MI-HET-830

Ladder Symmetries and Love Numbers of
Reissner–Nordström Black Holes

Mudit Rai,a,b and Luca Santonic

aPittsburgh Particle Physics, Astrophysics, and Cosmology Center,

Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, USA

bMitchell Institute for Fundamental Physics and Astronomy,

Department of Physics and Astronomy, Texas A&M University, College Station,
USA

cUniversité Paris Cité, CNRS, Astroparticule et Cosmologie,

10 Rue Alice Domon et Léonie Duquet, F-75013 Paris, France

###### Abstract

It is well known that asymptotically flat black holes in general relativity
have vanishing tidal Love numbers. In the case of Schwarzschild and Kerr black
holes, this property has been shown to be a consequence of a hidden structure
of ladder symmetries for the perturbations. In this work, we extend the ladder
symmetries to non-rotating charged black holes in general relativity. As
opposed to previous works in this context, we adopt a more general definition
of Love numbers, including quadratic operators that mix gravitational and
electromagnetic perturbations in the point-particle effective field theory. We
show that the calculation of a subset of those couplings in full general
relativity is affected by an ambiguity in the split between source and
response, which we resolve through an analytic continuation. As a result, we
derive a novel master equation that unifies scalar, electromagnetic and
gravitational perturbations around Reissner–Nordström black holes. The
equation is hypergeometric and can be obtained from previous formulations via
nontrivial field redefinitions, which allow to systematically remove some of
the singularities and make the presence of the ladder symmetries more
manifest.

###### Contents

  1. 1 Introduction
  2. 2 Perturbations of Reissner–Nordström black holes: axial sector
  3. 3 Master hypergeometric equation for Reissner–Nordström black hole perturbations
  4. 4 Analytic continuation and the Reissner–Nordström Love numbers
  5. 5 Polar sector: Chandrasekhar duality and equality of Love numbers
  6. 6 Scalar perturbations on Reissner–Nordström spacetime
  7. 7 Ladder symmetries of Reissner–Nordström black holes
  8. 8 Matching to point-particle effective field theory
  9. 9 Conclusions
  10. A Linearized perturbations of Reissner–Nordström black holes
    1. A.1 Linearized perturbations: odd sector
    2. A.2 Linearized perturbations: even sector
  11. B Reissner–Nordström perturbations and removable singularities

##  1 Introduction

The direct detection of gravitational waves by the LIGO/Virgo interferometers
[1] from merging binary systems has opened a new epoch in the study of gravity
and compact objects in the strong-field regime [2]. To maximize the science
return from the increasing number of gravitational-wave observations and the
discovery potential of future detectors [3, 4, 5, 6], ever more accurate
waveform templates are necessary [7, 8, 9, 10]. This requires in particular to
have a precise understanding of the conservative and dissipative dynamics of
two-body systems [11]. In this respect, a relevant role is played by tidal
effects [12]. The tidal deformability of compact objects in a binary system
affects the inspiral and in turn the phase of the emitted gravitational-wave
signal. Measuring tidal effects will offer important insights on the interior
structure of compact objects: it will for instance allow to extract relevant
information about the equation of state of neutron stars [13, 14, 15, 16, 17,
18], reveal the existence of exotic compact objects [19, 20, 21, 22, 23, 24],
and potentially disclose unknown aspects of the physics at the horizon of
black holes.

The tidal deformation of a self-gravitating body is in general parametrized in
terms of complex coefficients. Their real parts are often called Love numbers
[25, 26] and describe the conservative induced response of the body. The
imaginary parts are instead associated to dissipation. It is well known, as a
result of an explicit calculation in general relativity, that the Love numbers
of asymptotically flat black holes in four spacetime dimensions are exactly
zero. For Schwarzschild black holes, the calculation was originally obtained
in [27, 28, 29] (see also [30, 31, 32, 33]). It was later generalized to the
case of Kerr black holes in [34, 35, 36, 37], where it was shown that, in the
presence of spin, the tidal response coefficients are purely imaginary. The
vanishing of the Love numbers of charged, non-rotating black holes in
Einstein–Maxwell theory in four spacetime dimensions was instead pointed out
in [19, 38].

The peculiar property of the vanishing of the induced static response of black
holes is even more mysterious when it is formulated in terms of a large-
distance effective description for the object [39, 40, 41, 42, 43, 44, 45, 46,
47]. The idea is based on the simple intuition that, from far away, any object
appears in first approximation as a point source. Effects associated to its
finite size and its nontrivial internal structure can then be account for in
terms of higher-dimensional operators attached to its worldline. In this
effective-field-theory (EFT) description of the source, the Love numbers
correspond to the coupling constants of higher-dimensional operators. This
provides a robust definition of tidal deformability that is free of
ambiguities related to the choice of coordinates and nonlinearities [43, 48,
49]. On the other hand, the EFT gives a systematic recipe for computing the
induced response of the object, based on a standard matching procedure between
the EFT and the full solution in general relativity. After the matching, the
black hole Love number couplings are found to be zero, appearing therefore to
be associated to a naturalness puzzle in the gravitational EFT [42, 50].
Recent investigations suggest that the vanishing of the Love number couplings
is perhaps not so puzzling in the end: underlying it, in fact, is a hidden
structure of symmetries for the linearized perturbations [51, 52, 53],111See
Refs. [54, 55] for a different proposal. which can be used to explain the
absence of induced response of black holes in four spacetime dimensions and
rule out Love number couplings in the EFT [51].

The goal of the present work is to extend the ladder symmetries of
Schwarzschild and Kerr spacetimes to Reissner–Nordström black holes, and show
that they can be analogously used to rule out Love number couplings. We start
off by clarifying some previously unnoticed properties of the equations
describing the dynamics of the gravito-electromagnetic perturbations. We will
adopt the Chandrasekhar formulation, where the equations of the gravitational
and electromagnetic perturbations are decoupled and take, in the static limit,
a Fuchsian form with four and six regular singular points for the axial and
polar sector, respectively. We will show that all but three singular points of
the Fuchsian differential equations are removable in four spacetime
dimensions, and we will provide the explicit expressions of the field
redefinitions that allow to recast the equations in the standard
hypergeometric form. We will then solve them analytically in terms of finite
polynomials and show in full generality that the Love numbers vanish.222Our
procedure provides an alternative and more systematic explanation of why the
solution in [38] in $D=4$, where $D$ is the number of spacetime dimensions,
was found to be in the form of a finite polynomial, despite the fact that the
equations in the vector sector are of the Heun type. Our results are in
agreement with the ones in [19, 38], although, in contrast to those
references, we adopt here a wider definition of tidal response coefficients,
based on the point-particle EFT (see eq. (49)), which includes also operators
that mix gravitational and electromagnetic perturbations. Related to this, in
the derivation of the static solutions, we will point out a subtlety related
to an ambiguity in the source/response split, which we will address via an
analytic continuation, and explicitly perform the matching with the point-
particle EFT. Finally, we will show that the symmetries of the hypergeometric
equation act on the static Reissner–Nordström perturbations as ladder symmetry
generators, which can be used to understand why the static solutions take the
form of finite polynomials and why the induced response vanishes. Our result
generalizes the one of [51] to black holes with charge, and the one of [56] to
spin-1 and spin-2 fields.

#### Outline:

The paper is organized as follows. In section 2 we briefly review the static
equations for the axial perturbations of Reissner–Nordström black holes. In
section 3 we explicitly show that one of the singularities of the radial
differential equations describing the axial sector can be removed, and we
derive a general master hypergeometric equation for the Reissner–Nordström
static perturbations. In section 4 we discuss a potential ambiguity in the
definition of the Love numbers, which we resolve by performing an analytic
continuation that allows us to unambiguously disentangle the response from the
subleading tail of the external tidal source. In sections 5 and 6 we extend
the results to the polar sector and to spin-0 perturbations, respectively,
showing that their dynamics is also described by the same master equation
introduced in section 3. The ladder symmetries are then derived in section 7.
Finally, we discuss the matching with the point-particle effective field
theory in section 8. Some technical aspects of the calculations are collected
in appendices A and B.

#### Conventions:

Throughout we use the mostly-plus metric signature, and work in natural units
$\hbar=c=1$. We also set $G=1$. We denote $4$-dimensional spacetime indices
using Greek letters, e.g., $\mu,\nu,\rho,\cdots$, denote $3$-dimensional
spatial indices by Latin letters from the beginning of the alphabet, e.g.,
$a,b,c,\cdots$, and we denote angular indices on the $2$-sphere using Latin
indices from the middle of the alphabet, e.g., $i,j,k,\cdots$.

##  2 Perturbations of Reissner–Nordström black holes: axial sector

Reissner–Nordström black holes are static solutions to the Einstein–Maxwell
equations [57]. They are described by the following background metric
$\bar{g}_{\mu\nu}$,

${\rm d}s^{2}=\bar{g}_{\mu\nu}{\rm d}x^{\mu}{\rm
d}x^{\mu}\equiv-\frac{\Delta(r)}{r^{2}}{\rm
d}t^{2}+\frac{r^{2}}{\Delta(r)}{\rm d}r^{2}+r^{2}\left({\rm
d}\theta^{2}+\sin^{2}\theta{\rm d}\varphi^{2}\right)\,,$ (1)

with

$\Delta(r)\equiv r^{2}-2Mr+Q^{2}\,,$ (2)

where $M$ and $Q$ denote the mass and the charge of the black hole
respectively, while the background Maxwell 4-potential is

$\bar{A}_{\mu}=\left(\frac{Q}{r},0,0,0\right)\,.$ (3)

The event and Cauchy horizons of the black hole are determined by the roots of
$\Delta(r)=0$. Denoting them with $r_{+}$ and $r_{-}$ respectively, they are
given by the following expressions in terms of the black hole mass and charge:

$r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}\,.$ (4)

The dynamics of the fluctuations can be studied by perturbing, in the
Einstein–Maxwell equations, the metric tensor $g_{\mu\nu}$ and the vector
potential $A_{\mu}$ around the background solutions $\bar{g}_{\mu\nu}$ and
$\bar{A}_{\mu}$ [58, 59, 60, 61]. The spherical symmetry of the background
ensures that the equations of motion of the axial and polar components of the
perturbations decouple at linear order. For this reason they can be studied
separately. We will start in this section by focusing on the axial components,
while we will discuss the polar sector in section 5.

Using the notation of [57], the radial equations for the axial sector can be
cast, in the static limit, in the form333The notation in eqs. (5), as well as
in eqs. (7) and (8) below, mirrors the one of Ref. [57] except for the fact
that, for simplicity, we drop a superscript $-$ to denote the axial fields
$H_{i}$ and $Z_{i}$. We will reintroduce it when needed below.

$\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta(r)}{r^{2}}H_{2}^{\prime}(r)\right]-\frac{2}{r}\left[\left(\frac{r\,\eta(\ell)}{2}-3M+\frac{2Q^{2}}{r}\right)H_{2}(r)+Q\mu(\ell)H_{1}(r)\right]$
$\displaystyle=0\,,$ (5a) $\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta(r)}{r^{2}}H_{1}^{\prime}(r)\right]-\frac{2}{r}\left[\left(\frac{r\,\eta(\ell)}{2}+\frac{2Q^{2}}{r}\right)H_{1}(r)+Q\mu(\ell)H_{2}(r)\right]$
$\displaystyle=0\,,$ (5b)

where $\eta(\ell)\equiv\ell(\ell+1)$ and $\mu(\ell)\equiv\sqrt{\eta(\ell)-2}$,
and where $H_{1}$ and $H_{2}$ are related to the perturbations of the metric
tensor, $\delta g_{\mu\nu}=g_{\mu\nu}-\bar{g}_{\mu\nu}$, and of the
electromagnetic vector potential, $\delta A_{\mu}=A_{\mu}-\bar{A}_{\mu}$. We
refer to [57] for the precise definitions of $H_{1}$ and $H_{2}$ (see also
appendix A for a short summary of the relation between $H_{1}$ and $H_{2}$ and
the physical perturbations $\delta g_{\mu\nu}$ and $\delta A_{\mu}$). It is
convenient to introduce the $\ell$-dependent quantities

$\displaystyle q_{1}$ $\displaystyle=3M+\sqrt{9M^{2}+4Q^{2}\mu^{2}(\ell)}\,,$
(6a) $\displaystyle q_{2}$
$\displaystyle=6M-q_{1}=3M-\sqrt{9M^{2}+4Q^{2}\mu^{2}(\ell)}\,,$ (6b)

which satisfy $q_{1}q_{2}=-4Q^{2}\mu^{2}(\ell)$. Using the following field
redefinition,444Note that the redefinitions (7) slightly differ with respect
to the ones introduced in [57]. The expressions of [57] can be recovered by
rescaling the field variable $Z_{2}$ in eqs. (7) as
$Z_{2}\rightarrow\frac{2Q\mu}{q_{1}}Z_{2}$.

$\displaystyle H_{1}(r)$
$\displaystyle=\frac{Z_{1}(r)-Z_{2}(r)}{q_{1}-q_{2}}\,,$ (7a) $\displaystyle
H_{2}(r)$
$\displaystyle=\frac{1}{2Q\mu}\frac{q_{1}Z_{2}(r)-q_{2}Z_{1}(r)}{q_{1}-q_{2}}\,,$
(7b)

eqs. (5) decouple, and boil down to [59, 57]

$r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta}{r^{2}}Z_{i}^{\prime}(r)\right]-\frac{1}{r}\left(r\eta(\ell)+\frac{4Q^{2}}{r}-q_{j}\right)Z_{i}(r)=0\,,\qquad(i,j\in\\{1,2\\},\,i\neq
j)\,.$ (8)

Eqs. (8) are second-order, ordinary differential equations with four regular
singular points at the locations $\\{0,r_{-},r_{+},\infty\\}$. Hence, they
belong to the Heun class [62, 63]. The presence of four singularities makes
the equations less amenable to be studied analytically, with respect for
instance to the Schwarzschild (i.e., $Q=0$) case, where they take instead a
hypergeometric form in the static limit. Connection formulas relating
Frobenius solutions at different singular points—which are necessary to
compute the induced static response—are more involved to obtain, and can often
be written analytically in closed form only perturbatively in some limits
(see, e.g., [64, 65, 66]). Although these aspects are well known in the
literature, the new twist that we will add here is that one of the
singularities of the Heun equations (8) can actually be removed, reducing (8)
to a hypergeometric equations. This will significantly simplify the analysis
of the static perturbations, allowing us to derive general analytic solutions,
as we shall now show explicitly.

##  3 Master hypergeometric equation for Reissner–Nordström black hole
perturbations

Fuchsian equations are linear differential equations with coefficients that
are holomorphic functions of the variable except for a finite number of
singular points. The Heun equation is an example, having four regular
singularities and corresponding therefore to the next case in the Fuchsian
series after the hypergeometric equation [63]. With their confluent subcases,
Fuchsian equations are found to describe a wide variety of problems in
physics.

There are special cases in which, in a given singular point of a Fuchsian
equation, both linearly independent solutions happen to be holomorphic. In
these cases, the singularity can be removed via a field redefiniton and, for
this reason, it is often called ‘apparent’ [67]. Removing a singular point can
be particularly convenient, as it allows one to more easily find the solutions
of the equation and study their properties. One notorious example is given by
the Zerilli equation for parity-even perturbations around a Schwarzschild
black hole [68]. In the static limit the equation is of the Heun type, but one
of the four singularities is apparent. As a result, the equation can be recast
in a hypergeometric form [69, 70, 32]. In four spacetime dimensions, the field
redefinition that removes the apparent singularity is related to the
Chandrasekhar transformation [71, 57]: the latter maps the Zerilli equation
into the Regge–Wheeler equation for the odd perturbations [72], which is in
fact hypergeometric in the static limit and can therefore be easily solved
analytically.

The goal of this section is to show that something similar happens with
Reissner–Nordström black holes. Although both the even and odd static
equations of Reissner–Nordström perturbations have, at first sight, more than
three singularities, they can all in fact be reduced to a single master
hypergeometric equation with three regular singular points only. We will start
here from the axial sector described by eqs. (8).

It is convenient to first perform the following change of variable,

$z\equiv\frac{r-r_{-}}{r_{+}-r_{-}}=\frac{r-\left(M-\sqrt{M^{2}-Q^{2}}\right)}{2\sqrt{M^{2}-Q^{2}}}\,,$
(9)

and introduce

$2\mathbb{\mathcal{R}}+1\equiv\frac{M}{\sqrt{M^{2}-Q^{2}}}\,,$ (10)

where $\mathbb{\mathcal{R}}$ is a non-negative, real number. As a result, the
singularities are shifted from $\\{0,r_{-},r_{+},\infty\\}$ to
$\\{-\mathcal{R},0,1,\infty\\}$ in the new variable $z$. Eqs. (8) now take the
form

$\partial_{z}^{2}Z_{i}+\frac{(2z\mathcal{R}+z-\mathcal{R})}{z(z-1)(z+\mathcal{R})}\partial_{z}Z_{i}+\frac{q_{j}(2\mathcal{R}+1)(z+\mathcal{R})-2M\left(\eta
z^{2}+2\mathcal{R}(\eta
z+2)+(\eta+4)\mathcal{R}^{2}\right)}{2M(z-1)z(z+\mathcal{R})^{2}}Z_{i}=0\,,$
(11)

where $i,j=1,2$ and $j\neq i$.

Following, e.g., Ref. [73], it is straightforward to check that the point
$z=-\mathcal{R}$ satisfies all the conditions that are sufficient for it to be
a removable singularity of the Heun equation (11). We leave the details of
this check to appendix B. Once it is established that a singularity is
apparent, it is not hard to find the field redefinition that removes it. In
the case of eq. (11), such field redefinition takes the form

$Z_{i}(r(z))=(z+\mathcal{R})^{-1}\left[\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{4}+A_{i}\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{3}+B_{i}\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{2}+C_{i}\left(z\frac{{\rm d}}{{\rm d}z}\right)+D\right]F(z)\,,$
(12)

where $A_{i}$, $B_{i}$, $C_{i}$ and $D$ are constant coefficients. Choosing
them as follows (see also appendix B),

$\displaystyle A_{i}$
$\displaystyle=-\frac{3(q_{i}+2q_{j}(\mathcal{R}+1))}{(\mathcal{R}+1)(q_{i}+q_{j})}\,,$
(13a) $\displaystyle B_{i}$
$\displaystyle=\frac{q_{i}(2-7\mathcal{R})+11q_{j}(\mathcal{R}+1)}{(\mathcal{R}+1)(q_{i}+q_{j})}\,,$
(13b) $\displaystyle C_{i}$
$\displaystyle=-\frac{3\left(q_{i}\mathcal{R}(\eta+2(\eta-1)\mathcal{R}-2)+2q_{j}(\mathcal{R}+1)^{2}\right)}{(\mathcal{R}+1)^{2}(q_{i}+q_{j})}\,,$
(13c) $\displaystyle D$
$\displaystyle=-\frac{\eta\,\mu^{2}\,\mathcal{R}^{2}}{(\mathcal{R}+1)^{2}}\,,$
(13d)

with $i,j=1,2$ and $j\neq i$, it is easy to check that the Heun equation (11)
is equivalent to the following simple hypergeometric equation:555Note that all
the information about the removed $z=-\mathcal{R}$ ($r=0$) singularity remains
fully encoded in the overall prefactor $(z+\mathcal{R})^{-1}$ of the field
redefiniton (12).

$z(1-z)F^{\prime\prime}(z)+(1+2z)F^{\prime}(z)+[\ell(\ell+1)-2]F(z)=0\,.$ (14)

We stress that there is no index $i$ on $F$, nor in the parameters of (14): in
other words, the axial perturbations $Z_{1}(r)$ and $Z_{2}(r)$ satisfy the
same master hypergeometric equation (14). Dependence on $q_{1}$ and $q_{2}$ is
only through the coefficients (13) of the field redefinition (12), which links
(14) to (11).

Note that the transformation (12) can be greatly simplified by getting rid of
the higher derivatives on $F$ using its equation (14). After straightforward
manipulations, one finds

$Z_{i}(r(z))=\frac{zF^{\prime}(z)}{z+\mathcal{R}}\left[z\frac{A_{i}(1-z)(\eta-(\eta+7)z+1)+3B_{i}(z-1)^{2}+2\eta+(6\eta+15)z^{2}-8(\eta+1)z-1}{(z-1)^{3}}+C_{i}\right]\\\
+\frac{F(z)}{z+\mathcal{R}}\left[z(\eta-2)\frac{A_{i}\left(3z^{2}-4z+1\right)+B_{i}(z-1)^{2}+(\eta+7)z^{2}-(\eta+6)z+1}{(z-1)^{3}}+D\right]\,,$
(15)

which is in the form of a generalized Darboux transformation, e.g. [74].

The master equation (14) can be easily solved in terms of hypergeometric
functions. Since $\ell$ is an integer, the equation is degenerate and the most
general solution of (14) is given by the following superposition of
independent hypergeometric functions [75]:

$F(z)=c_{\ell}(z-1)^{4}\mathbin{{}_{2}{\sf
F}_{1}}(2-\ell,\ell+3,1;z)+d_{\ell}\,(-z)^{1-\ell}\mathbin{{}_{2}{\sf
F}_{1}}(\ell-1,\ell-1,2\ell+2;z^{-1})\,,$ (16)

where $c_{\ell}$ and $d_{\ell}$ are arbitrary integration constants. Note
that, when (16) is used to write the solution for $Z_{i}$, there will be four
constants in total—which we will denote below with $c_{i,\ell}$ and
$d_{i,\ell}$ with $i=1,2$—two for each $Z_{i}$.

Using eqs. (7) and (76), it is then straightforward to write the solutions for
the gravitational and electromagnetic potentials $h_{0}$ and $u_{4}$ (see eqs.
(73) and (74) for their definitions) in full generality, for arbitrary $\ell$,
as

$\displaystyle h_{0}(r)$
$\displaystyle=\frac{2Q\Delta(r)}{\eta(\ell)\mu(\ell)r^{2}}(rH_{2}(r))^{\prime}=\frac{\Delta(r)}{\eta(\ell)\mu(\ell)^{2}r^{2}}\frac{\left[r(q_{1}Z_{2}(r)-q_{2}Z_{1}(r))\right]^{\prime}}{q_{1}-q_{2}}\,,$
(17) $\displaystyle u_{4}(r)$
$\displaystyle=QH_{1}(r)=Q\frac{Z_{1}(r)-Z_{2}(r)}{q_{1}-q_{2}}\,,$ (18)

where the solutions for $Z_{1}(r)$ and $Z_{2}(r)$ are given by eqs. (12) and
(16).

We are interested in the static response. Therefore, we are going to fix the
integration constants $c_{i,\ell}$ and $d_{i,\ell}$ of the boundary-value
problem in such a way that the solution (16) asymptotes a static tidal field
at large distances and is regular at the black hole horizon [27, 28, 29]. One
can easily check that this amounts to setting $d_{i,\ell}=0$, while
$c_{i,\ell}$ correspond to the tidal field amplitudes [19]. Using the series
representation of the hypergeometric function, we find666For an alternative
way of writing the solution for $Z_{i}(r)$, see appendix B.

$Z_{i}(r)=\frac{c_{i,\ell}}{r}\left[\left((r-r_{-})\frac{{\rm d}}{{\rm
d}r}\right)^{4}+A_{i}\left((r-r_{-})\frac{{\rm d}}{{\rm
d}r}\right)^{3}+B_{i}\left((r-r_{-})\frac{{\rm d}}{{\rm
d}r}\right)^{2}+C_{i}\left((r-r_{-})\frac{{\rm d}}{{\rm
d}r}\right)+D\right]\\\
\times(r-r_{+})^{4}\sum_{n=0}^{\ell-2}(-1)^{n}\frac{(\ell-2)!}{n!(\ell-n-2)!}\frac{\Gamma(\ell+n+3)}{\Gamma(n+1)\Gamma(\ell+3)}\left(\frac{r-r_{-}}{r_{+}-r_{-}}\right)^{n}\,,$
(19)

which is manifestly a finite polynomial in $r$ with only positive powers,
except for a $1/r$ term, for each $i=1,2$. At large distances, $Z_{i}(r)$
scale as $\sim r^{\ell+1}$, correctly reproducing the tidal field profile on
flat space. After plugging into the gravitational and electromagnetic
potentials (17) and (18), one can extract the static solutions for $h_{0}$ and
$u_{4}$, and the induced odd response of the Reissner–Nordström black hole.
For instance, as an illustrative example, let us fix $\ell=2$. Using the
explicit solution (19), the potentials (17) and (18) take the form

$\displaystyle h_{0}^{\ell=2}(r)$
$\displaystyle={\cal{A}}_{h_{0}}r^{3}-\left(2{\cal{A}}_{h_{0}}M+\frac{1}{2}{\cal{A}}_{u_{4}}Q\right)r^{2}+Q\left({\cal{A}}_{h_{0}}Q+{\cal{A}}_{u_{4}}M\right)r-\frac{{\cal{A}}_{u_{4}}Q^{3}}{3}\left(1+\frac{M}{r}-\frac{Q^{2}}{2r^{2}}\right),$
(20) $\displaystyle u_{4}^{\ell=2}(r)$
$\displaystyle={\cal{A}}_{u_{4}}r^{3}-\frac{3}{2}({\cal{A}}_{h_{0}}Q+{\cal{A}}_{u_{4}}M)r^{2}+\frac{3Q^{2}}{2}\left({\cal{A}}_{h_{0}}Q+{\cal{A}}_{u_{4}}M\right)-\frac{{\cal{A}}_{u_{4}}Q^{4}}{r}\,,$
(21)

where we conveniently redefined the arbitrary integration constants
$c_{i,\ell=2}$ into ${\cal{A}}_{h_{0}}$ and ${\cal{A}}_{u_{4}}$, corresponding
to the amplitudes of the gravitational and electromagnetic external fields,
respectively.777Assuming that the external source is a purely gravitational
quadrupolar tidal field, we shall set ${\cal{A}}_{u_{4}}=0$ in the solutions
(20) and (21), which reduce to simply
$h_{0}^{\ell=2}(r)={\cal{A}}_{h_{0}}r\Delta(r)$ and
$u_{4}^{\ell=2}(r)=-\frac{3}{2}{\cal{A}}_{h_{0}}Q(r^{2}-Q^{2})$, in agreement
with [19]. Note that, as opposed to the Schwarzschild case ($Q=0$) where the
solutions are polynomials with no inverse powers of $r$, the presence of a
nonzero charge is responsible here for a decaying falloff, which is
proportional to $Q$ and is induced by the mixing between the gravitational and
electromagnetic perturbations on the curved Reissner–Nordström background. The
falloff is at most $1/r^{2}$ for all $\ell$ (see eqs. (17), (18) and (19)):
this implies that, by simple power counting, the static response coefficients
vanish for $\ell\geq 3$, as we will more precisely assess by explicitly
performing the matching with the point-particle EFT in section 8. The case
$\ell=2$, written explicitly in eqs. (20) and (21), requires additional care.
In fact, the $1/r^{2}$ falloff in (20) resembles at face value an induced
quadrupole in the gravitational sector, generated by an external magnetic
field with amplitude ${\cal{A}}_{u_{4}}$. (In the language of the EFT, this
would correspond to a non-vanishing coupling of the quadratic $\ell=2$
operator the mixes the magnetic field and the odd component of the Weyl
tensor, see eq. (49) below.) In the following section, we show that this is
not true: by carefully performing an analytic continuation in the solution, we
will in fact conclude that such a falloff actually corresponds to a subleading
correction to the external source on the curved background, and should not be
interpreted as an induced response.

##  4 Analytic continuation and the Reissner–Nordström Love numbers

To answer the question of whether the $1/r^{2}$ falloff in (20) corresponds to
a nonzero quadrupolar tidal deformation, we will rederive the static solutions
for the perturbations by performing an analytic continuation in
$\ell\in\mathbb{R}$. The analytic continuation will allow us to disentangle
the tidal source from the response field, and provide a clear interpretation
for the decaying terms in (20) and (21). Note that a similar procedure has
been employed before in [34, 35, 37] to extract the dissipative response of
Kerr black holes in general relativity. To the best of our knowledge, this
subtlety for Reissner–Nordström black holes was not addressed before in the
literature.

Let us start again from our master equation (14). Instead of solving it for
integer $\ell$ as in (16), we will now perform an analytic continuation in
$\ell$ to real (non-integer) values.888We stress that the derivation of the
master equation (14), and in particular the procedure of removing the
singularity from the original Heun equation, did not depend at any point on
the fact that $\ell$ was integer or not. In the analytic continuation sense,
the hypergeometric equation is non-degenerate and the independent solutions
are given by the fundamental ones [76]:

$F(z)=(z-1)^{4}z^{-\ell-3}\left[\tilde{c}_{\ell}\,\mathbin{{}_{2}{\sf
F}_{1}}\left(\ell+3,\ell+3,2\ell+2;z^{-1}\right)-\tilde{d}_{\ell}\,z^{2\ell+1}\mathbin{{}_{2}{\sf
F}_{1}}\left(2-\ell,2-\ell,-2\ell;z^{-1}\right)\right]\,,$ (22)

where $\ell\in\mathbb{R}$, and where $\tilde{c}_{\ell}$ and $\tilde{d}_{\ell}$
are arbitrary integration constants. Regularity at the horizon, $z=1$, fixes

$\tilde{d}_{\ell}=\tilde{c}_{\ell}\frac{\Gamma(2-\ell)^{2}\Gamma(2\ell+2)}{\Gamma(-2\ell)\Gamma(\ell+3)^{2}}\,.$
(23)

Plugging it back into (22), the physical solution for $F(z)$ takes the form

$F(z)={c}_{\ell}(z-1)^{4}z^{-\ell-3}\bigg{[}z^{2\ell+1}\mathbin{{}_{2}{\sf
F}_{1}}\left(2-\ell,2-\ell,-2\ell;z^{-1}\right)\\\
-\frac{\Gamma(-2\ell)\Gamma(\ell+3)^{2}}{\Gamma(2-\ell)^{2}\Gamma(2\ell+2)}\mathbin{{}_{2}{\sf
F}_{1}}\left(\ell+3,\ell+3,2\ell+2;z^{-1}\right)\bigg{]}$ (24)

where we rescaled the overall amplitudes, $\tilde{c}_{\ell}\mapsto{c}_{\ell}$,
for practical convenience. Note that, taking the limit of integer $\ell$ in
(24), one correctly recovers the result (19) obtained in section 3. For
instance, for $\ell\rightarrow 2$, eq. (24) boils down to
$F_{\ell=2}(z)={c}_{\ell}(z-1)^{4}$, in agreement with eq. (16) with
$d_{\ell}=0$. The two approaches give consistent results at the level of the
solution $F(z)$; the crucial difference however is that, as we shall see
explicitly, the analytic continuation allows to correctly interpret the
falloffs, and unambiguously distinguish source and response, which might not
be so obvious from the expressions obtained before assuming integer $\ell$.

From eq. (24), we can obtain the $Z_{i}$ fields in (12) and in turn the odd
gravitational and magnetic potentials $h_{0}$ and $u_{4}$ in eqs. (17) and
(18). Let us focus on $h_{0}$ only, which is the one that for $\ell=2$
contains the $1/r^{2}$ falloff (see eq. (20)), which one could be potentially
tempted to interpret as an induced quadrupolar tidal deformation. After
plugging the explicit solution (24) for $F$, the solution for the
gravitational potential $h_{0}$ takes the following asymptotic expansion at
large distances, up to an overall irrelevant constant factor:

$h_{0}(r\rightarrow\infty)\sim\frac{(\ell+2)^{2}\mu(\ell)^{-2}}{2(\mathcal{R}+1)^{2}(q_{1}-q_{2})}\bigg{[}r^{\ell+1}\left(1+\dots\right)+\frac{\Gamma(1-2\ell)\Gamma(\ell+3)^{2}\rho_{\ell}}{2\ell(\ell+2)^{2}\Gamma(1-\ell)^{2}\Gamma(2\ell+2)}r^{-\ell}\left(1+\dots\right)\bigg{]},$
(25)

where dots denote subleading polynomial corrections of the form $1/r^{n}$
(where $n$ is a positive integer) to the corresponding leading branch of
solution, and where we defined for convenience the constant

$\rho_{\ell}\equiv
1+\frac{2(2\ell+1)(2\mathcal{R}+1)(c_{1,\ell}q_{2}-c_{2,\ell}q_{1})}{(c_{2,\ell}q_{1}-c_{1,\ell}q_{2})\left[\ell(4\mathcal{R}+2)+2\mathcal{R}+1\right]-(c_{2,\ell}q_{1}+c_{1,\ell}q_{2})\sqrt{4(2\ell+1)^{2}\mathcal{R}(\mathcal{R}+1)+9}}\,,$
(26)

where the second term on the right-hand side should be thought of as the ratio
between the external magnetic field amplitude and the amplitude of the
external gravitational tidal potential $h_{0}$. In (25), the piece
$r^{\ell+1}\left(1+\dots\right)$ corresponds to the series expansion of the
external tidal field, while the response is encoded in
$r^{-\ell}\left(1+\dots\right)$. Note that, if $\ell$ is a positive real, non-
integer number, it is clear from (25) that the subleading terms to the tidal
source $r^{\ell+1}$ will never overlap with the second series of terms in
(25); in particular, they will not generate any $r^{-\ell}$ falloff. An
overlap is possible only when $\ell$ is integer: in this case, there is a
mixing between the $r^{-\ell}$ series and the tail of the external tidal
field, introducing a degeneracy at the level of the source/response split and
making therefore the identification of the Love numbers potentially ambiguous.
Such an ambiguity does not arise however for real $\ell$, in which case we can
confidently identify the induced multipole moments as the coefficients

$\lambda_{\ell\in\mathbb{R}}=\frac{\Gamma(1-2\ell)\Gamma(\ell+3)^{2}\rho_{\ell}}{2\ell(\ell+2)^{2}\Gamma(1-\ell)^{2}\Gamma(2\ell+2)}\,.$
(27)

Once the tidal response coefficients are extracted for real $\ell$, we can
then safely take the limit of integer $\ell$ in $\lambda_{\ell\in\mathbb{R}}$
and compute the physical Love numbers, which are found to vanish for all
$\ell\in\mathbb{N}$, $\ell\geq 2$:

$\lambda_{\ell\in\mathbb{N}}=0\,.$ (28)

This shows clearly that the $1/r^{2}$ falloff in the solution (20) for
$\ell=2$ does not correspond to a gravitational quadrupole induced by an
external magnetic field via the $\delta g_{\mu\nu}$ – $A_{\mu}$ mixing, but it
rather belongs to the subleading tail of the source.

##  5 Polar sector: Chandrasekhar duality and equality of Love numbers

The conclusions of the previous sections—i.e., reduction of the perturbation
equations to the master equation (14), static solutions and analytic
continuation—can be easily generalized to the polar sector. Instead of
proceeding as prescribed in appendix B, a more convenient strategy is however
to take advantage of the Chandrasekhar duality [57]. Such duality provides a
mapping between the axial and polar equations, relating even solutions to odd
ones, and viceversa.

The static equations for the radial polar fields are (see appendix A for
definitions and, e.g., [19] for a derivation999We take the opportunity to
correct a typo in the equation for $u_{1}$ in Ref. [19].)

$\displaystyle\frac{{\rm d}}{{\rm
d}r}\left[\Delta(r)H_{0}^{\prime}(r)\right]+\left[\frac{4(Q^{2}-M^{2})}{\Delta(r)}-\frac{2Q^{2}}{r^{2}}-\eta(\ell)\right]H_{0}(r){-}\frac{4Q}{r}\left[u^{\prime}_{1}(r)+\frac{\left(Q^{2}-r^{2}\right)u_{1}(r)}{r\,\Delta(r)}\right]$
$\displaystyle=0\,,$ (29a) $\displaystyle
u_{1}^{\prime\prime}(r)+\frac{\frac{4Q^{2}}{r^{2}}-\eta(\ell)}{\Delta(r)}u_{1}(r){-}\frac{Q}{r}\left[H^{\prime}_{0}(r)-\frac{2\left(Q^{2}-Mr\right)}{r\Delta(r)}H_{0}(r)\right]$
$\displaystyle=0\,.$ (29b)

As in the case of Schwarzschild black holes, the linearized equations for the
axial and polar perturbations around the Reissner–Nordström solution are
related via a duality symmetry, which enforces a degeneracy between the two
sectors. At finite frequency this implies isospectrality properties of the
even and odd modes [57], while in the static limit it entails, as we will
emphasize below, the equality of the Love numbers. Such duality, originally
derived by Chandrasekhar [57], is commonly written in terms of the $Z_{1,2}$
fields (see section 2 and appendix A.2). Here, we re-express it, in the static
limit, in terms of the gravitational and electromagnetic potentials as [77]

$\displaystyle u_{1}(r)$
$\displaystyle=\frac{\Delta(r)}{r\,\eta(\ell)}\partial_{r}u_{4}(r)-\frac{Q}{r}h_{0}(r)\,,$
(30a) $\displaystyle H_{0}(r)$
$\displaystyle=\frac{\Delta(r)}{r^{2}}\partial_{r}\left[\frac{r^{2}\,h_{0}(r)}{\Delta(r)}\right]\,.$
(30b)

It is straightforward to check that the polar equations (29) are linked to the
axial ones (75) via the relations (30). Note that the transformation (30) maps
regular solutions at the horizon into regular solutions—recall that the
regular branch of $h_{0}$ scales as $h_{0}\sim\Delta$ at the
Reissner–Nordström horizon, see eqs. (17) and (19). As a result, it enforces
that the linear static response induced by external polar-type perturbations
must vanish as a consequence of the vanishing of the odd Love numbers [78].

As an example, we shall use the relations (30) to write the solutions for
$H_{0}$ and $u_{1}$ in the case $\ell=2$, starting from eqs. (20) and (21):

$\displaystyle H_{0}^{\ell=2}(r)$
$\displaystyle={\cal{A}}_{H_{0}}r^{2}-2\left({\cal{A}}_{H_{0}}M+{\cal{A}}_{u_{1}}Q\right)r+Q\left({\cal{A}}_{H_{0}}Q+4{\cal{A}}_{u_{1}}M\right)-\frac{2{\cal{A}}_{u_{1}}Q^{3}}{r},$
(31) $\displaystyle u_{1}^{\ell=2}(r)$
$\displaystyle={\cal{A}}_{u_{1}}r^{3}-\frac{1}{2}({\cal{A}}_{H_{0}}Q+6{\cal{A}}_{u_{1}}M)(r^{2}+Q^{2})+\left[{\cal{A}}_{H_{0}}M\,Q+2{\cal{A}}_{u_{1}}(M^{2}+Q^{2})\right]r+\frac{{\cal{A}}_{u_{1}}Q^{4}}{r}\,,$
(32)

where we redefined the amplitudes of the external polar gravitational and
electric potentials as ${\cal{A}}_{H_{0}}$ and ${\cal{A}}_{u_{1}}$,
respectively. Under the assumption that the external source is purely
gravitational (i.e., ${\cal{A}}_{u_{1}}=0$), one finds
$H_{0}^{\ell=2}(r)={\cal{A}}_{H_{0}}\Delta(r)$ and
$u_{1}^{\ell=2}(r)=\frac{1}{2}{\cal{A}}_{H_{0}}Q\Delta(r)$, in agreement with
[19].

Expressions for generic $\ell$ can be analogously obtained by plugging eqs.
(17) and (18) into (30):

$\displaystyle H_{0}(r)$
$\displaystyle=\frac{\Delta(r)}{r^{2}\eta(\ell)\mu(\ell)^{2}}\frac{\left[r\left(q_{1}Z_{2}(r)-q_{2}Z_{1}(r)\right)\right]^{\prime\prime}}{(q_{1}-q_{2})}\,,$
(33a) $\displaystyle u_{1}(r)$
$\displaystyle=\frac{Q\Delta(r)}{r\eta(\ell)(q_{1}-q_{2})}\left[Z_{1}^{\prime}(r)-Z_{2}^{\prime}(r)-\frac{(r\left(q_{1}Z_{2}(r)-q_{2}Z_{1}(r)\right)^{\prime}}{r^{2}\mu(\ell)^{2}}\right]\,,$
(33b)

where $Z_{1,2}$ are given in eq. (19).

##  6 Scalar perturbations on Reissner–Nordström spacetime

In this short section, we briefly recall the equation of a (static) massless
spin-0 field around a Reissner–Nordström black hole. The goal is to show that,
analogously to the gravitational and electromagnetic perturbations, the scalar
equation can be also recast in the form (14).101010See [56] for a thorough
study of the ladder symmetries of a scalar field on Reissner–Nordström.

Let us start from the Klein–Gordon equation

$\square\Phi=0\,,$ (34)

for a scalar field $\Phi$. In the static limit, after introducing the
coordinate $z$ in (9) and decomposing $\Phi$ in spherical harmonics as
$\Phi(r(z),\theta,\phi)=\sum_{\ell m}S_{\ell}(z)Y_{\ell}^{m}(\theta,\phi)$,
the equation for the radial profile $S_{\ell}(z)$ takes the hypergeometric
form,

$z(1-z)S_{\ell}^{\prime\prime}(z)+(1-2z)S_{\ell}^{\prime}(z)+\ell(\ell+1)S_{\ell}(z)=0\,.$
(35)

It is straightforward to check that the scalar equation (35) reduces to our
master equation (14) after performing the Darboux transformation111111Note
that (35) is a hypergeometric equation in the standard form i.e.,
$z(1-z)S_{\ell}^{\prime\prime}(z)+[c-(a+b+1)z]S_{\ell}^{\prime}(z)-abS_{\ell}(z)=0$,
with parameters $a=-\ell$, $b=\ell+1$ and $c=1$. The Darboux transformation
(36) simply corresponds to a Gauss’ contiguous relation, shifting $a\mapsto
a-2$ and $b\mapsto b-2$.

$S_{\ell}(z)=\frac{2z}{(z-1)^{3}}\left[2\ell(\ell+1)(z-1)+2z+1\right]F^{\prime}(z)+\frac{(\ell+2)(\ell-1)}{(z-1)^{3}}\left[\ell(\ell+1)(z-1)+2z\right]F(z)\,.$
(36)

##  7 Ladder symmetries of Reissner–Nordström black holes

Hypergeometric equations possess a large set of symmetry properties leading to
well-known and classified identities. One example of identities is given by
contiguous relations connecting hypergeometric functions with parameters
differing by integer values.

In the case of (14), the contiguous relations of the hypergeometric solutions
translate into ladder symmetries connecting physical perturbations of black
holes with different multipole number. Such ladder symmetries were introduced
in [51] for spin-$0$, $1$ and $2$ perturbations around Schwarzschild and Kerr
black holes, and were later generalized to the case of spin-$0$ field on
Reissner–Nordström spacetime in [56]. One main consequence of the symmetries
is that they constrain the static response of black holes and are responsible
for the vanishing of the black hole Love numbers in general relativity [51,
52, 53]. In this section, we show that ladder symmetries exist also for
electromagnetic and gravitational perturbations of Reissner–Nordström black
holes, completing the program initiated in [51] for non-rotating black holes.

To this end, it is convenient to introduce

$\xi_{\ell}(z)=(z-1)^{-2}F(z),$ (37)

where we made the $\ell$ dependence explicit in $\xi$. In the following, we
will always implicitly assume that $\ell\geq 2$.121212Recall that for $\ell=0$
there is no propagating degree of freedom, while for $\ell=1$ only the
electromagnetic sector contains a propagating mode. In the latter case, there
is only one dynamical equation, and the analysis reduces, in the static limit,
to the one of [51].  From (14), $\xi_{\ell}(z)$ satisfies the equation

$(1-z)^{-2}\partial_{z}\left[z(1-z)^{5}\partial_{z}\left(\frac{\xi_{\ell}(z)}{(z-1)^{2}}\right)\right]+(\ell-2)(\ell+3)\xi_{\ell}(z)=0\,.$
(38)

By setting $\ell=2$ in eq. (38), it is immediate to recognize the following
conserved quantity for $\xi_{2}(z)$:

$P_{2}\equiv
z(z-1)^{5}\partial_{z}\left(\frac{\xi_{2}(z)}{(z-1)^{2}}\right)=\text{constant}\,,$
(39)

which acts as a ‘horizontal symmetry’ for the $\ell=2$ solution and is
constant in $z$ [51]. The conserved charge $P_{2}$ is useful to connect
asymptotics. To see why, let us start by considering the large-$z$ limit. The
two possible falloffs satisfying (39) are, at large $z$, $\xi_{2}^{(1)}\sim
z^{2}$ and $\xi_{2}^{(2)}\sim z^{-3}$: the first one has vanishing $P_{2}$,
while the second one corresponds to a solution with nonzero $P_{2}$. From the
conservation of the horizontal charge $P_{2}$, it is immediate to see that
near the horizon (i.e., $z\rightarrow 1$) $\xi_{2}^{(1)}$ is finite, going as
$(z-1)^{2}$, while $\xi_{2}^{(2)}$ blows up as $(z-1)^{-2}$. One can then
easily show that the divergence of $\xi_{2}^{(2)}$ is associated to a
singularity at the horizon [19, 38], and therefore that $\xi_{2}^{(2)}$ is
unphysical.

The existence of the horizontal symmetry $P_{2}$ is perhaps not surprising, as
it is in fact just related to the Wronskian of the differential equation [51,
66].131313Let us consider the Wronskian ${\cal
W}[\xi_{2},\tilde{\xi}_{2}]\equiv
z(1-z)(\tilde{\xi}_{2}\partial_{z}\xi_{2}-\xi_{2}\partial_{z}\tilde{\xi}_{2})$.
By construction, ${\cal W}$ is constant, i.e. $\partial_{z}{\cal W}=0$, when
$\xi_{2}$ and $\tilde{\xi}_{2}$ solve (38). It is straightforward to check
that, up to an irrelevant proportionality constant, $P_{2}$ in (39) coincides
with the Wronskian ${\cal W}[\xi_{2},\tilde{\xi}_{2}]$ if $\tilde{\xi}_{2}$ is
taken to be the solution that is regular at the horizon, i.e.
$\tilde{\xi}_{2}=(z-1)^{2}$. More interesting is instead the existence of
another set of ladder symmetries, which connect solutions at different level
$\ell$ and allow to generalize the previous argument to higher multipoles. Let
us rewrite the equation of motion (38) as

${\mathscr{H}}_{\ell}\xi_{\ell}=0\,,\qquad{\mathscr{H}}_{\ell}\equiv
z(1-z)\left[(1-z)z\frac{\partial^{2}}{\partial
z^{2}}+(1-2z)\frac{\partial}{\partial z}+\ell(\ell+1)-\frac{4}{1-z}\right]\,,$
(40)

where ${\mathscr{H}}_{\ell}$ plays the role of a ‘Hamiltonian’. It is
straightforward to show that the operators

$\displaystyle D_{\ell}^{+}$
$\displaystyle\equiv-z(1-z)\frac{\partial}{\partial z}+f_{\ell}(z)\,,$ (41a)
$\displaystyle D_{\ell}^{-}$ $\displaystyle\equiv
z(1-z)\frac{\partial}{\partial z}+f_{\ell-1}(z)\,,$ (41b)

where

$f_{\ell}(z)\equiv(\ell+1)z-\frac{(\ell-1)(\ell+3)}{2(\ell+1)}\,,$ (42)

satisfy the following algebra with the Hamiltonian ${\mathscr{H}}_{\ell}$:

${\mathscr{H}}_{\ell+1}D_{\ell}^{+}=D_{\ell}^{+}{\mathscr{H}}_{\ell}\,,\qquad{\mathscr{H}}_{\ell-1}D_{\ell}^{-}=D_{\ell}^{-}{\mathscr{H}}_{\ell}\,.$
(43)

From (43) it is clear that $D_{\ell}^{\pm}$ act as ladder operators: they can
be used to generate solutions at levels $\ell\pm 1$ from a given solution
$\xi_{\ell}$ at level $\ell$, i.e. $\xi_{\ell\pm 1}=D_{\ell}^{\pm}\xi_{\ell}$.
From the definitions (41), one can also easily check that the ladder operators
satisfy the following relations:

$\displaystyle D_{\ell+1}^{-}D_{\ell}^{+}+{\mathscr{H}}_{\ell}$
$\displaystyle=\frac{(\ell-1)^{2}(\ell+3)^{2}}{4(\ell+1)^{2}}\,,$ (44)
$\displaystyle D_{\ell-1}^{+}D_{\ell}^{-}+{\mathscr{H}}_{\ell}$
$\displaystyle=\frac{\left(\ell^{2}-4\right)^{2}}{4\ell^{2}}\,.$ (45)

The generators $D_{\ell}^{\pm}$ are useful because, starting from a seed
solution $\xi_{\ell=2}$, we can construct the full tower of solutions for
$\xi_{\ell}$. In particular, they allow to extend the horizontal symmetries to
higher $\ell$ and find the connection coefficients for the differential
equation (38). Note that the coefficients in $D_{\ell}^{\pm}$ are smooth
functions of $z$ at $z=0,1$: this ensures that $D_{\ell}^{\pm}$ map regular
solutions at the singular points $z=0,1$ into regular solutions. In
particular, this implies that $\xi_{\ell}^{(1)}$, constructed by acting
$\ell-2$ times on $\xi_{2}^{(1)}$ with $D_{\ell}^{+}$, is regular at the
horizon $z=1$ and is a finite polynomial in $z$ growing as $\sim z^{\ell}$ at
$z\rightarrow\infty$. Conversely, the solution $\xi_{\ell}^{(2)}$, generated
from the seed $\xi_{2}^{(2)}$, with decaying falloff at large $z$, is singular
at $z=1$.

In conclusion, in analogy with the Schwarzschild case [51], the vanishing of
the Love numbers is a consequence of the fact that the $\xi_{\ell}$ solution
that is regular at the horizon is a polynomial with only positive powers in
$z$, while the branch with decaying $1/z^{\ell+1}$ profile is associated with
an unphysical divergence at the black hole horizon.

##  8 Matching to point-particle effective field theory

Any object, when seen from distances much larger than the its typical size,
appears in first approximation as a point source. This intuition is
systematized in the point-particle effective theory [39, 40, 41, 42, 43, 44,
45, 46, 47], which takes advantage of the separation of scales in the problem
and provides an organizing principle that allows to describe, in a robust way,
the finite-size properties of the object, as seen by a far-distance observer.
In particular, the EFT provides a conceptually clean definition of tidal Love
numbers that is free of ambiguities due to nonlinearities and gauge invariance
in the theory, which make the partition into tidal source and response in
general relativity less neat, as opposed to Newtonian gravity. In addition,
the EFT allows to more transparently relate phenomena and observables that are
controlled by the same operator coefficient, which might not be so clear in
different frameworks.

In the following, we consider the point-particle EFT of a massive, charged,
self-gravitating object. We will neglect dissipative effects and focus on the
leading order in the time-derivative expansion, which is enough to study the
static response in the absence of angular momentum. By performing the matching
with the full Reissner–Nordström solution in general relativity of sections 2
and 5, we will then derive the Love number couplings.

The EFT can be written as

$S_{\rm EFT}=S_{\rm bulk}+S_{\rm pp}+S_{\rm finite-size}\,,$ (46)

where

$S_{\rm bulk}=\frac{1}{16\pi}\int{\rm
d}^{4}x\sqrt{-g}\left(R-F_{\mu\nu}F^{\mu\nu}\right)\,,$ (47)

is the Einstein–Maxwell action that describes the dynamics of the
gravitational and electromagnetic fields in the bulk,

$S_{\rm pp}=\int{\rm d}\tau\left(-M\sqrt{-g_{\mu\nu}\frac{{\rm d}x^{\mu}}{{\rm
d}\tau}\frac{{\rm d}x^{\nu}}{{\rm d}\tau}}+QA_{\mu}\frac{{\rm d}x^{\mu}}{{\rm
d}\tau}\right)$ (48)

describes the free motion of the point particle, whose worldline is
parametrized by the coordinate $\tau$, and141414The notation is chosen to
mirror the one in [32].

$\begin{split}S_{\rm finite-size}=\sum_{\ell=1}^{\infty}\int{\rm
d}\tau&\Bigg{[}\frac{\lambda_{\ell}^{(E)}}{2\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}E_{a_{\ell})_{T}}\right)^{2}+\frac{\lambda_{\ell}^{(B)}}{4\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}B_{a_{\ell})_{T}b}\right)^{2}\\\
&+\frac{\lambda_{\ell}^{(C_{E})}}{2\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}E_{a_{\ell-1}a_{\ell})_{T}}^{(2)}\right)^{2}+\frac{\lambda_{\ell}^{(C_{B})}}{4\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}B_{a_{\ell-1}a_{\ell})_{T}|b}^{(2)}\right)^{2}\\\
&+\frac{\eta_{\ell}^{(E)}}{\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}E_{a_{\ell})_{T}}\right)\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}E_{a_{\ell-1}a_{\ell})_{T}}^{(2)}\right)\\\
&+\frac{\eta_{\ell}^{(B)}}{2\ell!}\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}B_{a_{\ell})_{T}b}\right)\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}B_{a_{\ell-1}a_{\ell})_{T}|b}^{(2)}\right)\Bigg{]}\end{split}$
(49)

captures the leading finite-size effects. The quantities $E$ and $B$ denote
the electric and magnetic fields, and the electric and magnetic components of
the Weyl tensor $C_{\mu\nu\rho\sigma}$, defined by

$E_{a}\equiv F_{0a}=\dot{A}_{a}-\partial_{a}A_{0}\,,\qquad B_{ab}\equiv
F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}\,,$ (50) $E^{(2)}_{ab}\equiv
C_{0a0b}=-\frac{1}{2}\partial_{a}\partial_{b}h_{00}\,,\qquad
B^{(2)}_{ab|c}\equiv
C_{0abc}=\frac{1}{2}\left(\partial_{a}\partial_{b}h_{c0}-\partial_{a}\partial_{c}h_{b0}\right)=\partial_{a}\partial_{[b}h_{c]0}\,,$
(51)

where we linearized in $h_{\mu\nu}$ around the flat spacetime metric,
$h_{\mu\nu}\equiv g_{\mu\nu}-\eta_{\mu\nu}$. Note that, in the standard spirit
of EFTs, in (49) we added all possible quadratic operators in the fields that
are compatible with the symmetries in the theory. In particular, in addition
to $\lambda_{\ell}^{(E,B)}$ and $\lambda_{\ell}^{(C_{E,B})}$, which correspond
to the Love numbers discussed e.g. in [19, 38], we have in (49) also
$\eta_{\ell}^{(E,B)}$. Such effective couplings capture the response, due to
the object’s finite size, induced by the gravitational-electromagnetic mixing.
In analogy with $\lambda_{\ell}^{(E,B)}$ and $\lambda_{\ell}^{(C_{E,B})}$,
$\eta_{\ell}^{(E,B)}$ can be determined by performing the matching between the
EFT and the Reissner–Nordström solution for the perturbations.

First of all, let us compute, in the effective theory (46), the static
response induced by an external, time-independent gravito-electromagnetic
field acting on the point particle. One way of solving this problem is to find
the solution of the equation for the response field in the presence of the
external source. In practice, we shall decompose the metric and the vector
fields as

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}^{(0)}+h_{\mu\nu}^{(1)},\qquad
A_{\mu}=A_{\mu}^{(0)}+A_{\mu}^{(1)}\,,$ (52)

where $h_{\mu\nu}^{(0)}$ and $A_{\mu}^{(0)}$ denote the external gravitational
and electromagnetic sources, while $h_{\mu\nu}^{(1)}$ and $A_{\mu}^{(1)}$ are
the response fields that we want to compute. By construction,
$h_{\mu\nu}^{(0)}$ and $A_{\mu}^{(0)}$ solve the bulk (static) equations of
motion. Choosing the de Donder gauge for the gravitational sector and the
Lorentz gauge for the electromagnetic one i.e., respectively

$\partial^{\mu}h_{\mu\nu}-\frac{1}{2}\partial_{\nu}h=0\,,\qquad\partial^{\mu}A_{\mu}=0\,,$
(53)

with $h\equiv h^{\alpha}_{\alpha}$, a certain number of simplifications occur
in the static limit in the bulk, e.g.,
$\vec{\nabla}^{2}h_{a}^{a}=\vec{\nabla}^{2}h_{00}=\vec{\nabla}^{2}h_{0a}=\partial_{a}h^{a0}=0$
and
$\partial_{a}h^{ab}=\tfrac{1}{2}\partial^{b}h_{c}^{c}-\tfrac{1}{2}\partial^{b}h_{00}$,
where $\vec{\nabla}^{2}\equiv\partial_{a}\partial^{a}$ is the spatial
laplacian. In the end, the static equations for $h_{\mu\nu}^{(0)}$ and
$A_{\mu}^{(0)}$ in the bulk boil down to

$\vec{\nabla}^{2}h_{\mu\nu}^{(0)}(\vec{x})=0\,,\qquad\vec{\nabla}^{2}A_{\mu}^{(0)}(\vec{x})=0\,.$
(54)

In the following, it will be enough to focus on the solutions for the
components [32]

$h^{(0)}_{00}(\vec{x})=c^{(E)}_{a_{1}\cdots a_{\ell}}x^{a_{1}}\cdots
x^{a_{\ell}}\,,\qquad h_{0a}^{(0)}(\vec{x})=c^{(B)}_{a|b_{1}\cdots
b_{\ell}}x^{b_{1}}\cdots x^{b_{\ell}}\,,$ (55)

and

$A_{0}^{(0)}(\vec{x})=d^{(E)}_{a_{1}\cdots a_{\ell}}x^{a_{1}}\cdots
x^{a_{\ell}}\,,\qquad A_{a}^{(0)}(\vec{x})=d^{(B)}_{a|b_{1}\cdots
b_{\ell}}x^{b_{1}}\cdots x^{b_{\ell}}\,,$ (56)

where $c^{(E)}_{a_{1}\cdots a_{\ell}}$, $c^{(B)}_{a|a_{1}\cdots a_{\ell}}$,
$d^{(E)}_{a_{1}\cdots a_{\ell}}$ and $d^{(B)}_{a|a_{1}\cdots a_{\ell}}$ are
traceless tensors in the spatial indices $a_{1}\cdots a_{\ell}$ and correspond
to the amplitudes of the external probing fields. Using that

$\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}B_{a_{\ell})b}=\partial_{a_{1}}\cdots\partial_{a_{\ell}}A_{b}-\frac{1}{\ell}\left(\partial_{b}\partial_{a_{2}}\cdots\partial_{a_{\ell}}A_{a_{1}}+(\ell-1)\times\text{perms.}\right)\,,$
(57)
$\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}C_{0a_{\ell-1}a_{\ell})b}=\frac{1}{2}\partial_{a_{1}}\cdots\partial_{a_{\ell}}h_{b0}-\frac{1}{2\ell}\left(\partial_{b}\partial_{a_{2}}\cdots\partial_{a_{\ell}}h_{a_{1}0}+(\ell-1)\times\text{perms.}\right)\,,$
(58)

which imply

$\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}B_{a_{\ell})b}\right)^{2}=2\frac{\ell+1}{\ell}(\partial_{a_{1}}\cdots\partial_{[a_{\ell}}A_{b]})^{2}\,,$
(59)
$\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}C_{0a_{\ell-1}a_{\ell})b}\right)^{2}=\frac{\ell+1}{2\ell}(\partial_{a_{1}}\cdots\partial_{[a_{\ell}}h_{b]0})^{2}\,,$
(60)
$\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-1}}B_{a_{\ell})b}\right)\left(\partial_{(a_{1}}\cdots\partial_{a_{\ell-2}}C_{0a_{\ell-1}a_{\ell})b}\right)=\frac{\ell+1}{\ell}(\partial_{a_{1}}\cdots\partial_{[a_{\ell}}A_{b]})(\partial_{a_{1}}\cdots\partial_{[a_{\ell}}h_{b]0})\,,$
(61)

and given the solutions (55) and (56), we can extract the response fields
$h_{\mu\nu}^{(1)}$ and $A_{\mu}^{(1)}$ by solving the inhomogeneous equations

$\displaystyle\vec{\nabla}^{2}A_{0}^{(1)}(\vec{x})$
$\displaystyle=-2\pi(-1)^{\ell}\left(2d^{(E)}_{a_{1}\cdots
a_{\ell}}\lambda_{\ell}^{(E)}+c^{(E)}_{a_{1}\cdots
a_{\ell}}\eta^{(E)}_{\ell}\right)\partial_{a_{1}}\cdots\partial_{a_{\ell}}\delta^{(3)}(\vec{x})\,,$
(62a) $\displaystyle\vec{\nabla}^{2}A_{b}^{(1)}(\vec{x})$
$\displaystyle=-2\pi(-1)^{\ell}\frac{\ell+1}{\ell}\left(2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(B)}+c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\partial_{a_{1}}\cdots\partial_{a_{\ell}}\delta^{(3)}(\vec{x})\,,$
(62b) $\displaystyle\vec{\nabla}^{2}h_{00}^{(1)}(\vec{x})$
$\displaystyle=-4\pi(-1)^{\ell}\left(c^{(E)}_{a_{1}\cdots
a_{\ell}}\lambda_{\ell}^{(C_{E})}+2d^{(E)}_{a_{1}\cdots
a_{\ell}}\eta^{(E)}_{\ell}\right)\partial_{a_{1}}\cdots\partial_{a_{\ell}}\delta^{(3)}(\vec{x})\,,$
(62c) $\displaystyle\vec{\nabla}^{2}h_{0b}^{(1)}(\vec{x})$
$\displaystyle=-8\pi(-1)^{\ell}\frac{\ell+1}{\ell}\left(c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(C_{B})}+2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\partial_{a_{1}}\cdots\partial_{a_{\ell}}\delta^{(3)}(\vec{x})\,.$
(62d)

#### Gravito-magnetic response.

Let us start by solving eqs. (62b) and (62d) for the odd components of the
fields. In Fourier space, we shall write [32]

$\displaystyle A_{b}^{(1)}(\vec{p})$
$\displaystyle=2\pi(-i)^{\ell}\frac{\ell+1}{\ell}\left(2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(B)}+c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\frac{p_{a_{1}}\cdots
p_{a_{\ell}}}{\vec{p}^{2}}\,,$ (63a) $\displaystyle h_{0b}^{(1)}(\vec{p})$
$\displaystyle=8\pi(-i)^{\ell}\frac{\ell+1}{\ell}\left(c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(C_{B})}+2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\frac{p_{a_{1}}\cdots
p_{a_{\ell}}}{\vec{p}^{2}}\,.$ (63b)

Transforming back to real space using the formula

$(-i)^{\ell}\int\frac{{\rm
d}^{3}\vec{p}}{(2\pi)^{3}}\,\operatorname{e}^{i\vec{p}\cdot\vec{x}}\,c^{a_{1}\cdots
a_{\ell}}\frac{p_{a_{1}}\cdots
p_{a_{\ell}}}{\vec{p}^{2}}=(-1)^{\ell}\frac{2^{\ell-2}}{\sqrt{\pi}\,\Gamma(\frac{1}{2}-\ell)}\,c_{a_{1}\cdots
a_{\ell}}\frac{x^{a_{1}}\cdots x^{a_{\ell}}}{|\vec{x}|^{2\ell+1}}\,,$ (64)

we find

$\displaystyle A_{b}^{(1)}(\vec{x})$
$\displaystyle=(-1)^{\ell}\frac{\ell+1}{\ell}\frac{2^{\ell-1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\left(2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(B)}+c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\frac{x^{a_{1}}\cdots
x^{a_{\ell}}}{|\vec{x}|^{2\ell+1}}\,,$ (65a) $\displaystyle
h_{0b}^{(1)}(\vec{x})$
$\displaystyle=(-1)^{\ell}\frac{\ell+1}{\ell}\frac{2^{\ell+1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\left(c^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\lambda_{\ell}^{(C_{B})}+2d^{(B)}_{[b|a_{1}]\cdots
a_{\ell}}\eta^{(B)}_{\ell}\right)\frac{x^{a_{1}}\cdots
x^{a_{\ell}}}{|\vec{x}|^{2\ell+1}}\,.$ (65b)

In order to match these expressions to the full solutions for the
gravitational and magnetic potentials computed in general relativity, we have
to account for the fact that the two computations have been carried out in
different gauges. One option is to explicitly change gauge in one of the
answers and then compare. Perhaps more easily, we will instead match the
magnetic field and the linearized Weyl tensor, which are gauge-invariant
quantities. Summing up the tidal field with the induced profiles, and plugging
into the definitions of the electromagnetic and Weyl tensors, we obtain in
spherical coordinates, for the angular components,

$\displaystyle B_{ij}$ $\displaystyle\propto
r^{\ell+1}\nabla_{[i}Y_{j]}^{(T)}{}^{\ell}_{m}\left[\bar{d}^{(B)}\left(1+(-1)^{\ell}\frac{\ell+1}{\ell}\frac{2^{\ell}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\lambda_{\ell}^{(B)}r^{-2\ell-1}\right)+\bar{c}^{(B)}(-1)^{\ell}\frac{\ell+1}{\ell}\frac{2^{\ell-1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\eta^{(B)}_{\ell}r^{-2\ell-1}\right],$
(66a) $\displaystyle C_{0rij}$ $\displaystyle\propto
r^{\ell}\nabla_{[i}Y_{j]}^{(T)}{}^{\ell}_{m}\Bigg{[}\bar{c}^{(B)}\left(1-(-1)^{\ell}\frac{(\ell+1)(\ell+2)}{\ell(\ell-1)}\frac{2^{\ell+1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\lambda_{\ell}^{(C_{B})}r^{-2\ell-1}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\bar{d}^{(B)}(-1)^{\ell}\frac{(\ell+1)(\ell+2)}{\ell(\ell-1)}\frac{2^{\ell+2}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\eta^{(B)}_{\ell}r^{-2\ell-1}\Bigg{]}\,,$
(66b)

where we redefined the external amplitudes as $\bar{c}^{(B)}$ and
$\bar{d}^{(B)}$, and where $Y_{j}^{(T)}{}^{\ell}_{m}$ are the transverse
vector spherical harmonics on the $2$-sphere, which can be written in terms of
the Levi–Civita symbol as
$Y_{i}^{(T)}{}_{\ell}^{m}=\epsilon_{ij}\nabla^{j}Y_{\ell}^{m}/\sqrt{\ell(\ell+1)}$
[32].

In order to find the response coefficients $\lambda^{(C_{B})}_{\ell}$,
$\lambda^{(B)}_{\ell}$ and $\eta^{(B)}_{\ell}$, we shall compare the EFT
expressions (66) with the full general relativity results. In the notation of
appendix A, we shall write

$\displaystyle B_{ij}$
$\displaystyle=-\frac{u_{4}(r)}{\ell(\ell+1)}\nabla_{[i}Y_{j]}^{(T)}{}^{\ell}_{m}\,,$
(67a) $\displaystyle C_{0rij}$
$\displaystyle=-\left(h_{0}^{\prime}(r)-\frac{2h_{0}(r)}{r}\right)\nabla_{[i}Y_{j]}^{(T)}{}^{\ell}_{m}\,,$
(67b)

where $u_{4}(r)$ and $h_{0}(r)$ are the solutions in the full theory, which
can be read off from our results in section 2. Given the explicit solution
(19) and the considerations in section 4, we can conclude that
$\lambda^{(C_{B})}_{\ell}$, $\lambda_{\ell}^{(B)}$ and $\eta_{\ell}^{(B)}$ all
vanish for all $\ell$. Note that, for $\ell\geq 3$, this follows
straightforwardly from simple dimensional analysis at the level of the full
solutions (67). The matching of the $\ell=2$ mode requires instead some
caution. Indeed, by naively comparing (66) with (67), where $h_{0}$ and
$u_{4}$ are given in (20) and (21) respectively, one could have reached a
wrong conclusion on the value of $\eta_{2}^{(B)}$. On the other hand, from our
analysis in section 4, we already know that the decaying falloff in (20) does
not belong to the response series and should therefore not be associated with
the Love number coupling $\eta_{2}^{(B)}$ in (49). In section 4, we reached
this conclusion by performing an analytic continuation in $\ell$, and it would
be interesting to recover the same result purely at the level of the worldline
EFT. Subleading terms in the full general relativity solutions can be
reconstructed, order by order in $M$ and $Q$, by computing the one-point
functions of $h_{0}$ and $u_{4}$ from Feynman diagrams with one external
source field and increasing number of insertions on the worldline. Since the
questioned $1/r^{2}$ falloff in the full solution (20) scales as $Q^{5}$, we
expect it to correspond to a four-loop diagram at order $Q^{5}$ in the EFT,
with five $A_{\mu}$ insertions on the worldline. We leave this explicit check
and the reconstruction of the full tidal field profiles (20) and (21) for
future work.

#### Gravito-electric response.

Similarly, we can solve eqs. (62a) and (62c) for the even components of the
fields. The solutions for the potentials $A_{0}$ and $h_{00}$ read,
respectively,

$\displaystyle A_{0}^{(1)}(\vec{x})$
$\displaystyle=(-1)^{\ell}\frac{2^{\ell-1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\left(2d^{(E)}_{a_{1}\cdots
a_{\ell}}\lambda_{\ell}^{(E)}+c^{(E)}_{a_{1}\cdots
a_{\ell}}\eta^{(E)}_{\ell}\right)\frac{x^{a_{1}}\cdots
x^{a_{\ell}}}{|\vec{x}|^{2\ell+1}}\,,$ (68a) $\displaystyle
h_{00}^{(1)}(\vec{x})$
$\displaystyle=(-1)^{\ell}\frac{2^{\ell}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\left(c^{(E)}_{a_{1}\cdots
a_{\ell}}\lambda_{\ell}^{(C_{E})}+2d^{(E)}_{a_{1}\cdots
a_{\ell}}\eta^{(E)}_{\ell}\right)\frac{x^{a_{1}}\cdots
x^{a_{\ell}}}{|\vec{x}|^{2\ell+1}}\,.$ (68b)

Summing up the tidal field with the solution for the induced profile, we
obtain, up to an overall irrelevant constant, the following expressions for
the radial electric field and the linearized Weyl tensor:

$\displaystyle E_{r}$ $\displaystyle\propto r^{\ell-1}Y^{\ell
m}\left[\bar{d}^{(E)}\left(1-(-1)^{\ell}\frac{2^{\ell}\sqrt{\pi}(\ell+1)}{\ell\,\Gamma(\frac{1}{2}-\ell)}\lambda_{\ell}^{(E)}r^{-2\ell-1}\right)-\bar{c}^{(E)}(-1)^{\ell}\frac{2^{\ell-1}\sqrt{\pi}(\ell+1)}{\ell\,\Gamma(\frac{1}{2}-\ell)}\eta^{(E)}_{\ell}r^{-2\ell-1}\right]\,,$
(69a) $\displaystyle C_{0r0r}$ $\displaystyle\propto r^{\ell-2}Y^{\ell
m}\Bigg{[}\bar{c}^{(E)}\left(1+(-1)^{\ell}\frac{(\ell+1)(\ell+2)}{\ell(\ell-1)}\frac{2^{\ell}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\lambda_{\ell}^{(C_{E})}r^{-2\ell-1}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\bar{d}^{(E)}(-1)^{\ell}\frac{(\ell+1)(\ell+2)}{\ell(\ell-1)}\frac{2^{\ell+1}\sqrt{\pi}}{\Gamma(\frac{1}{2}-\ell)}\eta^{(E)}_{\ell}r^{-2\ell-1}\Bigg{]}\,.$
(69b)

These expressions should be compared with the solutions for the electric field
and the Weyl tensor in the full calculation in general relativity. For the
radial component of the electric field we simply have

$E_{r}=-\partial_{r}\left(\frac{u_{1}(r)}{r}\right)Y^{\ell m}\,,$ (70)

where $u_{1}$ is given in eq. (33b). Recall that $Z_{i}$ has the form (see eq.
(105))

$Z_{i}(r)=c_{i,\ell}\left\\{\lambda_{i}(\ell)\left[\frac{24Q^{4}}{\eta(\ell)}\frac{1}{r}-\frac{6Q^{2}q_{i}}{\eta(\ell)}+q_{i}r^{2}-r^{3}(\eta(\ell)-2)\right]+\ldots+O(r^{\ell+1})\right\\}\,,$
(71)

where we wrote explicitly only the lowest powers in $r$, while the dots
contain terms from $r^{4}$ up to $r^{\ell+1}$. After plugging into (33b), it
is straightforward to check that $u_{1}$ is a polynomial in $r$ with only
positive powers, except for a $1/r$ term. Comparing the resulting $E_{r}$ with
(69a), this implies, by simple power counting, that
$\eta_{\ell}^{(E)}=\lambda_{\ell}^{(E)}=0$.

The $C_{0r0r}$ component of the Weyl tensor can instead be written as

$C_{0r0r}=-\frac{1}{2}\partial_{r}^{2}\left(\frac{\Delta}{r^{2}}H_{0}(r)\right)+\frac{2Q}{r^{2}}\partial_{r}\left(\frac{u_{1}(r)}{r}\right)\,.$
(72)

Using (33), after comparing with the EFT result (69b), one infers that
$\lambda_{\ell}^{(C_{E})}=0$. Note that, similarly to $u_{1}$, the full
solution for $H_{0}(r)$, obtained from (33a), is also a finite polynomial with
only positive powers except for a $1/r$ term (see eq. (31) for an explicit
example with $\ell=2$). This implies that $C_{0r0r}$ contains subleading
decaying powers of $r$, at most of the form $1/r^{5}$. Despite resembling an
induced $\ell=2$ deformation of the black hole (see eq. (69b)), similarly to
the axial case above, this term does not correspond to a nonzero Love number,
but it should be interpreted as belonging to the subleading tail of the
external source. From a UV perspective, this can be seen by doing an analytic
continuation in $\ell$ in the equations of $Z_{1}$ and $Z_{2}$, as we did in
section 4, and by using (33). Within the EFT, such falloff can be recovered by
computing a two-loop diagram, which corresponds to the order-$Q^{3}$
correction to the $H_{0}$ tidal field solution.

##  9 Conclusions

Black holes are notoriously among the simplest macroscopic objects in nature.
As it is well known, asymptotically flat solutions in general relativity are
completely fixed in terms of just three parameters: mass, spin and charge.
Such uniqueness and simplicity of the solution is in turn inherited by the
perturbations, whose dynamics is highly constrained by the symmetries in the
theory. Some examples include isospectrality and the vanishing of the Love
numbers. The former means that, despite a gravitational wave carries two
helicity modes, there is in the end a single degenerate observable spectrum of
emission, as a consequence of a duality symmetry that relates them [79, 80,
57, 81, 82, 83]. In addition, hidden ladder symmetries in general relativity
establish that the tidal response coefficients of an asymptotically flat, four
dimensional black hole must vanish identically [51, 52], making the black
hole, to a far-distance observer, indistinguishable from a point particle, as
long as static perturbations are concerned.

Although there are good reasons to expect that astrophysical black holes are
neutral (e.g., [84]), studying charged solution turns out to be a very useful
laboratory to clarify the special property of general relativity and shed
light on the symmetry structure of gravity. Our main results here can be
summarized as follows. First, we proved in full generality that, in the static
limit, perturbations of Reissner–Nordström black holes are described by a
single, decoupled master hypergeometric equation, which captures both
gravitational and electromagnetic, as well as spin-0, perturbations. This has
been obtained through a nontrivial (generalized) Darboux-like field
redefinition that brings the number of singularities in the original Fuchsian
differential equation down to three. In the case of the axial sector, our
result agrees with the findings of [38], although it holds more in general and
provides a more explicit and systematic recipe for the removal of the singular
points. The master equation allows in the end to more easily derive general
solutions, with respect to previous formulations in the literature, valid for
generic $\ell$, and unify at the same time different spins.

Second, we pointed out a subtlety in the identification of the tidal response
coefficients that correspond to mixed quadratic operators in the point-
particle EFT, which were not considered in previous analyses [19, 38].
Background nonlinearities of the Reissner–Nordström solution, in addition to
inducing gravitational-electromagnetic mixing, are also responsible for
generating a decaying tail in the full solution of the tidal source. By
performing an analytic continuation, we showed explicitly that such a falloff
should not be ascribed to the black hole’s response—as a naive power counting
might have erroneously led to think—confirming that the Reissner–Nordström
Love number couplings, including those of mixed type, vanish in four spacetime
dimensions.

Finally, we showed that underlying the Reissner–Nordström perturbations is a
hidden structure of ladder symmetries, which is responsible for the explicit
form of the static solutions and the vanishing of the tidal response.

We envision various directions that will be worth exploring in the future on
the subject. First, it would be interesting to check which of the results
derived here—for instance the removal of the singular points or the ladder
symmetries—can be extended to higher $D$-dimensions, in particular to the
gravitational scalar and electric sectors [38, 85]. It is worth recalling
that, even though there is no known Chandrasekhar duality for
Schwarzschild–Tangherlini black holes in $D>4$, the Heun equation for the
gravitational scalar (i.e., even) perturbations can still be recast into
hypergeometric form [86, 87, 32], and it would be interesting to see whether
this remains true in the presence of charge. In addition, it would be
interesting to explicitly perform the matching with the point-particle EFT for
charged black holes in $D>4$ [30, 32, 88], including spin [89, 90] as well as
magnetic charge [91]. To date, the static response of Kerr–Newman black holes,
for generic spin values, is yet unknown. One difficulty to analytically
computing it is that the equations for the Kerr–Newman perturbations do not
seem to be separable—not even in the static limit (e.g., [92]). In the future,
it will interesting to see whether one can find a formulation that allows to
simplify the static equations and obtain an analytic formula for the response
coefficients.

#### Acknowledgements:

We thank Roman Berens and David Pereñiguez for comments on the draft. The work
of LS was supported by the Programme National GRAM of CNRS/INSU with INP and
IN2P3 co-funded by CNES. The work of MR was supported by DOE Grant DE-
SC0010813. MR thanks ICTP, Trieste for hospitality while this work was being
initiated.

##  Appendix A Linearized perturbations of Reissner–Nordström black holes

In this appendix, we briefly review some details and notation of the
linearized equations for the perturbations of a Reissner–Nordström black hole.
The background Reissner–Nordström metric $\bar{g}_{\mu\nu}$ and Maxwell
4-potential $\bar{A}_{\mu}$ can be read off from eqs. (1) and (3)
respectively. The tensor and vector perturbations, $\delta
g_{\mu\nu}=g_{\mu\nu}-\bar{g}_{\mu\nu}$ and $\delta
A_{\mu}=A_{\mu}-\bar{A}_{\mu}$, around the background solution can be
conveniently separated in parity-even (polar) and parity-odd (axial) parts,
and can be parametrized as (see, e.g., [57, 77, 93, 19, 32, 38])

$\begin{split}\delta
g_{\mu\nu}=&\left(\begin{array}[]{cccc}\frac{\Delta(r)}{r^{2}}H_{0}^{\ell
m}(t,r)Y^{\ell m}&{\cal H}_{1}^{\ell m}(t,r)Y^{\ell m}&0&0\\\ {\cal
H}_{1}^{\ell m}(t,r)Y^{\ell m}&\frac{r^{2}}{\Delta(r)}{\cal H}_{2}^{\ell
m}(t,r)Y^{\ell m}&0&0\\\ 0&0&r^{2}K^{\ell m}(t,r)Y^{\ell m}&0\\\
0&0&0&r^{2}\sin^{2}{\theta}K^{\ell m}(t,r)Y^{\ell m}\end{array}\right)\\\
&+\left(\begin{array}[]{cccc}0&0&h_{0}^{\ell m}(t,r)S_{\theta}^{\ell
m}&h_{0}^{\ell m}(t,r)S_{\varphi}^{\ell m}\\\ 0&0&h_{1}^{\ell
m}(t,r)S_{\theta}^{\ell m}&h_{1}^{\ell m}(t,r)S_{\varphi}^{\ell m}\\\
h_{0}^{\ell m}(t,r)S_{\theta}^{\ell m}&h_{1}^{\ell m}(t,r)S_{\theta}^{\ell
m}&0&0\\\ h_{0}^{l\ell m}(t,r)S_{\varphi}^{\ell m}&h_{1}^{\ell
m}(t,r)S_{\varphi}^{\ell m}&0&0\end{array}\right)\,,\end{split}$ (73)

and

$\delta A_{\mu}=\left(\frac{u_{1}^{\ell m}(r,t)}{r}Y^{\ell
m},\frac{u_{2}^{\ell m}(r,t)}{\Delta(r)/r}Y^{\ell m},\frac{u_{3}^{\ell
m}(r,t)}{\eta(l)}Y_{b}^{\ell m}\right)+\left(0,0,\frac{u_{4}^{\ell
m}(r,t)}{\eta(\ell)}S_{b}^{\ell m}\right)\,,$ (74)

where we decomposed in spherical harmonics $Y^{\ell m}(\theta,\varphi)$, and
where we introduced the notations $Y_{b}^{\ell m}\equiv(Y_{,\theta}^{\ell
m},Y_{,\varphi}^{\ell m})$ and $S_{b}\equiv\left(S_{\theta}^{\ell
m},S_{\varphi}^{\ell m}\right)\equiv(-Y_{,\varphi}^{\ell
m}/\sin{\theta},\,\sin{\theta}\,Y_{,\theta}^{\ell m})$. In (73) we fixed the
Regge–Wheeler gauge [72], which allows to remove four independent components
in $\delta g_{\mu\nu}$. Gauge invariance in the spin-1 sector allows in
addition to set $u_{3}(r,t)=0$ in $\delta A_{\mu}$.

###  A.1 Linearized perturbations: odd sector

Plugging (73) and (74) into the Einstein–Maxwell equations, after some
manipulations one can derive the equations for the propagating degrees of
freedom. For instance, in the odd sector, one finds, in the static limit, the
system of equations [19, 93, 77]:

$\displaystyle\Delta(r)\left[r^{2}h_{0}^{\prime\prime}(r)-\frac{4Q}{\eta(\ell)}u_{4}^{\prime}(r)\right]-h_{0}(r)\left[r(r\,\eta(\ell)-4M)+2Q^{2}\right]$
$\displaystyle=0\,,$ (75a)
$\displaystyle\Delta(r)u_{4}^{\prime\prime}(r)+2\left(M-\frac{Q^{2}}{r}\right)u_{4}^{\prime}(r)-\eta(\ell)\,u_{4}(r)+Q\,\eta(\ell)\left[\frac{2h_{0}(r)}{r}-h_{0}^{\prime}(r)\right]$
$\displaystyle=0\,,$ (75b)

which couples the spin-2 and spin-1 degrees of freedom. It is straightforward
to check that, with the identification

$\displaystyle u_{4}(r)$ $\displaystyle=Q\,H_{1}^{-}(r)\,,$ (76a)
$\displaystyle h_{0}(r)$
$\displaystyle=\frac{2Q\,\Delta(r)}{\eta(\ell)\,\mu(\ell)\,r^{2}}(rH_{2}^{-}(r))^{\prime}\,,$
(76b)

the coupled equations (75) are compatible with (5) [57].151515To stress the
difference from the polar fields in the subsection below, in eqs. (76) we
restored the $-$ superscript in $H_{1}^{-}$ and $H_{2}^{-}$, which we had
dropped in eqs. (5) for ease of notation.

###  A.2 Linearized perturbations: even sector

The polar equations can be derived analogously and take the form

$\displaystyle\frac{{\rm d}}{{\rm
d}r}\left[\Delta(r)H_{0}^{\prime}(r)\right]+\left[\frac{4(Q^{2}-M^{2})}{\Delta(r)}-\frac{2Q^{2}}{r^{2}}-\eta(\ell)\right]H_{0}(r){-}\frac{4Q}{r}\left[u^{\prime}_{1}(r)+\frac{\left(Q^{2}-r^{2}\right)u_{1}(r)}{r\,\Delta(r)}\right]$
$\displaystyle=0\,,$ (77a) $\displaystyle
u_{1}^{\prime\prime}(r)+\frac{\frac{4Q^{2}}{r^{2}}-\eta(\ell)}{\Delta(r)}u_{1}(r){-}\frac{Q}{r}\left[H^{\prime}_{0}(r)-\frac{2\left(Q^{2}-Mr\right)}{r\Delta(r)}H_{0}(r)\right]$
$\displaystyle=0\,.$ (77b)

A different, equivalent formulation of the polar equations is given by [57]:

$\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta}{r^{2}}\frac{{\rm d}}{{\rm
d}r}H_{2}^{+}(r)\right]-\frac{1}{r}\left[U(r)H_{2}^{+}(r)+W(r)(-3MH_{2}^{+}(r)+2Q\mu(\ell)H_{1}^{+}(r))\right]$
$\displaystyle=0\,,$ (78a) $\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta}{r^{2}}\frac{{\rm d}}{{\rm
d}r}H_{1}^{+}(r)\right]-\frac{1}{r}\left[U(r)H_{1}^{+}(r)+W(r)(3MH_{1}^{+}(r)+2Q\mu(\ell)H_{2}^{+}(r))\right]$
$\displaystyle=0\,,$ (78b)

where $H_{1}^{+}$ and $H_{2}^{+}$ denote the polar fields and

$\displaystyle U(r)$
$\displaystyle\equiv\left(\mu^{2}(\ell)r+3M\right)W(r)+\varpi-\frac{\mu^{2}(\ell)r}{2}-M+\frac{\mu^{2}(\ell)\Delta(r)}{\varpi}\,,$
(79) $\displaystyle W(r)$
$\displaystyle\equiv\frac{\Delta(r)}{r\varpi^{2}}\left(\mu^{2}(\ell)r+3M\right)+\frac{1}{\varpi}\left(\frac{\mu^{2}(\ell)r}{2}+M\right),$
(80)

with $\varpi\equiv\frac{1}{2}\mu^{2}(\ell)r+3M-\frac{2}{r}Q^{2}$. We refer to
[57] for details. Using the same field redefinition (7),

$\displaystyle H_{1}^{+}(r)$
$\displaystyle=\frac{Z_{1}^{+}(r)-Z_{2}^{+}(r)}{q_{1}-q_{2}}\,,$ (81a)
$\displaystyle H_{2}^{+}(r)$
$\displaystyle=\frac{1}{2Q\mu}\frac{q_{1}Z_{2}^{+}(r)-q_{2}Z_{1}^{+}(r)}{q_{1}-q_{2}}\,,$
(81b)

the equations for $H_{1}^{+}$ and $H_{2}^{+}$ decouple, and one finds

$\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta(r)}{r^{2}}\frac{{\rm d}}{{\rm
d}r}Z_{1}^{+}(r)\right]-\frac{1}{r}\left[U(r)+\frac{1}{2}\left(q_{1}-q_{2}\right)W(r)\right]Z_{1}^{+}(r)$
$\displaystyle=0\,,$ (82a) $\displaystyle r^{2}\frac{{\rm d}}{{\rm
d}r}\left[\frac{\Delta(r)}{r^{2}}\frac{{\rm d}}{{\rm
d}r}Z_{2}^{+}(r)\right]-\frac{1}{r}\left[U(r)-\frac{1}{2}\left(q_{1}-q_{2}\right)W(r)\right]Z_{2}^{+}(r)$
$\displaystyle=0\,.$ (82b)

In terms of $z$, defined in (9), the polar equations (82) can be equivalently
rewritten as

$0=\partial_{z}^{2}Z^{+}_{i}+\frac{(2z\mathcal{R}+z-\mathcal{R})}{z(z-1)(z+\mathcal{R})}\partial_{z}Z^{+}_{i}+\Bigg{[}\frac{\mathcal{R}-z(2\mathcal{R}+1)}{(z-1)z(z+\mathcal{R})^{2}}-\frac{2(\eta-2)M}{(2\mathcal{R}+1)w(z)(z+\mathcal{R})}\\\
-\frac{M\left(z\left(6z-4z^{2}-3\right)+\eta^{2}(z+\mathcal{R})^{3}-\mathcal{R}\right)(M(4(\eta-2)z+4(\eta+1)\mathcal{R}+6)+(2\mathcal{R}+1)(q_{i}-q_{j}))}{4(z-1)z(2\mathcal{R}+1)^{2}w(z)^{2}(z+\mathcal{R})^{2}}\Bigg{]}Z_{i}^{+},$
(83)

where $i,j=1,2$ with $j\neq i$, and where

$w(z)={-}\frac{M\left(\eta(z+\mathcal{R})^{2}+z(-2z+2\mathcal{R}+3)-\mathcal{R}\right)}{(2\mathcal{R}+1)(z+\mathcal{R})}\,.$
(84)

Note that, with respect to the the axial sector (see eq. (11)), the equations
(83) for the polar fields $Z_{i}^{+}$ have two more singularities,
corresponding to the roots of $w(z)$ in (84). Instead of going through the
Fuchsian analysis of appendix B, a more convenient way to study the static
response of polar-type perturbations is to take advantage of the even-odd
duality, discussed in section 5.

##  Appendix B Reissner–Nordström perturbations and removable singularities

In this appendix we focus on the equation (11) describing the axial spin-1 and
spin-2 perturbations of a Reissner–Nordström black hole. The equation has four
regular singularities in the variable $z$ at the locations
$\\{-\mathcal{R},0,1,\infty\\}$, with $\mathcal{R}>0$. As such it belongs to
the Heun class. Via the rescaling

$Z_{i}(r(z))=z^{-\frac{1}{2}}(1-z)^{-\frac{1}{2}}(z+\mathcal{R})\psi_{i}(z)\,,$
(85)

it can be brought to the Heun normal form

$\psi_{i}^{\prime\prime}(z)+\left[\frac{1-\text{$\alpha_{1}^{2}$}}{4z^{2}}+\frac{1-\text{$\alpha_{2}^{2}$}}{4(z-1)^{2}}+\frac{1-\text{$\alpha_{3}^{2}$}}{4(z+\mathcal{R})^{2}}+\frac{\text{$\beta_{1}$}}{z}+\frac{\text{$\beta_{2}$}}{z-1}+\frac{\text{$\beta_{3}$}}{z+\mathcal{R}}\right]\psi_{i}(z)=0\,,$
(86)

where the parameters are

$\text{$\alpha_{1}^{2}$}=0\,,\qquad\text{$\,\alpha_{2}^{2}$}=0\,,\qquad\text{$\alpha_{3}^{2}$}=25\,,$
(87)

and

$\displaystyle\beta_{1}$
$\displaystyle=\eta-\frac{q_{j}(2\mathcal{R}+1)}{2M\mathcal{R}}+\frac{5}{\mathcal{R}}+\frac{9}{2}\,,$
(88a) $\displaystyle\beta_{2}$
$\displaystyle=-\eta+\frac{q_{j}(2\mathcal{R}+1)}{2M(\mathcal{R}+1)}+\frac{5}{\mathcal{R}+1}-\frac{9}{2}\,,$
(88b) $\displaystyle\beta_{3}$
$\displaystyle=-\frac{(2\mathcal{R}+1)(10M-q_{j})}{2M\mathcal{R}(\mathcal{R}+1)}\,,$
(88c)

which satisfy the standard constraint $\beta_{1}+\beta_{2}+\beta_{3}=0$.

Note that $\alpha_{3}^{2}$ in (87) is a positive integer number. As we shall
see, this is associated to the fact that the singularity at $z=-\mathcal{R}$
can be removed via a suitable field redefinition. A discussion on how to test
whether a given singularity is apparent can be found e.g. in [94, 73].
Operationally, we shall first rewrite the equation (86) by expanding the
coefficient of $\psi_{i}(z)$ in series around $z=-\mathcal{R}$:

$(z+\mathcal{R})^{2}\psi_{i}^{\prime\prime}(z)+\left(\sum_{k=0}^{\infty}x_{k}(z+\mathcal{R})^{k}\right)\psi_{i}(z)=0\,,$
(89)

where it is convenient to rewrite the coefficients $x_{k}$ of the Taylor
series as

$x_{k}\equiv\frac{1-l^{2}_{k}}{4}\,,$ (90)

where in particular $l_{0}\equiv\alpha_{3}$. The singular point
$z=-\mathcal{R}$ is apparent if and only if $l_{0}=\alpha_{3}$ is a positive
integer and the following determinant vanishes [73]:

$Y_{l_{0}}(x_{1},x_{2},\cdots,x_{l_{0}})\equiv\text{det}\left[\begin{array}[]{ccccc}x_{1}&1(1-l_{0})&0&\cdots&0\\\
x_{2}&x_{1}&2(2-l_{0})&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
x_{l_{0}-1}&x_{l_{0}-2}&x_{l_{0}-3}&\cdots&(l_{0}-1)(-1)\\\
x_{l_{0}}&x_{l_{0}-1}&x_{l_{0}-2}&\cdots&x_{1}\end{array}\right]=0\,.$ (91)

In our case, $\alpha_{3}=5$ and

$\displaystyle x_{0}$ $\displaystyle=-6\,,$ (92a) $\displaystyle x_{1}$
$\displaystyle=\beta_{3}\,,$ (92b) $\displaystyle x_{2}$
$\displaystyle=\frac{1}{4}\left(\frac{1-\alpha_{1}^{2}}{\mathcal{R}^{2}}+\frac{1-\text{$\alpha_{2}^{2}$}}{(\mathcal{R}+1)^{2}}-\frac{4\text{$\beta_{1}$}}{\mathcal{R}}-\frac{4\text{$\beta_{2}$}}{\mathcal{R}+1}\right)\,,$
(92c) $\displaystyle x_{3}$
$\displaystyle=\frac{1}{2}\left(\frac{1-\alpha_{1}^{2}}{\mathcal{R}^{3}}+\frac{1-\text{$\alpha_{2}^{2}$}}{(\mathcal{R}+1)^{3}}-\frac{2\text{$\beta_{1}$}}{\mathcal{R}^{2}}-\frac{2\text{$\beta_{2}$}}{(\mathcal{R}+1)^{2}}\right)\,,$
(92d) $\displaystyle x_{4}$
$\displaystyle=\frac{1}{4}\left(\frac{3(1-\alpha_{1}^{2})}{\mathcal{R}^{4}}+\frac{3(1-\text{$\alpha_{2}^{2}$})}{(\mathcal{R}+1)^{4}}-\frac{4\text{$\beta_{1}$}}{\mathcal{R}^{3}}-\frac{4\text{$\beta_{2}$}}{(\mathcal{R}+1)^{3}}\right)\,,$
(92e) $\displaystyle x_{5}$
$\displaystyle=\frac{1}{4}\left(\frac{4(1-\text{$\alpha_{1}^{2}$})}{\mathcal{R}^{5}}+\frac{4(1-\text{$\alpha_{2}^{2}$})}{(\mathcal{R}+1)^{5}}-\frac{4\text{$\beta_{1}$}}{\mathcal{R}^{4}}-\frac{4\text{$\beta_{2}$}}{(\mathcal{R}+1)^{4}}\right)\,,$
(92f)

and it is straightforward to check that, with the values (92), the condition
(91) is indeed satisfied:

$Y_{5}(x_{1},x_{2},x_{3},x_{4},x_{5})=0\,.$ (93)

Having established that $z=-\mathcal{R}$ is an apparent singularity, one can
explicitly find the transformation that removes it and recasts the equation
into a new one with one less singular point. To this end, it is convenient to
start from the Heun standard form:161616The parameter $m$ in (94) should not
be confused with the magnetic quantum number of the spherical harmonics
$Y^{\ell m}(\theta,\varphi)$ used in the previous sections.

$y_{i}^{\prime\prime}(z)-\left(\frac{\alpha}{z}+\frac{\beta}{z-1}+\frac{m}{z+\mathcal{R}}\right)y_{i}^{\prime}(z)+\frac{d_{0}+\gamma\delta
z}{z(z-1)(z+\mathcal{R})}y_{i}(z)=0\,,$ (94)

with $\alpha+\beta+m+\gamma+\delta+1=0$, and where $y_{i}(z)$ is related to
the normal-form $\psi_{i}(z)$ via

$\psi_{i}(z)=z^{-\frac{\alpha}{2}}(z-1)^{-\frac{\beta}{2}}(z+\mathcal{R})^{-\frac{m}{2}}y_{i}(z)\,.$
(95)

The parameters in (94) are related to the ones in (86) via the standard
mapping:

$\frac{1}{4}(1-\alpha_{1}^{2})=\frac{\alpha}{2}\left(-\frac{\alpha}{2}-1\right),\quad\frac{1}{4}(1-\alpha_{2}^{2})=\frac{\beta}{2}\left(-\frac{\beta}{2}-1\right),\quad\frac{1}{4}(1-\alpha_{3}^{2})=\frac{m}{2}\left(-\frac{m}{2}-1\right),$
(96) $\beta_{1}=\frac{\alpha\beta}{2}-\frac{d_{0}}{\mathcal{R}}-\frac{\alpha
m}{2\mathcal{R}},\quad\beta_{2}=-\frac{\alpha\beta}{2}+\frac{\gamma\delta+d_{0}}{\mathcal{R}+1}-\frac{\beta
m}{2(\mathcal{R}+1)},\quad\beta_{3}=-\beta_{2}-\beta_{1}.$ (97)

Therefore, in our case, we find

$\alpha=-1\,,\qquad\beta=-1\,,\qquad
m=4\,,\qquad\gamma\delta=2-\eta(\ell)\,,\qquad
d_{0}=-\frac{q_{i}(2\mathcal{R}+1)}{2M}-(\eta-2)\mathcal{R}\,.$ (98)

Given the standard form (94), the transformation that removes $z=-\mathcal{R}$
should be looked for in the form $y(z)={\cal Q}(z\frac{{\rm d}}{{\rm
d}z})F(z)$, where ${\cal Q}(z\frac{{\rm d}}{{\rm d}z})$ is a polynomial of the
differential operator $z\frac{{\rm d}}{{\rm d}z}$, with constant coefficients,
of degree fixed by the characteristic exponent at the singularity
$z=-\mathcal{R}$ [73]. In the present case, the degree of the polynomial is
$m=4$ and the field redefinition takes the form

$y_{i}(z)=\left[\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{4}+A_{i}\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{3}+B_{i}\left(z\frac{{\rm d}}{{\rm
d}z}\right)^{2}+C_{i}\left(z\frac{{\rm d}}{{\rm d}z}\right)+D\right]F(z)\,,$
(99)

where $A_{i}$, $B_{i}$, $C_{i}$ and $D$ are (with $i,j=1,2$ and $j\neq i$)

$\displaystyle A_{i}$
$\displaystyle=-\frac{3(q_{i}+2q_{j}(\mathcal{R}+1))}{(\mathcal{R}+1)(q_{i}+q_{j})}\,,$
(100a) $\displaystyle B_{i}$
$\displaystyle=\frac{q_{i}(2-7\mathcal{R})+11q_{j}(\mathcal{R}+1)}{(\mathcal{R}+1)(q_{i}+q_{j})}\,,$
(100b) $\displaystyle C_{i}$
$\displaystyle=-\frac{3\left(q_{i}\mathcal{R}(\eta+2(\eta-1)\mathcal{R}-2)+2q_{j}(\mathcal{R}+1)^{2}\right)}{(\mathcal{R}+1)^{2}(q_{i}+q_{j})}\,,$
(100c) $\displaystyle D$
$\displaystyle=-\frac{\eta\,\mu^{2}\,\mathcal{R}^{2}}{(\mathcal{R}+1)^{2}}\,.$
(100d)

It is straightforward to check that (94), with (99) and (100), is equivalent
to

$z(1-z)F^{\prime\prime}(z)+(1+2z)F^{\prime}(z)+[\eta(\ell)-2]F(z)=0\,,$ (101)

which is indeed a hypergeometric equation with singularities at
$z=0,1,\infty$. The most general solution to the hypergeometric equation (101)
is given by the linear superposition (16). Then, as we argued in the main
text, requiring regularity of the solution at the black hole event horizon
fixes the integration constants $d_{\ell}$ to zero. The physical solution for
$Z_{i}$ can then be cast in the form (19). In the following, we provide an
equivalent alternative way of writing the solution (19) for $Z_{i}$.

Using standard derivative identities for the hypergeometric function and the
explicit expressions (100) for the constants of $A_{i}$, $B_{i}$, $C_{i}$ and
$D$, we shall rewrite the physical solution for $Z_{i}$ as

$\displaystyle Z_{i}(r(z))$
$\displaystyle=\frac{c_{i,\ell}\,\eta\,(\eta-2)}{24(\mathcal{R}+1)^{2}(z+\mathcal{R})}\Biggl{[}12\,\mathcal{R}(2\mathcal{R}+1)\,\frac{q_{i}}{M}\,z\mathbin{{}_{2}{\sf
F}_{1}}(-\ell-1,\ell,2;z)-24\mathcal{R}^{2}\mathbin{{}_{2}{\sf
F}_{1}}(-\ell-2,\ell-1,1;z)$ $\displaystyle\quad+\eta
z^{3}(\mathcal{R}+1)\Bigl{(}\left(\eta-2\right)(\mathcal{R}+1)\,z\mathbin{{}_{2}{\sf
F}_{1}}(2-\ell,\ell+3,5;z)-2\,(2\mathcal{R}+1)\,\frac{q_{i}}{M}\mathbin{{}_{2}{\sf
F}_{1}}(1-\ell,\ell+2,4;z)\Bigr{)}\Biggr{]}$
$\displaystyle=\frac{c_{i,\ell}\,\eta\,(\eta-2)}{24(\mathcal{R}+1)^{2}(z+\mathcal{R})}\sum_{m=0}^{\ell+2}(-z)^{m}\Biggl{[}\Theta(m-4)\eta(\eta-2)(\mathcal{R}+1)^{2}\,{\ell-2\choose
m-4}\frac{(\ell+3)_{m-4}}{(5)_{m-4}}$
$\displaystyle\quad-\frac{\Theta(m-1)\,12\,\mathcal{R}(2\mathcal{R}+1)\,q_{i}}{M}\,{\ell+1\choose
m-1}\frac{(\ell)_{m-1}}{(2)_{m-1}}-24\mathcal{R}^{2}\,{\ell+2\choose
m}\frac{(\ell-1)_{m}}{(1)_{m}}$
$\displaystyle\quad+\frac{\Theta(m-3)\,2q_{i}\,\eta(\mathcal{R}+1)(2\mathcal{R}+1)}{M}\,{\ell-1\choose
m-3}\,\frac{(\ell+2)_{m-3}}{(4)_{m-3}}\Biggr{]}$ (102)

where $q_{1,2}$ are defined in eqs. (6), ${n\choose x}=\frac{n!}{x!(n-x)!}$ is
the standard binomial coefficient, $(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(a)}$ is
the Pochhammer symbol, and $\Theta(n)$ is the (discrete) Heaviside step
function, $\Theta(n)=1$ for all $n\geq 0$.

As argued in section 3, the series representation of the physical solution for
$Z_{i}$ is finite. In particular, $(z+\mathcal{R})Z_{i}(r(z))$ is a finite
polynomial in $z$ of degree $(\ell+2)$. Since
$z+\mathcal{R}=\frac{r}{2\sqrt{M^{2}-Q^{2}}}=r\,\frac{(1+2\mathcal{R})}{2M}$,
$rZ_{i}(r)$ is also a polynomial of degree $(\ell+2)$ in $r$. Explicitly, we
shall rewrite it as

$\begin{split}Z_{i}(r)&=\frac{c_{i,\ell}\,\eta\,(\eta-2)\,M}{12\,r\,(\mathcal{R}+1)^{2}(1+2\mathcal{R})}\sum_{m=0}^{\ell+2}\alpha_{m,i}\,\mathcal{R}^{m}\,\sum_{a=0}^{m}\,{m\choose
a}\,r^{a}\left(\frac{1+2\mathcal{R}}{2M}\right)^{a}\\\
&=\frac{c_{i,\ell}\,\eta\,(\eta-2)\,}{24\,(\mathcal{R}+1)^{2}}\sum_{a=0}^{\ell+2}r^{a-1}\,\left(\frac{1+2\mathcal{R}}{2M}\right)^{a-1}\,\mathcal{(-R)}^{-a}\,\sum_{m=a}^{\ell+2}\alpha_{m,i}\,\mathcal{R}^{m}\,{m\choose
a}\,,\end{split}$ (103)

where

$\begin{split}\alpha_{m,i}=\Biggl{[}&\theta(m-4)\eta(\eta-2)(\mathcal{R}+1)^{2}\,{\ell-2\choose
m-4}\frac{(\ell+3)_{m-4}}{(5)_{m-4}}\\\
&-\frac{\theta(m-1)\,12\,\mathcal{R}(2\mathcal{R}+1)\,q_{i}}{M}\,{\ell+1\choose
m-1}\frac{(\ell)_{m-1}}{(2)_{m-1}}-24\mathcal{R}^{2}\,{\ell+2\choose
m}\frac{(\ell-1)_{m}}{(1)_{m}}\\\
&+\frac{\theta(m-3)\,2q_{i}\,\eta(\mathcal{R}+1)(2\mathcal{R}+1)}{M}\,{\ell-1\choose
m-3}\,\frac{(\ell+2)_{m-3}}{(4)_{m-3}}\Biggr{]}\,.\end{split}$ (104)

Performing the summation over $m$, we arrive at the final result:

$\begin{split}Z_{i}(r)&=c_{i,\ell}\,\Biggl{\\{}\lambda_{i}(\ell)\,\left[\frac{24Q^{4}}{\eta}\frac{1}{r}-\frac{6Q^{2}q_{i}}{\eta}+q_{i}r^{2}-r^{3}(\eta-2)\right]\\\
&+\frac{\eta\,(\eta-2)\,}{2\,(\mathcal{R}+1)^{2}}\sum_{a>4}^{\ell+2}(-1)^{a}r^{a-1}\left(\frac{1+2\mathcal{R}}{2M}\right)^{a-1}\frac{\Gamma(a+\ell-1)}{\Gamma(a+1)\Gamma(-a+\ell+3)}\times\\\
&\times\Biggl{[}\frac{q_{i}(2\mathcal{R}+1)}{M}\left(-\eta\mathcal{R}\frac{\mathbin{{}_{2}{\sf
F}_{1}}(a-\ell-2,a+\ell-1,a;-\mathcal{R})}{\Gamma(a)}+(\mathcal{R}+1)\frac{\mathbin{{}_{2}{\sf
F}_{1}}(a-\ell-2,a+\ell-1,a-2;-\mathcal{R})}{\Gamma(a-2)}\right)\\\
&+\frac{2(\mathcal{R}+1)^{2}\mathbin{{}_{2}{\sf
F}_{1}}(a-\ell-2,a+\ell-1,a-3;-\mathcal{R})}{\Gamma(a-3)}-\frac{2\eta(\eta-2)\mathcal{R}^{2}\mathbin{{}_{2}{\sf
F}_{1}}(a-\ell-2,a+\ell-1,a+1;-\mathcal{R})}{\Gamma(a+1)}\Biggr{]}\Biggr{\\}}\end{split}$
(105)

where

$\begin{split}\lambda_{i}(\ell)&=\frac{\eta\,(\eta-2)\,(\ell+1)(2\mathcal{R}+1)^{3}}{192M^{4}\mathcal{R}(\mathcal{R}+1)^{4}}\Bigl{[}\mathbin{{}_{2}{\sf
F}_{1}}(-\ell,\ell,2;-\mathcal{R})(q_{i}(2\mathcal{R}+1)-2M(2\mathcal{R}+3))\\\
&\quad-(M(4(\ell-1)\mathcal{R}-6)+q_{i}(2\mathcal{R}+1))\mathbin{{}_{2}{\sf
F}_{1}}(-\ell-1,\ell,2;-\mathcal{R})\Bigr{]}\,.\end{split}$ (106)

We remind that the above solution has been derived under the assumption that
$\ell>1$, which is the physically interesting case where the gravitational
modes are propagating.

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