}\to\hat{\ell}\pm 1$, implying that the operator
$\mathcal{D}_{k_{\pm}}$ generates another scalar field
$\mathcal{D}_{k_{\pm}}\Psi_{\hat{\ell}}$ with $\hat{\ell}\to\hat{\ell}\pm 1$
keeping the potential function $P$.

Now, we seek for ladder operators $\mathcal{D}_{k_{\pm}}$ and potentials $P$
such that the commutation relation (84) holds. We begin with introducing a
form of a first-order differential operator,

$\mathcal{D}_{k_{\pm}}=V^{a}\nabla_{a}+\mathcal{K},$ (87)

where $V^{a}$ and ${\cal K}$ are a vector field and a function in ${\rm
AdS}_{2}$, respectively. Then, the left-hand side of the commutation relation
(84) is calculated to

$\left[\Box_{{\rm
AdS_{2}}}-P,\mathcal{D}_{k_{\pm}}\right]=2\left(\nabla^{a}V^{b}\right)\nabla_{a}\nabla_{b}+\left(\Box_{{\rm
AdS}_{2}}V^{b}+V_{a}R^{ab}+2\nabla^{b}{\cal K}\right)\nabla_{b}+\Box_{{\rm
AdS}_{2}}{\cal K}+V^{a}\nabla_{a}P,$ (88)

where $R_{ab}$ is the Ricci tensor of ${\rm AdS}_{2}$. Here, we have assumed
that the commutation relation acts on a smooth function. The right-hand side
is calculated to

$\begin{split}&-2k_{\pm}\mathcal{D}_{k_{\pm}}+2Q\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)+P\right)\right]\\\ &=2Q\Box_{{\rm
AdS}_{2}}-2k_{\pm}V^{a}\nabla_{a}+\left[-2k_{\pm}{\cal
K}-2Q\left(k_{\pm}(k_{\pm}+1)+P\right)\right].\end{split}$ (89)

With these explicit forms, the commutation relation (84) is divided into the
following three conditions:

$\nabla_{a}V_{b}+\nabla_{b}V_{a}=2Qg_{ab},$ (90) $\Box_{{\rm
AdS}_{2}}V^{a}+R^{ab}V_{b}+2\nabla^{a}{\cal K}=-2k_{\pm}V^{a},$ (91)
$\Box_{{\rm AdS}_{2}}{\cal K}+V^{a}\nabla_{a}P=-2k_{\pm}{\cal
K}-2Q\left[k_{\pm}(k_{\pm}+1)+P\right].$ (92)

Let us investigate these conditions one by one. We focus on the case $Q\neq
0$.151515For the case $Q=0$, Eq. (90) corresponds to a Killing equation. All
the Killing vector fields are not compatible with the conditions (91) and (92)
for the time-independent potential $P(z)$. Then, Eq. (90) corresponds to a
conformal Killing equation. Taking the trace of Eq. (90), we obtain

$Q=\frac{1}{2}\nabla_{a}V^{a}.$ (93)

For the conformal Killing vector field $V^{a}$ in two dimensions, we have

$\Box_{{\rm AdS}_{2}}V_{a}+R_{ab}V^{b}=0.$ (94)

Therefore, Eq. (91) is reduced to

$\nabla^{a}{\cal K}=-k_{\pm}V^{a}.$ (95)

Acting $\epsilon^{ab}\nabla_{a}$ with the Levi-Civita tensor $\epsilon^{ab}$
from the left-hand side leads to

$\epsilon^{ab}\nabla_{a}\nabla_{b}{\cal
K}=-k_{\pm}\epsilon^{ab}\nabla_{a}V_{b}.$ (96)

The left-hand side vanishes, thereby yielding a condition,

$\nabla_{a}V_{b}=\nabla_{b}V_{a}.$ (97)

This means that the conformal Killing vector fields $V^{a}$ are “closed”. In
${\rm AdS}_{2}$, there exist three independent closed conformal Killing vector
fields. Then, the conditions (90)–(92) reduce to

$\nabla_{a}V_{b}=\left(\nabla_{c}\zeta^{c}\right)g_{ab},$ (98)

which is the closed conformal Killing equation, and

$\nabla^{a}{\cal K}=-k_{\pm}V^{a},$ (99)

which is the same as Eq. (95), and

$\Box_{{\rm AdS}_{2}}{\cal K}+V^{a}\nabla_{a}P=-2k_{\pm}{\cal
K}-\left(\nabla_{a}\zeta^{a}\right)\left[k_{\pm}(k_{\pm}+1)+P\right].$ (100)

###  C.3 Ladder operators from spacetime conformal symmetry

From the conditions (98)–(100), we derive ladder operators and possible
potential forms. Although there exist three independent closed conformal
Killing vector fields satisfying Eq. (98), we use here solely:161616The
assumption of the time independence of the potential forces one to choose the
closed conformal Killing vector field (101) only. The other closed conformal
Killing vector fields are not compatible with the conditions (99) and (100);
however, if we admit a time dependence of the potential, a ladder operator can
be constructed in the almost parallel manner as in the present subsection.

$\begin{split}V^{a}\frac{\partial}{\partial
x^{a}}=&z(z-1)\frac{\partial}{\partial{z}}.\end{split}$ (101)

Then, integrating Eq. (99), ${\cal K}$ is calculated to

${\cal K}=-k_{\pm}\left(z-c_{0}\right),$ (102)

where $c_{0}$ is an integration constant. Thus, the operator $\mathcal{D}_{k}$
takes a form,

$\mathcal{D}_{k_{\pm}}=z(z-1)\partial_{z}-k_{\pm}\left(z-c_{0}\right).$ (103)

This ladder operator works for the potential $P$ that satisfies the last
condition (100), which is rewritten as

$\begin{split}z\left(z-1\right)\frac{dP}{dz}+\left(2z-1\right)P+\left(-1+2c_{0}\right)k_{\pm}^{2}=0.\end{split}$
(104)

Solving this differential equation for $P$, we obtain

$P=\frac{k_{\pm}^{2}\left(1-2c_{0}\right)z+c_{p}}{z(z-1)},$ (105)

where $c_{p}$ is an integration constant. A regularity for $P$ at $z=1$
requires $c_{0}=1/2+c_{p}/(2k_{\pm}^{2})$. Then, the ladder operator and the
potential take forms,

$\mathcal{D}_{k_{\pm}}=z\left(z-1\right)\partial_{z}-\frac{k_{\pm}}{2}\left(2z-1-\frac{c_{p}}{k_{\pm}^{2}}\right),$
(106)

and

$P=-\frac{c_{p}}{z},$ (107)

respectively. Thus, the ladder operator (106) exists in the system,

$\left[\Box_{{\rm
AdS}_{2}}-\left(k_{\pm}(k_{\pm}+1)-\frac{c_{p}}{z}\right)\right]\Phi_{\hat{\ell}}=0.$
(108)

###  C.4 Application: Perturbations to the Schwarzschild-Tangherlini black
hole

We discuss the application of the above ladder operators to a problem of a
weakly time-dependent perturbation to the Schwarzschild-Tangherlini black hole
in $(n+2)$ dimensions, which is described by Tangherlini:1963bw

$ds^{2}=-\left(1-\frac{1}{\hat{z}^{n-1}}\right)dt^{2}+\frac{1}{1-\frac{1}{\hat{z}^{n-1}}}d\hat{z}^{2}+\hat{z}^{2}d\Omega_{n}^{2},$
(109)

where $\hat{z}$ is a dimensionless areal coordinate and $d\Omega_{n}^{2}$ is
the line element of the unit sphere in $n$ dimensions.

Let us first consider the four-dimensional case $n=2$, i.e., the Schwarzschild
black hole case. As seen in Appendix B (see also Sec. III), the weakly time-
dependent spin-$s$ field perturbation can be reduced to a set of infinite
time-dependent scalar fields satisfying an equation (see Eqs. (75) and (21)),
which take the form of Eq. (108) with $c_{p}=s^{2}$, $z=\hat{z}$, and
$\hat{\ell}=\ell$. Then, the ladder operators in Eq. (106) with $c_{p}=s^{2}$
shift the multipole index $\ell$ into $\ell\pm 1$.

Let us next consider the higher-dimensional case $n\geq 3$. A weakly time-
dependent scalar field perturbation can be reduced to a set of infinite time-
dependent scalar fields satisfying an equation that takes the form of Eq.
(108) with $c_{p}=0$, $z=\hat{z}^{n-1}$, and ${\hat{\ell}}=\ell/(n-1)$ (c.f.,
discussion around Eq. (4.48) in Ref. Charalambous:2022rre ). Then, the ladder
operators (106) with $c_{p}=0$ shift the multipole index ${\hat{\ell}}$ into
${\hat{\ell}}\pm 1$. Thus, in the higher dimensions, the ladder operator
relates a mode with a given $\ell$ with not all modes but with another mode
with $\ell\propto n-1$ for the Schwarzschild-Tangherlini black hole.

##  Appendix D Vanishing Love for polar-type perturbations from the hidden
supersymmetric structure

Here, we show no static response for the polar-type tidal field perturbations
in terms of a radially conserved quantity associated with a symmetric
structure. We map the polar-type perturbation into the axial-type one with the
Chandrasekhar transformation Chandrasekhar:1975zza  and then analyze the
problem by following the strategy in Sec. IV.

###  D.1 Construction of axial-type static tidal fields

In the parallel manner as in Appendix A, we generate axial-type perturbations
$\Phi_{\ell}^{-}$ from a polar-type perturbation $\Phi_{\ell}^{+}$ with the
Chandrasekhar transformation Chandrasekhar:1975zza ,

$\Phi_{\ell}^{+}\to\tilde{\Phi}_{\ell}^{-}=D_{-}\Phi_{\ell}^{+},$ (110)

where

$D_{-}:=\frac{\Delta}{z^{2}}\frac{d}{dz}-\frac{\Delta}{z^{3}\left(1+\lambda
z/3\right)}-\frac{\lambda(\lambda+2)}{6},$ (111)

with $\lambda=\ell^{2}+\ell-2$. One can then show that the function
$\tilde{\Phi}_{\ell}^{-}$ satisfies the static Regge-Wheeler equation (63).
Introducing a new variable,

$\tilde{\phi_{\ell}}(z):=\frac{\tilde{\Phi}_{\ell}^{-}}{z},$ (112)

the static Regge-Wheeler equation for $\tilde{\Phi}_{\ell}^{-}$ leads to

$\displaystyle\left[\square_{{\rm
AdS}_{2}}-\left(\ell(\ell+1)-\frac{4}{z}\right)\right]\tilde{\phi_{\ell}}=$
$\displaystyle 0.$ (113)

This is the same form as Eq. (21).

###  D.2 Vanishing Love from the hidden supersymmetric structure

We construct construct a radially conserved quantity for $\tilde{\phi_{\ell}}$
in the same manner as in Sec. IV:

$Y_{\ell}^{+}:=z^{2}\mathcal{D}_{3}\cdots\mathcal{D}_{\ell-1}\mathcal{D}_{\ell}\tilde{\phi_{\ell}},$
(114)

which satisfies

$\frac{d}{dz}Y_{\ell}^{+}=0.$ (115)

In the same manner as in Sec. IV, one can show that the regular solution at
the horizon connects to the purely growing solution at large distances in
terms of the radially conserved quantity $Y_{\ell}^{+}$, showing the vanishing
of the tidal Love numbers for the polar-type tidal field perturbation for all
$\ell$.

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