# Normal modes of Proca fields in AdSd spacetime

David Lopes  Centro de Astrofísica e Gravitação - CENTRA, Departamento de
Física, Instituto Superior Técnico - IST, Universidade de Lisboa - UL, Avenida
Rovisco Pais 1, 1049-001 Lisboa, Portugal  Tiago V. Fernandes  Centro de
Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior
Técnico - IST, Universidade de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001
Lisboa, Portugal  José P. S. Lemos  Centro de Astrofísica e Gravitação -
CENTRA, Departamento de Física, Instituto Superior Técnico - IST, Universidade
de Lisboa - UL, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal

###### Abstract

The normal modes of Proca field perturbations in $d$-dimensional anti-de
Sitter spacetime, AdSd for short, with reflective Dirichlet boundary
conditions, are obtained exactly. Within the Ishibashi-Kodama framework, we
decompose the Proca field in scalar-type and vector-type components, according
to their tensorial behavior on the $(d-2)$-sphere $\mathcal{S}^{d-2}$. Two of
the degrees of freedom of the Proca field are described by scalar-type
components, which in general are coupled due to the mass of the field, but in
AdSd we show that they can be decoupled. The other $d-3$ degrees of freedom of
the field are described by a vector-type component that generically decouples
completely. The normal modes and their frequencies for both the scalar-type
and vector-type components of the Proca field are then obtained analytically.
Additionally, we analyze the normal modes of the Maxwell field as the massless
limit of the Proca field. We find that for scalar-type perturbations in $d=4$
there is a discontinuity in the massless limit, in $d=5$ the massless limit is
well defined using Dirichlet-Neumann rather than Dirichlet boundary
conditions, and in $d>5$ the massless limit is completely well defined, i.e.,
it is obtained smoothly from the massless limit of the scalar-type
perturbations of the Proca field. For vector-type perturbations the Maxwell
field limit is obtained smoothly for all $d$ from the massless limit of the
vector-type perturbations of the Proca field.

Anti-de Sitter, Proca field, normal modes, higher dimensions

##  I Introduction

The anti-de Sitter spacetime Calabi:1962 ; Penrose:1968 ; hawking , or AdS for
short, is the maximally symmetric vacuum solution to the Einstein field
equations with a negative cosmological constant. This spacetime can be
obtained by performing the universal cover of the AdS universe. As a result,
the AdS spacetime is not a globally hyperbolic spacetime and it possesses a
timelike boundary at spatial infinity.

Due to the properties at spatial infinity, there is an intrinsic interest in
asymptotically AdS spacetimes as they can describe systems in a gravitational
box. In particular, pure AdS spacetime is of special importance in the
construction of quantum field theories avis , where two reflective and one
transparent boundary conditions to the fields are possible at infinity. AdS
spacetime is also essential in the formulation of supergravity theories, where
it acts as a natural arena for quantum supersymmetric fields, including the
possibility that these might have negative mass Breitenlohner:1982 . In this
connection, AdS plays a fundamental role in the AdS/CFT conjecture, that
establishes a duality between supergravity as a low energy phenomenon of
string theory in AdS, and a conformal field theory at its boundary maldacena .

Given the importance of the AdS spacetime it is relevant to study and
understand its stability. To linear perturbations AdS is stable, as we will
see below, but to nonlinear perturbations it seems that AdS is unstable. This
instability was explored in bizon1 , where the evolution of the Einstein-
Klein-Gordon system was considered and a large class of arbitrary small
amplitudes of the scalar field was found to have an evolution leading to black
hole formation. It was further reinforced by analyzing a complex scalar field
Buchel:2012 , yet it does not occur for all classes of initial data as there
are islands of stability Masachs:2019 .

Linear stability of a spacetime is also important. To study it, one must
analyze linear perturbations which are described by normal modes for the case
of pure AdS and quasinormal modes in black hole spacetimes be they
asymptotically flat, AdS, or otherwise. In spherical symmetry, the
perturbations are classified as scalar-type, vector-type, and tensor-type
perturbations, which regards their tensorial behavior on the 2-sphere. This
decomposition allows to write the linearized field equations as a radial
Schrödinger-like equation with an effective given potential for each type of
perturbation. Moreover, to obtain normal modes and quasinormal modes, one must
impose boundary conditions at the center or at the black hole horizon if there
is one, and at spatial infinity. The linear stability of a Schwarzschild black
hole spacetime was solved first trough the Regge-Wheeler formalism by
carefully expanding the perturbations regge_wheeler ; zerilli . In ckjpsl
boundary conditions and all types of perturbations in the Schwarzschild-AdS
black hole were imposed and worked out. In konoplya  it was analyzed Proca
massive vector field perturbations in a Schwarzschild background. In redkov
spherical waves of spin-1 particle in anti-de Sitter space-time were
performed. In dolan  massive vector fields on the Schwarzschild spacetime were
further analyzed. For other boundary conditions based on the vanishing of the
energy flux see Herdeiro2 . In fhlc  quasinormal modes of Proca fields in a
Schwarzschild-AdS spacetime were found. In tf  the normal modes of Proca
fields in AdS spacetime were found.

Higher dimensions are important. The universe may be higher dimensional
somewhere, higher dimensions may have existed at some time in the very early
universe, or perhaps they can be constructed or detected in a future
experiment. The study of the physics in higher dimensions also provides a
means of understanding what is intrinsic and important to $d=4$. AdS spacetime
can be extended to higher dimensions which is referred AdSd, where $d$ is the
number of dimensions. Normal modes and quasinormal modes have also been
studied in AdSd spacetimes. In spherical symmetry in higher dimensions, the
perturbations are also classified as scalar-type, vector-type, and tensor-type
perturbations, which regards their behavior now on the $(d-2)$-sphere. Again,
the decomposition allows to write the linearized field equations as a radial
Schrödinger-like equation for each type of perturbation, whose potential now
also depends on the dimension of the spacetime. The normal modes of AdSd were
firstly obtained in burgess  for a scalar field. The problem of linear
stability in higher-dimensional spacetimes was considered in the Ishibashi-
Kodama formalism by expanding the perturbations in higher-dimensional scalar,
vector and tensor spherical harmonics, which independently form a complete
basis on a $(d-2)$-sphere ishibashi_kodama1 . In ishibashiwald  it was found
that in some cases the Dirichlet boundary conditions are not the only suitable
boundary conditions and generalized Robin boundary conditions are also
permitted, a general analysis of the equations that the scalar,
electromagnetic and gravitational perturbations obey in AdSd was made, and a
general formula for the eigenfrequencies in a range of parameters of the
equation was obtained. Moreover, scalar-type, vector-type and tensor-type
gravitational perturbations were studied and the eigenfrequencies given
imposing Dirichlet boundary conditions natario . The result was extended to
the scalar-type and vector-type Maxwell electromagnetic perturbations in
ortega . The problem of linear stability in higher-dimensional spacetimes was
further addressed by expanding perturbations in higher-dimensional scalar,
vector and tensor spherical harmonics, which independently form a complete
basis on a $(d-2)$-sphere, now with applications to black hole spacetimes
ishibashi_kodama2 . In Herdeiro  the wave equation for a Proca field in
$d$-dimensional spherically symmetric black hole spacetimes was obtained with
interest in understanding the Hawking radiation for a Proca field in
$d$-dimensions. In ueda  massive vector field perturbations on extremal and
near-extremal static black holes were analyzed. A study of the Proca
perturbations in $d$-dimensional pure AdS is thus of interest. The mass of the
Proca field introduces a coupling between two scalar-type degrees of freedom.
For black hole spacetimes in general, a decoupling of these two degrees of
freedom does not seem to be analytically allowed. For pure AdS in four
dimensions, however, the scalar-type degrees of freedom can be decoupled by
making a transformation of the fields, and it is thus of interest to know if
this occurs in AdSd.

In this work we obtain the exact expression for normal modes of linear Proca
perturbations in AdSd background, using the Ishibashi-Kodama formalism. We
find that the scalar-type degrees of freedom decouple in AdSd, by making a
linear transformation to the relevant fields. We also study the
electromagnetic perturbations in order to understand the $\mu=0$ limit of the
Proca field. We consulted results in lovenumbers ; myers ; higuchi  on
$(d-2)$-sphere, and use results of the priceless manual abramowitz .

The work is organized as follows. In Sec. II we introduce the field equations
for a Proca field minimally coupled to curved spacetime background. In Sec.
III, we obtain the equations for the Proca field perturbations in pure AdS
spacetime by introducing the Ishibashi-Kodama formalism and further
decomposing the Proca field in scalar-type and vector-type components,
according to their tensorial behavior on the $(d-2)$-sphere. We also decouple
the scalar-type components by making a linear transformation to the relevant
fields. In Sec. IV, we obtain the normal mode eigenfrequencies and
eigenfunctions as a solution to the Proca field equations, which can be put
into Schrödinger-like equations. In Sec. V, we rederive the normal mode
frequencies in AdSd of the Maxwell electromagnetic perturbations and analyze
it in the context of the zero mass limit of the Proca field. In Sec. VI, we
conclude. In Appendix A, we review the properties of spherical harmonics on
the $(d-2)$-sphere. In Appendix B, we study the solutions of the
hypergeometric differential equation.

##  II Proca field in curved spacetime

The action of a Proca field, i.e., a massive vector field, minimally coupled
to the metric field of a generic $d$-dimensional curved spacetime with
negative cosmological constant can be written as

$\displaystyle S=S_{\rm EH}+S_{\rm P}\,\,,$ (1)

where

$\displaystyle S_{\rm EH}=\int d^{d}x\sqrt{-g}\,\frac{R-2\Lambda}{16\pi}\,\,,$
(2)

is the Einstein-Hilbert action,

$\displaystyle S_{\rm P}=-\int
d^{d}x\sqrt{-g}\left(\frac{1}{2}\mu^{2}A_{\mu}A^{\mu}+\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right)\,\,,$
(3)

is the Proca action, $g$ is the determinant of the metric $g_{\mu\nu}$,
${R}=R_{\mu\nu}g^{\mu\nu}$ is the Ricci scalar defined as the trace of the
Ricci tensor $R_{ab}$ composed by the metric itself and its first and second
derivatives, $\Lambda=-\frac{(d-1)(d-2)}{2l^{2}}$ is the cosmological constant
with $l$ being the characteristic AdS length, $A_{\mu}$ is the Proca field
with mass $\mu$ and $F_{\mu\nu}\equiv\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$
is the Proca field strength. Spacetime indices are denoted by Greek letters,
e.g., $\mu$, $\nu$, run from 0 to $d-1$, where $0$ is the time index, and 1 to
$d-1$ specify the spatial indices. The field equations for the metric and the
Proca field are obtained by applying the variational principle to the action
Eq. (1). The field equations for the metric are then the Einstein equations
given by

$G_{\mu\nu}-\frac{(d-1)(d-2)}{2l^{2}}g_{\mu\nu}=8\pi T_{\mu\nu}\,,$ (4)

where $G_{\mu\nu}={R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{R}$ is the Einstein
tensor and $T_{\mu\nu}$ is the Proca stress-energy tensor, given by

$\displaystyle T_{\mu\nu}$
$\displaystyle=g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}+\mu^{2}A_{\mu}A_{\nu}$
$\displaystyle-
g_{\mu\nu}\left(\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}+\frac{\mu^{2}}{2}A_{\alpha}A^{\alpha}\right)\,.$
(5)

Moreover, the Einstein tensor obeys the Bianchi identities
$\nabla^{\mu}G_{\mu\nu}=0$, which in turn imply the conservation law for
$T_{\mu\nu}$, i.e., $\nabla^{\mu}T_{\mu\nu}=0$. The Proca field equations are
obtained either by the conservation law for $T_{\mu\nu}$ or by varying the
action with respect to the Proca field $A_{\mu}$ and can be written as

$\nabla_{\nu}F^{\mu\nu}+\mu^{2}A^{\mu}=0\,.$ (6)

Due to $F_{\mu\nu}$ being an antisymmetric tensor, one can calculate the
divergence of Eq. (6) to obtain a Bianchi identity for $A_{\mu}$

$\nabla^{\mu}A_{\mu}=0\,.$ (7)

It must be noted that Eq. (7) is a direct consequence of the Proca field
equation, Eq. (6), when $\mu\neq 0$. Thus, $A_{\mu}$ is a physical field and
describes $d-1$ degrees of freedom, as one component of the vector field can
always be obtained from the others by integrating Eq. (7). For $\mu=0$,
$A_{\mu}$ corresponds to the Maxwell field and the field equation Eq. (6)
becomes invariant under the gauge transformation $A^{\mu}\longrightarrow
A^{\mu}+\partial^{\mu}h$, where $h$ is an arbitrary scalar field. The Bianchi
identity for Proca fields, Eq. (7), ceases to be a consequence of the field
equation and becomes the usual Lorenz gauge condition of the Maxwell field.
Even after imposing this gauge, a residual gauge freedom remains as Eq. (7) is
invariant under $A^{\mu}\longrightarrow A^{\mu}+\partial^{\mu}h$ if
$\nabla_{\mu}\nabla^{\mu}h=0$, i.e., if $h$ obeys the Klein-Gordon equation.
Hence, the Maxwell field describes $d-2$ degrees of freedom. In $d=4$, the
previous discussion implies that while the Proca field describes three degrees
of freedom, corresponding to two transversal polarizations and one
longitudinal polarization, the residual gauge freedom of the Maxwell field
eliminates the longitudinal polarization, so that in total the Maxwell field
describes two degrees of freedom, corresponding to the two transversal
polarizations.

To formally describe the spacetime permeated by the Proca field, one would
have to solve Eq. (4) for the metric $g_{\mu\nu}$ and Eq. (6) for the Proca
field $A_{\mu}$ simultaneously. Here we are only interested in linear
perturbations of the Proca field $A_{\mu}$ around the trivial solution
$A_{\mu}=0$. As a result, we only consider Eqs. (4) and (6) up to first order
in perturbations of the Proca field. Since $T_{\mu\nu}$ in Eq. (II) is of
second order in $A_{\mu}$, linear perturbations in the Proca field induce a
curvature perturbation on $g_{\mu\nu}$ only at second order and $T_{\mu\nu}$
can be neglected in Eq. (4) at first order, reducing Eq. (4) to the vacuum
Einstein equations with negative cosmological constant. Thus, $g_{\mu\nu}$
corresponds to the background metric, as if the Proca field $A_{\mu}$ was
absent. Moreover, the linear Proca perturbations obey the Proca equation, Eq.
(6), with the connection $\Gamma^{\mu}_{\nu\sigma}$ being associated to the
background metric.

In what follows, we apply the Ishibashi-Kodama formalism to the Proca equation
in a $d$-dimensional AdS spacetime. We firstly write the Proca equation and
decompose the $A_{\mu}$ field in the AdS$d$ background.

##  III Linear Proca field perturbations in AdSd

###  III.1 The factorization of AdSd spacetime and the Proca equation

We consider the $d$-dimensional AdS spacetime which is a solution of the
vacuum Einstein equation with a cosmological constant, Eq. (4). The
$d$-dimensional AdS spacetime manifold, $\mathcal{M}^{d}\equiv{\rm AdS}^{d}$,
$d\geq 4$, can be written as a warped product of a submanifold
$\mathcal{N}^{2}$ of dimension two with a $(d-2)$-sphere, $\mathcal{S}^{d-2}$,
i.e., $\mathcal{M}^{d}=\mathcal{N}^{2}\times\mathcal{S}^{d-2}$. Its associated
line element $g_{\mu\nu}dx^{\mu}dx^{\nu}$ in coordinates $x^{\mu}$,
$\mu=0,...,d-1$, is thus decomposed as

$\displaystyle
g_{\mu\nu}dx^{\mu}dx^{\nu}={\tilde{g}}_{ab}dy^{a}dy^{b}+r^{2}{\hat{g}}_{ij}d\theta^{i}d\theta^{j}\,,$
(8)

where

$\displaystyle{\tilde{g}}_{ab}dy^{a}dy^{b}=-f(r)dt^{2}+\dfrac{dr^{2}}{f(r)}\,,\quad
f(r)=1+\frac{r^{2}}{l^{2}}\,,$ (9)

is the line element of the $\mathcal{N}^{2}$ submanifold, written in
coordinates $y^{a}=(y^{0},y^{1})=(t,r)$, with $t$ and $r$ being the time and
radial coordinates, respectively, and

$\displaystyle{\hat{g}}_{ij}d{\theta}^{i}d{\theta}^{j}=(d\theta^{2})^{2}+\sum_{i=3}^{d-1}\prod^{i-1}_{k=2}\sin^{2}(\theta^{k})(d\theta^{i})^{2}\,,$
(10)

which is the line element of the $(d-2)$-sphere, $\mathcal{S}^{d-2}$,
sometimes represented as $(d\Omega^{d-2})^{2}$, where the $\theta^{i}$ are the
angular coordinates and $i,j,k=2,...,d-1$. Note that in this convention
$\theta_{2}$ would be the usual $\theta$ and $\theta_{3}$ would be the usual
$\phi$ in $d=4$. As it should be clear by now, to allow one to distinguish
between tensors living on the different manifolds $\mathcal{M}^{d}$,
$\mathcal{N}^{2}$, and $\mathcal{S}^{d-2}$, we use Greek indices $\mu,\nu,...$
for tensors on $\mathcal{M}^{d}$, latin indices in the range $a,b,...,h$ for
tensors on $\mathcal{N}^{2}$ and latin indices in the range $i,j,...$ for
tensors on $\mathcal{S}^{d-2}$.

In order to factorize AdSd spacetime, one also must separate the connection
$\nabla_{\mu}$ associated to the manifold $(\mathcal{M}^{d},g)$ into the
connection $\tilde{\nabla}_{a}$ associated to the manifold
$(\mathcal{N}^{2},\tilde{g})$ and the connection $\hat{\nabla}_{i}$ associated
to the manifold $(\mathcal{S}^{d-2},\hat{g})$. This can be accomplished by
using the following relations between the Christoffel symbols associated to
each manifold, those are

$\displaystyle\Gamma^{a}_{\ bc}=\tilde{\Gamma}^{a}_{\ bc}\,,\quad\Gamma^{a}_{\
ij}=-r\left(\partial^{a}r\right){\hat{g}}_{ij}\,,\quad$
$\displaystyle\Gamma^{i}_{\
aj}=\frac{\partial_{a}r}{r}\delta^{i}_{j}\,,\quad\Gamma^{i}_{\
jk}=\hat{\Gamma}^{i}_{\ jk}\,,$ (11)

where ${\Gamma}^{\mu}_{\ \nu\rho}$ are the Christoffel symbols associated with
the metric $g_{\mu\nu}$, with the greek indices spanning through the indices
in $\mathcal{N}^{2}$ and in $\mathcal{S}^{d-2}$, e.g. $\mu=\\{a,i\\}$, and
$\tilde{\Gamma}^{a}_{\ bc}$ and $\hat{\Gamma}^{i}_{\ jk}$ are the Christoffel
symbols associated with the metrics ${\tilde{g}}_{ab}$ and ${\hat{g}}_{ij}$,
respectively.

The projections of the Proca equation, Eq. (6), into $\mathcal{N}^{2}$ and
$\mathcal{S}^{d-2}$ are written as

$\tilde{\nabla}_{b}F^{ab}+(d-2)\frac{\partial_{b}r}{r}F^{ab}+\hat{\nabla}_{j}F^{aj}+\mu^{2}A^{a}=0\,,$
(12)
$\tilde{\nabla}_{b}F^{ib}+(d-2)\frac{\partial_{b}r}{r}F^{ib}+\hat{\nabla}_{j}F^{ij}+\mu^{2}A^{i}=0\,,$
(13)

respectively. These equations are supplemented with the Bianchi identity, Eq.
(7), which is now

$\tilde{\nabla}_{a}A^{a}+(d-2)\frac{\partial_{a}r}{r}A^{a}+\hat{\nabla}_{i}A^{i}=0\,.$
(14)

###  III.2 The decomposition of the Proca field and spherical harmonics
expansion

To simplify the field equations for $A_{\mu}$, one exploits the spherical
symmetry of the background metric in Eq. (8). The strategy is to project the
field $A_{\mu}$ into components that are orthogonal to $\mathcal{S}^{d-2}$ and
components that are tangent to $\mathcal{S}^{d-2}$. The $A_{\mu}$ field can be
written as

$A_{\mu}dx^{\mu}={\tilde{A}}_{a}dy^{a}+{\hat{A}}_{i}d{\theta}^{i}\,,$ (15)

where ${\tilde{A}}_{a}$ denotes the projection of $A_{\mu}$ orthogonal to the
cotangent space of $\mathcal{S}^{d-2}$ and ${\hat{A}}_{i}$ denotes the
projection of $A_{\mu}$ tangent to the cotangent space of $\mathcal{S}^{d-2}$.
The latter can be further decomposed using the Helmoltz-Hodge decomposition
ishibashi_kodama1 , which allows one to write uniquely a dual vector field on
$\mathcal{S}^{d-2}$, ${\hat{A}}_{i}$, as the sum of a scalar field on
$\mathcal{S}^{d-2}$, ${\hat{A}}^{(s)}$, and a transverse covector field on the
cotangent space of $\mathcal{S}^{d-2}$, ${\hat{A}}^{(v)}_{i}$, in the
following way

${\hat{A}}_{i}={\hat{A}}^{(v)}_{i}+\hat{\nabla}_{i}{\hat{A}}^{(s)}\quad\,,\quad\quad\hat{\nabla}_{i}{{\hat{A}}^{(v)\,i}}=0\,.$
(16)

Since ${\tilde{A}}_{a}$ and ${\hat{A}}^{(s)}$ behave as scalars on
$\mathcal{S}^{d-2}$, these are called the scalar-type components of $A_{\mu}$,
while ${\hat{A}}^{(v)\,i}={\hat{g}}^{ij}{\hat{A}}^{(v)}_{j}$, a vector on the
tangent space of $\mathcal{S}^{d-2}$, is called the vector-type component of
$A_{\mu}$. Moreover, the scalar-type components of the Proca field transform
as scalars under the SO($d-1$) rotation group, and the vector-type component
of the Proca field transforms as a vectors under the SO($d-1$) rotation group.
Since the correspondence between ${\hat{A}}^{(v)\,i}$ and
${\hat{A}}^{(v)}_{i}$ is one to one, we shall make the abuse of language that
${\hat{A}}^{(v)}_{i}$ corresponds to the vector-type component of $A_{\mu}$ as
well. In the literature, scalar-type is also referred to as polar- or even-
type, whereas vector-type is called axial- or odd-type. This has to do with
the transformation properties of these components under parity
transformations, see regge_wheeler ; zerilli . The scalar-type components of
$A_{\mu}$ can be expanded in scalar harmonics $Y_{\vec{k}_{s}}$, where
$\vec{k}_{s}$ is a vector containing the the angular momentum number $\ell$
and the $d-3$ azimuthal numbers, which form a complete basis on
$\mathcal{S}^{d-2}$, satisfying

$\displaystyle\left(\hat{\Box}+k_{s}^{2}\right)Y_{\vec{k}_{s}}=0\,,\quad\quad\int
d{\Omega}^{d-2}Y_{\vec{k}_{s}}Y_{\vec{k}_{s}^{\prime}}=\delta_{\vec{k}_{s}\vec{k}_{s}^{\prime}}\,,$
(17)

where $\hat{\Box}={\hat{g}}^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}$,

$k_{s}^{2}=\ell(\ell+d-3)\,,\quad\ell=0,1,2,...\,,$ (18)

and $d{\Omega}^{d-2}$ is here the volume element over the sphere given by
$d{\Omega}^{d-2}=\sqrt{\hat{g}}d\theta^{2}...d\theta^{d-1}$, $\hat{g}$ being
the determinant of the metric ${\hat{g}}_{ij}$. Similarly, the vector-type
component of $A_{\mu}$ can be expanded in vector harmonics
${Y_{\vec{k}_{v}\,i}}$ which also form a complete basis on
$\mathcal{S}^{d-2}$, where $\vec{k}_{v}$ is a vector containing the angular
momentum number $\ell$ and $d-3$ azimuthal numbers, that in principle are
different from the azimuthal numbers of the scalar harmonics. The vector
harmonics ${Y_{\vec{k}_{v}\,i}}$ then satisfy

$\displaystyle\left(\hat{\Box}+k_{v}^{2}\right){Y_{\vec{k}_{v}\,i}}=0\,,\quad\hat{\nabla}_{i}{Y_{\vec{k}_{v}}^{i}}=0\,\,,$
$\displaystyle\int
d{\Omega}^{d-2}{\hat{g}}^{ij}{Y_{\vec{k}_{v}\,i}}{Y_{\vec{k}_{v}^{\prime}\,j}}=\delta_{\vec{k}_{v}\vec{k}_{v}^{\prime}}\,,$
(19)

with

$k_{v}^{2}=\ell(\ell+d-3)-1\,,\quad\ell=1,2,3,...\,,$ (20)

see Appendix A for more details.

Thus, the scalar-type components $\\{{\tilde{A}}_{a},{\hat{A}}^{(s)}\\}$ can
be expanded in terms of the $Y_{\vec{k}_{s}}(\theta)$, and vector-type
component $\\{{\hat{A}}_{i}^{(v)}\\}$ can be expanded in terms of the
${Y_{\vec{k}_{v}\,i}}(\theta)$, where
$\theta\equiv(\theta^{2}...,\theta^{i},...\theta^{d-1})$. Indeed,
$\\{{\tilde{A}}_{a},{\hat{A}}^{(s)}\\}$ can be expanded as
${\tilde{A}}_{a}(y,{\theta})=\sum_{\vec{k}_{s}}{{\tilde{A}}_{\vec{k}_{s}\,a}}(y)Y_{\vec{k}_{s}}({\theta})$,
where $y\equiv(t,r)$, and
${\hat{A}}^{(s)}(y,{\theta})=\sum_{\vec{k}_{s}}{{\hat{A}}^{(s)}_{\vec{k}_{s}}}(y)Y_{\vec{k}_{s}}(\theta)$,
respectively, and $\\{{\hat{A}}_{i}^{(v)}\\}$ as
${\hat{A}}^{(v)}_{i}(y,{\theta})=\sum_{\vec{k}_{v}}{\hat{A}}^{(v)}_{\vec{k}_{v}}(y){Y_{\vec{k}_{v}\,i}}({\theta})$.
Note that
$\hat{\nabla}_{i}{\hat{A}}^{(s)}(y,{\theta})=\sum_{\vec{k}_{s}}{{\hat{A}}^{(s)}_{\vec{k}_{s}}}(y)\hat{\nabla}_{i}Y_{\vec{k}_{s}}(\theta)$.
Since ${\tilde{A}}_{\vec{k}_{s}\,a}$, ${\hat{A}}^{(s)}_{\vec{k}_{s}}$, and
${\hat{A}}^{(v)}_{\vec{k}_{v}}$ are cumbersome symbols to carry along, we
define $\psi_{\vec{k}_{s}\,a}\equiv{\tilde{A}}_{\vec{k}_{s}\,a}$,
$\phi_{\vec{k}_{s}}\equiv{\hat{A}}^{(s)}_{\vec{k}_{s}}$, and
$\chi_{\vec{k}_{v}}\equiv{\hat{A}}^{(v)}_{\vec{k}_{v}}$. so that for the
scalar-type components $\\{{\tilde{A}}_{a},{\hat{A}}^{(s)}\\}$, one writes

${\tilde{A}}_{a}(y,{\theta})=\sum_{\vec{k}_{s}}{\psi_{\vec{k}_{s}\,a}}(y)Y_{\vec{k}_{s}}({\theta})\,,$
(21)
${\hat{A}}^{(s)}(y,{\theta})=\sum_{\vec{k}_{s}}{\phi_{\vec{k}_{s}}}(y)Y_{\vec{k}_{s}}(\theta)\,,$
(22)

while for the vector-type component $\\{{\hat{A}}_{i}\\}$ one writes

${\hat{A}}^{(v)}_{i}(y,{\theta})=\sum_{\vec{k}_{v}}\chi_{\vec{k}_{v}}(y){Y_{\vec{k}_{v}\,i}}({\theta})\,.$
(23)

Here, ${\psi_{\vec{k}_{s}\,a}}(y)$ is a vector field on $\mathcal{N}^{2}$, and
$\phi_{\vec{k}_{s}}(y)$ and $\chi_{\vec{k}_{v}}(y)$ are scalar fields on
$\mathcal{N}^{2}$. The scalar-type components cover two degrees of freedom,
whereas the vector-type component covers $d-3$ degrees of freedom as can be
noted from the transverse condition of ${Y_{\vec{k}_{v}\,i}}$ in Eq. (III.2).
As Eq. (7) needs to be satisfied, these variables cover in total $d-1$ degrees
of freedom, as expected for a Proca field. In summary, the complete expression
of the Proca field decomposed in spherical harmonics is

$\displaystyle
A_{\mu}dx^{\mu}\hskip-2.56064pt=\hskip-2.56064pt\sum_{\vec{k}_{s}}\hskip-2.56064pt\left[{\psi_{\vec{k}_{s}\,a}}Y_{\vec{k}_{s}}dy^{a}\hskip-2.56064pt+\hskip-2.56064pt\phi_{\vec{k}_{s}}\hat{\nabla}_{i}Y_{\vec{k}_{s}}d\theta^{i}\right]\hskip-2.56064pt+\hskip-2.56064pt\sum_{\vec{k}_{v}}\chi_{\vec{k}_{v}}Y_{\vec{k}_{v}\,i}d\theta^{i}.$
(24)

In terms of the expansion in Eqs. (21)-(23), i.e., Eq. (24), the components of
the Proca field strength tensor are written as

$F^{ab}=\sum_{\vec{k}_{s}}\left[{\tilde{\nabla}}^{a}\psi_{\vec{k}_{s}}^{b}-{\tilde{\nabla}}^{b}\psi_{\vec{k}_{s}}^{a}\right]{Y}_{\vec{k}_{s}}\,,$
(25) $\displaystyle F^{ai}=$
$\displaystyle\sum_{\vec{k}_{s}}\left[\frac{1}{r^{2}}\left(\nabla^{a}\phi_{\vec{k}_{s}}-\psi_{\vec{k}_{s}}^{a}\right)\right]\hat{\nabla}^{i}{Y}_{\vec{k}_{s}}$
$\displaystyle+$
$\displaystyle\sum_{\vec{k}_{v}}\left[\frac{1}{r^{2}}\nabla^{a}\chi_{\vec{k}_{v}}\right]{Y_{\vec{k}_{v}}^{i}},$
(26)
$F^{ij}=\sum_{\vec{k}_{v}}\left[\frac{\chi_{\vec{k}_{v}}}{r^{4}}\right]\left(\hat{\nabla}^{i}Y_{\vec{k}_{v}}^{j}-\hat{\nabla}^{j}Y_{\vec{k}_{v}}^{i}\right)\,,$
(27)

where the covariant derivative $\nabla_{a}$ can be swapped to a partial
derivative $\partial_{a}$ when acting on scalars.

With these decompositions, it is possible to decouple only partially the
equations in general. We will see that the scalar-type perturbations are
completely decoupled from the vector-type perturbations. Moreover, the
separation of the Proca equations are achieved due to the separation of the
fields into functions that purely depend on the coordinates $(t,r)$ and the
spherical harmonics that depend only on the spherical angles.

###  III.3 Separation of the Proca equations

####  III.3.1 Proca equations after spherical harmonics expansion

The expansion on spherical harmonics of the Proca field in Eqs. (21)-(23),
i.e., Eq. (24), and the corresponding expansion of the strength field tensor
in Eqs. (25)-(27) can be inserted into Eqs. (12)-(14), which allows the
separation of the Proca equations in AdSd. In this way, the Proca equations
are separated into three sums, one in terms of the spherical harmonics
$Y_{\vec{k}_{s}}$ over each $\vec{k}_{s}$, i.e
$\sum_{\vec{k}_{s}}{\tilde{W}}_{\vec{k}_{s}}^{a}Y_{\vec{k}_{s}}$, another in
terms of the gradients of the spherical harmonics
$\hat{\nabla}_{i}Y_{\vec{k}_{s}}$ over each $\vec{k}_{s}$, i.e.,
$\sum_{\vec{k}_{s}}{\hat{W}}_{\vec{k}_{s}}^{(s)}\hat{\nabla}^{i}Y_{\vec{k}_{s}}$,
and the remaining sum in terms of the vector spherical harmonics
$Y_{\vec{k}_{v}\,i}$ over each $\vec{k}_{v}$, i.e.,
$\sum_{\vec{k}_{v}}{\hat{W}}_{\vec{k}_{v}}^{(v)}Y_{\vec{k}_{v}}^{i}$, where
the coefficients of all the three sums only depend on the coordinates $t$ and
$r$, i.e., ${\tilde{W}}_{\vec{k}_{s}}^{a}={\tilde{W}}_{\vec{k}_{s}}^{a}(t,r)$,
${\hat{W}}_{\vec{k}_{s}}^{(s)}={\hat{W}}_{\vec{k}_{s}}^{(s)}(t,r)$ and
${\hat{W}}_{\vec{k}_{v}}^{(v)}={\hat{W}}_{\vec{k}_{v}}^{(v)}(t,r)$. We must
note that $Y_{\vec{k}_{s}}$, $\hat{\nabla}_{i}Y_{\vec{k}_{s}}$ and
$Y_{\vec{k}_{v}\,i}$ are orthogonal between each other. An argument can be
made that these span different representations of the rotation group
$SO(d-1)$, for $d>4$, while for $d=4$, the scalar and vector spherical
harmonics span different representations of the rotation group $O(3)$, see
ishibashiwald  and also Appendix A. Therefore, the Proca equations separate
into the equations $\tilde{W}_{\vec{k}_{s}}^{a}=0$, $W_{\vec{k}_{s}}^{(s)}=0$
and $W_{\vec{k}_{v}}^{(v)}=0$, described only by the coordinates $t$ and $r$.
Since the Proca equation is a linear differential equation in $A_{\mu}$, there
is no mixing between the different $\vec{k}_{v}$ and $\vec{k}_{s}$ modes.
Therefore, without loss of generality and for convenience, we drop the sum on
all the $\vec{k}_{s}$ and $\vec{k}_{v}$. In the rest of the section, we treat
the scalar spherical harmonics as $Y$ and the vector spherical harmonics as
$Y_{i}$.

We now show the equations obtained from the separation of the Proca equations.
The projection of the Proca equation into $\mathcal{N}^{2}$, Eq. (12), can be
written in terms of a sum of spherical harmonics $Y$, in which the associated
coefficients must satisfy ${\tilde{W}}_{\vec{k}_{s}}^{a}=0$, or explicitly,

$\displaystyle
2\tilde{\nabla}_{b}\tilde{\nabla}^{[a}\psi^{b]}+2(d-2)\frac{\tilde{\nabla}_{b}r}{r}\tilde{\nabla}^{[a}\psi^{b]}$
$\displaystyle+\left(\frac{\ell(\ell+d-3)}{r^{2}}+\mu^{2}\right)\psi^{a}-\frac{\ell(\ell+d-3)}{r^{2}}\partial^{a}\phi=0\,,$
(28)

for each $\vec{k}_{s}$, where it was used that $k^{2}_{s}=\ell(\ell+d-3)$,
$\tilde{\nabla}^{[a}\psi^{b]}=\frac{1}{2}(\tilde{\nabla}^{a}\psi^{b}-\tilde{\nabla}^{b}\psi^{a})$
and that $\hat{\nabla}_{i}Y^{i}=0$, which avoids the appearance of $\chi$ in
Eq. (28). The projection of the Proca equation into $\mathcal{S}^{d-2}$, Eq.
(13), can be written in terms of two sums. One sum is in terms of the gradient
of the spherical harmonics $\hat{\nabla}_{i}Y$, whose coefficients satisfy
${\hat{W}}_{\vec{k}_{s}}^{(s)}=0$ or explicitly

$\displaystyle\tilde{\Box}{\phi}\hskip-1.42271pt+\hskip-1.42271pt(d-4)\frac{\partial_{b}r}{r}\partial^{b}\phi\hskip-1.42271pt-\hskip-1.42271pt\mu^{2}\phi\hskip-1.42271pt-\hskip-1.42271pt{\tilde{\nabla}}_{b}\psi^{b}\hskip-1.42271pt-\hskip-1.42271pt(d-4)\frac{\partial_{b}r}{r}\psi^{b}\hskip-1.42271pt=\hskip-1.42271pt0,$
(29)

for each $\vec{k}_{s}$, where
$\tilde{\Box}=\tilde{g}^{ab}\tilde{\nabla}_{a}\tilde{\nabla}_{b}$. The other
sum is in terms of the vector spherical harmonics $Y_{i}$, whose coefficients
must satisfy ${\hat{W}}_{\vec{k}_{v}}^{(v)}=0$ or explicitly

$\displaystyle\tilde{\Box}\chi\hskip-1.42271pt+\hskip-1.42271pt\frac{d-4}{r}(\partial^{b}r)(\partial_{b}\chi)\hskip-1.42271pt-\hskip-1.42271pt\left(\frac{\ell(\ell+d-3)\hskip-1.42271pt+\hskip-1.42271ptd\hskip-1.42271pt-\hskip-1.42271pt4}{r^{2}}\hskip-1.42271pt+\hskip-1.42271pt\mu^{2}\right)\hskip-1.42271pt\chi\hskip-1.42271pt=\hskip-1.42271pt0,$
(30)

for each $\vec{k}_{v}$, where $k^{2}_{v}=\ell(\ell+d-3)-1$ was used.
Furthermore, to obtain Eq. (30), the commutator
$2\hat{\nabla}_{[j}\hat{\nabla}_{i]}Y^{j}=\hat{R}_{mi}Y^{m}$ was used, where
$\hat{R}_{mi}=(d-3)\hat{g}_{mi}$ is the Ricci tensor of $\mathcal{S}^{d-2}$.
Finally, the Bianchi identity Eq. (14) can be written as a sum of spherical
harmonics $Y$, whose coefficients satisfy

$\displaystyle\tilde{\nabla}_{b}\psi^{b}+(d-2)\frac{\partial_{b}r}{r}\psi^{b}-\frac{\ell(\ell+d-3)}{r^{2}}\phi=0\,\,,$
(31)

for each $\vec{k}_{s}$, where again $k_{s}^{2}=\ell(\ell+d-3)$ and
$\hat{\nabla}_{i}Y^{i}=0$ were used. We see that Eqs. (28)-(30) form a set of
four equations, two component equations in Eq. (28) and two equations in Eqs.
(29) and (30). There is a fifth equation, the Proca Bianchi identity equation
given in Eq. (31). So in total we can play with five equations.

####  III.3.2 Proca equations decoupled: The important equations for
quasinormal modes

The equations for the scalar-type components of the Proca field, i.e., Eqs.
(28) and (29) are coupled in $\psi_{t}$, $\psi_{r}$, $\phi$, and Eq. (30) for
the vector-type component represented by $\chi$ is decoupled. The Bianchi
identity Eq. (31) is also coupled. By defining new variables $q_{0}$, $q_{1}$,
$q_{2}$, and $q_{3}$ as functions of $\psi_{t}$, $\psi_{r}$, $\phi$, and
$\chi$ it is possible to decouple the equations. We give first the result and
then show the steps to obtain it. So, Eqs. (28)-(30) in the new variables are

$\displaystyle\hat{\mathcal{D}}_{\ell}q_{0}+\frac{2r}{l^{2}}\left(\partial_{t}q_{1}+\partial_{t}q_{2}-\partial_{r_{*}}q_{0}\right)=0\,,$
(32) $\displaystyle\hat{\mathcal{D}}_{j_{k}}q_{k}=0\,,$ (33)

and the Bianchi identity Eq. (31) is now

$\displaystyle\partial_{t}q_{0}\hskip-2.56064pt-\hskip-2.56064pt\partial_{r_{*}}(q_{1}\hskip-2.56064pt+\hskip-2.56064ptq_{2})\hskip-2.56064pt=\hskip-2.56064pt\frac{f}{r}\hskip-2.56064pt\left(\hskip-2.56064pt\frac{d\hskip-2.56064pt-\hskip-2.56064pt2}{2}\hskip-2.56064pt-\hskip-2.56064pt(\ell\hskip-2.56064pt+\hskip-2.56064ptd\hskip-2.56064pt-\hskip-2.56064pt3)\hskip-2.56064pt\right)\hskip-2.56064ptq_{2}\hskip-2.56064pt+\hskip-2.56064pt\frac{f}{r}\hskip-2.56064pt\left(\ell\hskip-2.56064pt+\hskip-2.56064pt\frac{d\hskip-2.56064pt-\hskip-2.56064pt2}{2}\right)\hskip-2.56064ptq_{1},$
(34)

where here $k\in\\{1,2,3\\}$, $j_{k}=(j_{1},j_{2},j_{3})$, $j_{1}=\ell+1$ with
$\ell\in\mathbb{N}_{0}$, $j_{2}=\ell-1$ with $\ell\in\mathbb{N}$ and
$j_{3}=\ell$ with $\ell\in\mathbb{N}$, $r_{*}$ is defined as
$r_{*}=l\arctan\left(\frac{r}{l}\right)$, the operator
$\hat{\mathcal{D}}_{\ell}$ is defined as

$\displaystyle\hat{\mathcal{D}}_{\ell}=-\partial_{t}^{2}+\partial_{r_{*}}^{2}$
$\displaystyle-f\left[\frac{\ell(\ell+d-3)}{r^{2}}+\mu^{2}+\frac{(d-2)(d-4)}{4l^{2}}\left(1+\frac{l^{2}}{r^{2}}\right)\right]\,\,,$
(35)

and $q_{0}(t,r)$, $q_{1}(t,r)$, $q_{2}(t,r)$, and $q_{3}(t,r)$ are defined by

$\displaystyle q_{0}(t,r)=\psi_{t}(t,r)\,r^{\frac{d}{2}-1}\,,$ (36)
$\displaystyle
q_{1}(t,r)\hskip-2.56064pt=\hskip-2.56064pt\frac{(\ell\hskip-2.56064pt-\hskip-2.56064ptd\hskip-2.56064pt-\hskip-2.56064pt3)\psi_{r}(t,r)\,f(r)-\ell(\ell+d-3)\frac{\phi(t,r)}{r}}{2\ell+d-3}\,r^{\frac{d}{2}-1},$
(37) $\displaystyle
q_{2}(t,r)=\frac{\ell\psi_{r}(t,r)\,f(r)+\ell(\ell+d-3)\frac{\phi(t,r)}{r}}{2\ell+d-3}\,r^{\frac{d}{2}-1},$
(38) $\displaystyle q_{3}(t,r)=\frac{\chi(t,r)}{r}\,r^{\frac{d}{2}-1}\,.$ (39)

We see that Eqs. (32)-(33) provide four equations, the number we had
originally, and there is still the Proca Bianchi identity Eq. (34), yielding
five equations in total.

####  III.3.3 Proof of the decoupling of Proca equations

Now we show how to obtain Eqs. (32)-(34) together with the definitions Eqs.
(35)-(39). The equations for the scalar-type components of the Proca field,
i.e., Eqs. (28) and (29) are coupled in $\psi_{t}$, $\psi_{r}$, and $\phi$.
The Proca Bianchi identity equation Eq. (31) is also coupled in $\psi_{t}$,
$\psi_{r}$, and $\phi$. We now show that in the case of AdSd spacetime, it is
indeed possible to manipulate these equations and decouple these components
through further transformations. We start by differentiating Eq. (31) to
obtain

$\displaystyle\ell(\ell+d-3)\partial^{a}\phi=\tilde{\nabla}^{a}(r^{2}\tilde{\nabla}_{b}\psi^{b})+(d-2)\tilde{\nabla}^{a}(r\psi^{b}\partial_{b}r)\,.$
(40)

Using Eq. (40) in Eq. (28) we obtain a coupled equation for the $\psi^{a}$
components given by

$\displaystyle\tilde{\Box}\psi^{a}-{\tilde{R}}^{a}_{b}\psi^{b}+(d-2)\frac{\partial^{b}r}{r}{\tilde{\nabla}}_{b}\psi^{a}$
$\displaystyle-\left(\frac{\ell(\ell+d-3)}{r^{2}}+\mu^{2}\right)\psi^{a}+(d-2){\tilde{\nabla}}^{a}\left(\frac{\partial_{b}r}{r}\right)\psi^{b}$
$\displaystyle+\frac{2\partial^{a}r}{r}\left({\tilde{\nabla}_{b}}\psi^{b}+(d-2)\frac{\partial_{b}r}{r}\psi^{b}\right)=0\,,$
(41)

where ${\tilde{R}}^{a}_{b}=-\frac{f^{\prime\prime}}{2}\delta^{a}_{b}$ is the
Ricci tensor of $\mathcal{N}^{2}$ and the commutator
$2\tilde{\nabla}_{[a}\tilde{\nabla}_{b]}\psi^{a}=\tilde{R}_{ab}\psi^{a}$ was
used. Now, Eqs. (29) (31), and (41) can be further simplified into

$\displaystyle\hat{\mathcal{D}}_{\ell}u_{0}+\frac{2r}{l^{2}}\left(\partial_{t}u_{1}-\partial_{r_{*}}u_{0}\right)=0\,,$
(42)
$\displaystyle\hat{\mathcal{D}}_{\ell}u_{1}-\frac{2f}{r^{2}}\left(\frac{d-2}{2}u_{1}-u_{2}\right)=0\,,$
(43)
$\displaystyle\hat{\mathcal{D}}_{\ell}u_{2}+\frac{2f}{r^{2}}\left(\left(\frac{d}{2}-2\right)u_{2}+{\ell(\ell+d-3)}u_{1}\right)=0\,,$
(44)
$\displaystyle\partial_{t}u_{0}-\partial_{r_{*}}u_{1}=\frac{f}{r}\left(\frac{d-2}{2}u_{1}-u_{2}\right)\,,$
(45)

where again $r_{*}$ is defined such as $\frac{dr_{*}}{dr}=\frac{1}{f(r)}$ with
$f(r)=1+\frac{r^{2}}{l^{2}}$, i.e., $r_{*}=l\arctan\left(\frac{r}{l}\right)$,
$\hat{\mathcal{D}}_{\ell}$ is the operator already given in Eq. (35), i.e.,
$\hat{\mathcal{D}}_{\ell}=-\partial_{t}^{2}+\partial_{r_{*}}^{2}-f\left[\frac{\ell(\ell+d-3)}{r^{2}}+\mu^{2}+\frac{(d-2)(d-4)}{4l^{2}}\left(1+\frac{l^{2}}{r^{2}}\right)\right]$,
$u_{0}(t,r)$, $u_{1}(t,r)$ and $u_{2}(t,r)$ are defined by

$\displaystyle u_{0}(t,r)=\psi_{t}(t,r)\,r^{\frac{d}{2}-1}\,,$ (46)
$\displaystyle u_{1}(t,r)=\psi_{r}(t,r)\,f(r)\,r^{\frac{d}{2}-1}\,,$ (47)
$\displaystyle
u_{2}(t,r)=\frac{\phi(t,r)}{r}\,{\ell(\ell+d-3)}\,r^{\frac{d}{2}-1}\,,$ (48)

and we also have used Eq. (45) in the last two terms of Eq. (43). Finally, one
can notice that the coupling terms in Eqs. (43) and (44) are constants
multiplied by $\frac{2f}{r^{2}}$. Therefore, it is possible to further
decouple Eqs. (43) and (44) by making the transformations

$\displaystyle u_{0}=q_{0}\,,$ (49) $\displaystyle u_{1}=q_{1}+q_{2}\,,$ (50)
$\displaystyle u_{2}=(\ell+d-3)q_{2}-\ell q_{1}\,.$ (51)

Inserting Eqs. (49)-(51) into Eqs. (42)-(44) yield Eqs. (32)-(33) with
$k=1,2$, and inserting Eqs. (49)-(51) into Eq. (45) yields Eq. (34). The
vector-type component of the Proca field is described by Eq. (30) and it is
completely decoupled from the scalar-type components of the Proca field.
Equation (30) can be further simplified into

$\displaystyle\hat{\mathcal{D}}_{\ell}u_{3}=0\,\,,$ (52)

where

$\displaystyle u_{3}=\frac{\chi}{r}\,r^{\frac{d}{2}-1}\,,$ (53)

Defining trivially

$\displaystyle u_{3}=q_{3}\,.$ (54)

yields Eq. (33) with $k=3$. Thus, Eqs. (32)-(34) together with the definitions
Eqs. (35)-(39) have been obtained.

We confirm the results obtained in ueda ; Herdeiro , for the particular case
of pure AdS, where the vector-type component of the Proca field yields $d-3$
degrees of freedom, whereas the scalar-type component describes two degrees of
freedom, which are coupled by the mass of the field. We find that, similarly
to what was found in tf  for AdS4, the scalar-type degrees of freedom decouple
in AdSd, by making a particular linear transformation to the relevant fields.

##  IV Normal modes of Proca perturbations in the AdSd background

###  IV.1 Initial considerations and boundary conditions

The normal modes in AdSd are dynamical solutions of the Proca equations
described by $q_{0}$, $q_{1}$, $q_{2}$ and $q_{3}$ in Eqs. (32)-(34).
Moreover, the Bianchi identity in Eq. (34) can be used to describe $q_{0}$ in
terms of $q_{1}$, $q_{2}$ and $q_{3}$, which in turn obey Eq. (33). With the
relation given by Eq. (34), the equation for $q_{0}$, Eq. (32), is satisfied
and so the picture is consistent. The normal modes can then be obtained solely
by solving Eq. (33) for the $q_{1}$, $q_{2}$ and $q_{3}$, with the appropriate
boundary conditions at the origin and at spatial infinity. By making an
extension to the complex numbers, one can assume for the $q_{k}(t,r)$ an
ansatz of the form $q_{k}(t,r)=q_{k}(r)e^{-i\omega_{k}t}$, where $\omega_{k}$
is the normal mode frequency of the mode $q_{k}$, which is analogous to
performing a Fourier transformation from the time domain to the frequency
domain. Of course, when one wants to treat the real field, one must project
the complex field into the real axis. The equations given in Eq. (33) can then
be written for the $q_{k}(r)$ as

$\displaystyle\partial_{r_{*}}^{2}q_{k}+\left(\omega^{2}_{k}-V_{j_{k}}\right)q_{k}=0\,\,,$
(55) $\displaystyle
V_{j_{k}}=f\Bigg{[}\frac{4j_{k}(j_{k}+d-3)+(d-2)(d-4)}{4r^{2}}+\mu^{2}$
$\displaystyle\hskip 125.00018pt+\frac{(d-2)(d-4)}{4l^{2}}\Bigg{]}\,\,,$ (56)

for $k\in\\{1,2,3\\}$. Equation (55) is a Schrödinger-like equation for the
normal modes, $q_{k}$, with associated normal mode frequencies, $\omega_{k}$.
A general class of Schrödinger-like equations governing the dynamics of fields
in AdS with all their possible boundary conditions at spatial infinity, were
analyzed in ishibashiwald . Here, we provide the analysis for the Proca field
equation given in Eq. (55) and its solutions. We start by studying the
behavior of the solutions near the origin and near spatial infinity. We first
deal with the behavior at $r=0$. One can pinpoint the boundary conditions by
checking when the $q_{k}$ are square-integrable in the sense of
$\int{\bar{q}}_{k}q_{k}dr_{*}$, where ${\bar{q}}_{k}$ means complex conjugate
of $q_{k}$. This implies that the regularity conditions are such that
$q_{k}\propto r^{s}$ as $r\rightarrow 0$, for some $s$ with $s>-\frac{1}{2}$.
The functions $q_{k}$ near $r=0$ have the behavior

$\displaystyle
q_{k}=\alpha^{r=0}_{k}r^{j_{k}+\frac{d-2}{2}}+\beta^{r=0}_{k}r^{-j_{k}-\frac{d-4}{4}}\,\,,$
(57)

where $\alpha^{r=0}_{k}$ and $\beta^{r=0}_{k}$ are constants, and where the
equality is valid in first order near $r=0$. In all the cases except the case
$d=4$, $k=2$, and $j_{2}=0$, the functions $q_{k}$ are not square integrable
if $\beta^{r=0}_{k}$ is finite. Therefore, the regularity condition is
$\beta^{r=0}_{k}=0$. In the case of $d=4$, $k=2$, and $j_{2}=0$, the function
$q_{2}$ is square integrable if $\beta^{r=0}_{2}$ is finite, since
$q_{2}=\alpha^{r=0}_{2}r+\beta^{r=0}_{2}$. Still, this solution means that the
Proca field goes as $A_{\mu}\sim\frac{1}{r}$ and it is rather a solution to
the Proca equations but with a delta dirac distribution as a source term, see
Appendix B. For this reason, this particular solution cannot be considered and
the regularity condition $\beta^{r=0}_{k}=0$ is maintained. Such argument for
the regularity condition is also present for the scalar field in AdS4 with
$\ell=0$, see ishibashiwald . Note also that in the case of the scalar-type
Proca field with $d=4$ and $j_{2}=0$, the asymptotic expansion at $r=0$ of the
integrand of the energy $E$ defined as
$E=\int_{t}T_{\mu\nu}\xi^{\mu}t^{\nu}\frac{r^{d-2}}{\sqrt{f}}drd\Omega$, for a
constant $t$ slice, where $t^{\mu}$ is its normal vector, $\xi^{\mu}$ is the
timelike Killing vector, and $d\Omega$ is the line element of the unit
2-sphere, seems to have divergent terms $r^{-1}$, $r^{-2}$ and $r^{-4}$, which
only vanish if $\beta^{r=0}_{2}=0$.

Now, we analyze the boundary conditions at spatial infinity, $r\to\infty$. To
do that, one looks at the behavior of the Proca field near
$r\rightarrow+\infty$. For the functions $q_{k}$ to be square integrable they
must behave as $q_{k}\sim r^{s}$, for some $s$ with $s<\frac{1}{2}$. From the
Proca field equations, we get a behavior of the $q_{k}$ near
$r\rightarrow+\infty$ as

$\displaystyle q_{k}$
$\displaystyle=\alpha^{r=\infty}_{k}r^{-\frac{1}{2}(1+\sqrt{(d-3)^{2}+4\mu^{2}l^{2}})}$
$\displaystyle+\beta^{r=\infty}_{k}r^{-\frac{1}{2}(1-\sqrt{(d-3)^{2}+4\mu^{2}l^{2}})}\,\,,$
(58)

where $\alpha^{r=\infty}_{k}$ and $\beta^{r=\infty}_{k}$ are constants, and
where the equality is valid in first order near $r=\infty$. For the case
$d>4$, the functions $q_{k}$ are only square integrable if
$\beta^{r=\infty}_{k}=0$, which is the Dirichlet boundary condition. For $d=4$
and $(\mu l)^{2}\geq\frac{3}{4}$, the same rationale applies. However, for
$d=4$ and $0<(\mu l)^{2}<\frac{3}{4}$, the functions $q_{k}$ are square
integrable for finite $\beta^{r=\infty}_{k}$. This is an interesting case
because the potential does diverge for positive $\mu^{2}$. According to
ishibashiwald , it is then possible to impose a one parameter family of
boundary conditions. Still, by the calculation of the usual energy of the
Proca field, the condition $\beta^{r=\infty}_{k}=0$ is the only condition that
ensures that the energy is finite and does not diverge on $t$ constant slices.
Although the usual definition of the energy has been chosen, we note that
there are different valid definitions of the energy function for the fields
where the energy is finite and conserved, see ishibashiwald . Nevertheless, we
admit Dirichlet boundary conditions for the $q_{k}$, which means that
$q_{k}(r\rightarrow+\infty)=0$ for every possible case.

###  IV.2 Solutions of the Proca equations and normal mode frequencies

The Proca equations in Eq. (55) can be put in the form

$\displaystyle\partial_{r_{*}}^{2}q_{k}+\left(\omega_{k}^{2}-\frac{G_{k}}{\sin^{2}\left(\frac{r_{*}}{l}\right)}-\frac{H_{k}}{\cos^{2}\left(\frac{r_{*}}{l}\right)}\right)q_{k}=0\,,$
(59)

where $k\in\\{1,2,3\\}$, $r_{*}=l\arctan\left(\frac{r}{l}\right)$,
$G_{k}=\frac{4j_{k}(j_{k}+d-3)+(d-2)(d-4)}{4l^{2}}$ and
$H_{k}=\frac{(d-2)(d-4)+4\mu^{2}l^{2}}{4l^{2}}$. This is a second order
partial differential equation which is linear and it has three regular
singularity points at $\frac{r_{*}}{l}=-\frac{\pi}{2}$, $\frac{r_{*}}{l}=0$,
and $\frac{r_{*}}{l}=+\frac{\pi}{2}$. Therefore, this Fuchsian differential
equation can be transformed into an hypergeometric equation, see details in
Appendix B. The solutions of Eq. (59) that satisfy the regularity conditions
at $r=0$ and the Dirichlet boundary conditions at $r\rightarrow+\infty$ are

$\displaystyle
q_{k}=a_{k}\left(\frac{r}{l}\right)^{j_{k}+\frac{d-2}{2}}\left(1+\frac{r^{2}}{l^{2}}\right)^{n-\frac{\omega_{k}l}{2}}$
$\displaystyle\times\prescript{}{2}{F}_{1}\left[-n+\omega_{k}l,-n,j_{k}+\frac{d-1}{2};\frac{\frac{r^{2}}{l^{2}}}{1+\frac{r^{2}}{l^{2}}}\right]\,\,,$
(60)

where $k\in\\{1,2,3\\}$, $a_{k}$ is a constant, and with the normal mode
frequencies $\omega_{k}$ being

$\displaystyle\omega_{k}l=2n+j_{k}+\frac{d-1}{2}+\frac{1}{2}\sqrt{(d-3)^{2}+4\mu^{2}l^{2}}\,\,,$
(61)

where $n\in\mathbb{N}_{0}$, $j_{1}=\ell+1$ with $\ell\in\mathbb{N}_{0}$,
$j_{2}=\ell-1$ with $\ell\in\mathbb{N}$ and $j_{3}=\ell$ with
$\ell\in\mathbb{N}$. Notice that the monopole case of the Proca field is
described by $q_{1}$ with $j_{1}=1$ or $\ell=0$. Setting $d=4$, the expression
of the normal mode frequencies given in Eq. (61) agrees with the expression
given in tf .

Although we have not analyzed the case of negative $\mu^{2}l^{2}$, the
asymptotic behavior in Eq. (IV.1) and the mode frequencies in Eq. (61)
indicate that the eigenvalue problem is well defined even for negative field
masses, as along as they obey the inequality

$\displaystyle(\mu l)^{2}\geq-\frac{(d-3)^{2}}{4}\,.$ (62)

The bound given in Eq. (62) is the Proca field analog of the Breitenlohner-
Freedman bound. In $d=4$ the Breitenlohner-Freedman bound for Proca fields in
AdS4 is $(\mu l)^{2}\geq-\frac{1}{4}$. The Breitenlohner-Freedman bound was
found originally for a massive scalar field in pure AdS and is given by $(\mu
l)^{2}\geq-\frac{9}{4}$ in $d=4$ Breitenlohner:1982 , and is given by $(\mu
l)^{2}\geq-\frac{(d-1)^{2}}{4}$ for generic $d$ ishibashiwald .

##  V Normal modes of the Maxwell field in AdSd as the $\mu=0$ limit of Proca
normal modes

###  V.1 Maxwell equations and boundary conditions

####  V.1.1 Maxwell equations

Maxwellian electromagnetic perturbations, also called Maxwell perturbations,
can be viewed as the limit of Proca perturbations when $\mu=0$. However, in
Maxwell theory, the field equations are gauge invariant and the field loses
one physical degree of freedom, which becomes a pure gauge one. Thus, one
cannot simply set $\mu=0$ for $A_{\mu}$ in the results above, as one of the
degrees of freedom becomes spurious. Indeed, the identity on the field
$A_{\mu}$, Eq. (7), no longer follows directly from the field equations,
becoming simply a gauge choice. One finds that the gauge freedom in $A_{\mu}$
is only scalar, with the vector-type sector of the Maxwell field being gauge
invariant. We display first the final equations that are of interest here and
then we show how to obtain them from the Proca equations.

The Maxwell field is described by $d-1$ components. The time component
$q_{0}(t,r)=q_{0}(r)e^{-i\omega_{12}t}$, the scalar-type component
$q_{12}(t,r)=q_{12}(r)e^{-i\omega_{12}t}$ and the $d-3$ vector-type components
$q_{3}(t,r)=q_{3}(r)e^{-i\omega_{3}t}$, where $\omega_{12}$ is a normal mode
frequency of $q_{12}$ and $\omega_{3}$ is a normal mode frequency of $q_{3}$.
The component $q_{0}$ is given by

$q_{0}(r)=\frac{i}{\omega_{12}r^{\frac{d}{2}-2}}\partial_{r_{*}}\left(q_{12}r^{\frac{d}{2}-2}\right)\,,$
(63)

and is completely determined by the scalar-type component $q_{12}$, which in
turn obeys

$\partial_{r_{*}}^{2}q_{12}+\left(\omega_{12}^{2}-V_{12}(r)\right)q_{12}=0\,,$
(64) $\displaystyle{V}_{12}(r)=$ $\displaystyle
f\Bigg{[}\frac{4\ell(\ell+d-3)+(d-4)(d-2)}{4r^{2}}$ $\displaystyle\hskip
65.00009pt+\frac{(d-4)(d-6)}{4l^{2}}\Bigg{]}\,.$ (65)

This scalar-type component of the Maxwell field has been denominated as
$q_{12}$ since it is the corresponding mode to the Proca $q_{1}$ and $q_{2}$
modes when the mass of the field is zero $\mu=0$. The vector-type components
covers the same $d-3$ degrees of freedom as in the massive case and so they
are governed by Eq. (59), for $k=3$ and $\mu=0$, i.e.,

$\displaystyle\partial_{r_{*}}^{2}q_{3}+\left(\omega_{3}^{2}-V_{3}\right)q_{3}=0\,\,,$
(66) $\displaystyle V_{3}=f\Bigg{[}\frac{4\ell(\ell+d-3)+(d-2)(d-4)}{4r^{2}}$
$\displaystyle\hskip 105.00015pt+\frac{(d-2)(d-4)}{4l^{2}}\Bigg{]}\,\,,$ (67)

where these vector-type components of the Maxwell field have been denominated
as $q_{3}$ since they are the corresponding modes to the Proca $q_{3}$ modes
when the mass of the field is zero $\mu=0$. Surely, there is no possibility of
confusion, now we are dealing with Maxwell modes.

Let us derive the above equations. In Maxwell theory, the field equations are
gauge invariant which means that the Proca field loses one physical degree of
freedom when $\mu=0$. Thus, in the perturbed quantities there is one mode that
becomes nonphysical. In order to distinguish between the physical degrees of
freedom and the pure gauge ones, it is useful to work with gauge-invariant
variables rather than with $A_{\mu}$. Under the gauge transformation, using
the notation of Eqs. (15) and (16), the fields transform as

$\displaystyle{\tilde{A}}_{a}\to{\tilde{A}}_{a}+\partial_{a}h\,,$
$\displaystyle{\hat{A}}_{i}={\hat{A}}^{(v)}_{i}+\hat{\nabla}_{i}{\hat{A}}^{(s)}\to{\hat{A}}^{(v)}_{i}+\hat{\nabla}_{i}\left({\hat{A}}^{(s)}+h\right)\,,$
(68)

for some gauge function $h$. One sees that the gauge freedom in $A_{\mu}$ is
only scalar, with the vector-type sector of the Maxwell field being gauge
invariant. We start to play with the scalar-type components. For the scalar-
type components, one has to go back to Eqs. (28) and (29). Setting $\mu=0$ in
them one has

$\displaystyle\tilde{\nabla}_{b}\tilde{\nabla}^{[a}\psi^{b]}+(d-2)\frac{\partial_{b}r}{r}\tilde{\nabla}^{[a}\psi^{b]}+\frac{k_{s}^{2}}{2r^{2}}\left(\psi^{a}-\partial^{a}\phi\right)=0\
,$ (69)
$\displaystyle\tilde{\nabla}_{b}\left(\psi^{b}-\partial^{b}\phi\right)+\left(d-4\right)\frac{\partial_{b}r}{r}\left(\psi^{b}-\partial^{b}\phi\right)=0\,.$
(70)

This motivates the definition of the field $\zeta^{a}$ given by

$\displaystyle\zeta^{a}=\psi^{a}-\partial^{a}\phi\,,$ (71)

which is gauge invariant. Indeed, one can expand the gauge function $h$ as
$h(y,\theta)=h(y)Y(\theta)$, so that, under a gauge transformation, one has
$\psi_{a}\rightarrow\psi_{a}+\partial_{a}h$ and $\phi\rightarrow\phi+h$, where
the $\vec{k}_{s}$ indices were omitted for convenience. In terms of
$\zeta^{a}$ the equations of motion, Eqs. (69) and (70) become

$2\tilde{\nabla}_{b}\tilde{\nabla}^{[a}\zeta^{b]}+2(d-2)\frac{\partial_{b}r}{r}\tilde{\nabla}^{[a}\zeta^{b]}+\frac{k_{s}^{2}}{r^{2}}\zeta^{a}=0\,,$
(72) $\tilde{\nabla}_{b}\left(r^{d-4}\zeta^{b}\right)=0\,.$ (73)

Note that this transformation completely removes a pure gauge degree of
freedom from the system, as $\\{\psi^{a},\phi\\}\rightarrow\\{\zeta^{a}\\}$.
This only happens in the massless case, where $\zeta^{a}$ factorizes. In the
background of AdSd spacetime, substituting Eq. (73) in the $r$ component of
Eq. (72), and further making the transformation

$\displaystyle
q_{12}(r)=\frac{\zeta_{r}(t,r)}{r}fr^{\frac{d}{2}-1}e^{i\omega_{12}t}\,,$ (74)

yields the equations,
$\partial_{r_{*}}^{2}q_{12}+\left(\omega_{12}^{2}-V_{12}(r)\right)q_{12}=0$,
with
${V}_{12}(r)=f\Bigg{[}\frac{4\ell(\ell+d-3)+(d-4)(d-2)}{4r^{2}}+\frac{(d-4)(d-6)}{4l^{2}}\Bigg{]}$,
which correspond to Eqs. (64) and (V.1.1). Making the transformation

$\displaystyle
q_{0}(r)=\frac{\zeta_{t}(t,r)}{r}r^{\frac{d}{2}-1}e^{i\omega_{12}t}\,,$ (75)

yields
$q_{0}(r)=\frac{i}{\omega_{12}r^{\frac{d}{2}-2}}\partial_{r_{*}}\left(q_{12}r^{\frac{d}{2}-2}\right)$,
which corresponds to Eq. (63) and is completely determined from the scalar-
type component $q_{12}$. Finally, since the gauge freedom in $A_{\mu}$ is only
scalar, with the vector-type sector of the Maxwell field being gauge
invariant, see Eq. (68), this means that, in the $\mu=0$ limit, the vector-
type component of $A_{\mu}$ covers the same $d-3$ degrees of freedom as in the
massive case and so they are governed by Eq. (59), for $k=3$ and $\mu=0$,
i.e., $\partial_{r_{*}}^{2}q_{3}+\left(\omega_{3}^{2}-V_{3}\right)q_{3}=0$
with
$V_{3}=f\Bigg{[}\frac{4\ell(\ell+d-3)+(d-2)(d-4)}{4r^{2}}+\frac{(d-2)(d-4)}{4l^{2}}\Bigg{]}$,
which corresponds to Eqs. (66) and (67).

All this agrees with the Maxwell field having $d-2$ degrees of freedom.
Indeed, while the vector-type component covers $d-3$ degrees of freedom, the
scalar-type component only covers one degree of freedom, which in this case
was chosen to be $\zeta_{r}(t,r)$ or $q_{12}(r)$. Note also that we are
treating only dynamical solutions of the Maxwell’s equations, and so the
frequencies $\omega$ must be non-zero.

####  V.1.2 Boundary conditions

In summary, the Maxwell field is comprised of a scalar-type component $q_{12}$
that satisfies the Maxwell equation Eq. (64). and of vector-type components
$q_{3}$ that satisfy the Maxwell equation Eq. (66).

With respect to the regularity conditions, both $q_{12}$ and $q_{3}$ have the
same behavior as the Proca $q_{k}$ in Eq. (57) near $r=0$. Thus,

$\displaystyle
q_{i}=\alpha^{r=0}_{i}r^{\ell+\frac{d-2}{2}}+\beta^{r=0}_{i}r^{-\ell-\frac{d-4}{2}}\,\,,$
(76)

where now $i\in\\{12,3\\}$, and $\alpha^{r=0}_{i}$ and $\beta^{r=0}_{i}$ are
constants. For all cases of the electromagnetic field, the $q_{i}$ are square
integrable if $\beta^{r=0}_{i}=0$, and so we admit this condition as the
regularity condition.

In relation to the boundary conditions at spatial infinity, some care must be
taken. In the massive case, both scalar-type and vector-type perturbations
have the same effective mass. This contrasts with the massless case, where the
effective mass for the the scalar-type component,
$\mu_{\mathrm{eff}}^{2}=\frac{(d-4)(d-6)}{4l^{2}}$, and the effective mass for
the vector-type component, $\mu_{\mathrm{eff}}^{2}=\frac{(d-2)(d-4)}{4l^{2}}$,
are functionally different, being only the same in $d=4$. This means that the
asymptotic behavior at spatial infinity is different for the scalar-type
component and for the vector-type components. Near spatial infinity, the
electromagnetic fields behave as

$\displaystyle q_{12}$
$\displaystyle=\begin{cases}\dfrac{\alpha_{12}^{r=\infty}}{r}+\beta_{12}^{r=\infty}&\mathrm{for}\,\,d=4\,\,,\\\
\\\ \dfrac{\alpha_{12}^{r=\infty}}{\sqrt{r}}+\dfrac{\beta_{12}^{r=\infty}\ln
r}{\sqrt{r}}&\mathrm{for}\,\,d=5\,\,,\\\ \\\
\dfrac{\alpha_{12}^{r=\infty}}{r^{\frac{d}{2}-2}}+\beta_{12}^{r=\infty}r^{\frac{d}{2}-3}&\mathrm{for}\,\,d\geq
6\,\,,\\\ \end{cases}$ (77) $\displaystyle q_{3}$
$\displaystyle=\dfrac{\alpha_{3}^{r=\infty}}{r^{\frac{d}{2}-1}}+\beta_{3}^{r=\infty}r^{\frac{d}{2}-2}\
\ \ \mathrm{for}\,\,d\geq 4\,\,,$ (78)

where $\alpha_{i}^{r=\infty}$ and $\beta_{i}^{r=\infty}$ are constants, with
$i\in\\{12,3\\}$.

For the scalar-type perturbation, $q_{12}$, in $d=4$ the field is square
integrable for nonzero and finite $\beta_{i}$ when $r\to\infty$. The
requirement that the usual definition of the energy is finite and independent
of the $t$ constant slice allows for the two reflective boundary conditions,
the Dirichlet and the Neumann, with the Dirichlet imposing
$\beta_{i}^{r=\infty}=0$ and the Neumann imposing $\alpha_{i}^{r=\infty}=0$.
Also, other boundary conditions for these cases are also possible, see
ishibashiwald ; Herdeiro2 . Nevertheless, we impose $\beta_{i}^{r=\infty}=0$,
which corresponds to the Dirichlet boundary condition. For the scalar-type
perturbation, $q_{12}$, in $d=5$, the field is square integrable for every
$\alpha_{12}^{r=\infty}$, $\beta_{12}^{r=\infty}$. Moreover, the Dirichlet
boundary condition, which imposes the field to vanish at
$r\rightarrow+\infty$, does not restrict the asymptotic coefficients and
leaves the eigenvalue problem ill-defined. In order to have well-defined
dynamics for the field in the sense of ishibashiwald , one needs to choose
$\alpha_{12}^{r=\infty}$ and $\beta_{12}^{r=\infty}$ carefully. The boundary
condition that keeps the usual definition of the energy to be finite and time
independent is that $\beta_{12}^{r=\infty}=0$, which removes the dominant
logarithmic term in Eq. (77), and in this case, since it involves the field
and first derivatives of the field, is a Dirichlet-Neumann boundary condition.
For the scalar-type perturbation, $q_{12}$, in $d=6$, the field is square
integrable for nonzero and finite $\beta_{i}^{r=\infty}$, as in the case
$d=4$. Here, it seems that the requirement that the usual definition of the
energy is finite and independent of the $t$ constant slice only allows the
Dirichlet boundary condition. Also notice that other boundary conditions for
this cases are also possible, see ishibashiwald ; Herdeiro2 . Nevertheless, we
impose $\beta_{i}^{r=\infty}=0$, which corresponds to the Dirichlet boundary
condition. For the scalar-type perturbations, $q_{12}$, in $d\geq 7$, the
field is square integrable only if $\beta_{i}^{r=\infty}=0$. Therefore,
Dirichlet boundary conditions must be imposed.

For the vector-type perturbation, $q_{3}$, in $d=4$, the field is square
integrable for nonzero and finite $\beta_{i}^{r=\infty}$. The requirement that
the usual definition of the energy is finite and independent of the $t$
constant slice allows also for the two reflective boundary conditions, the
Dirichlet and the Neumann, with the Dirichlet imposing
$\beta_{i}^{r=\infty}=0$ and the Neumann imposing $\alpha_{i}^{r=\infty}=0$.
As well, other boundary conditions for these cases are possible, see
ishibashiwald ; Herdeiro2 . Nevertheless, we impose $\beta_{i}^{r=\infty}=0$,
which corresponds to the Dirichlet boundary condition. For the vector-type
perturbations, $q_{3}$, in $d\geq 5$, the fields are only square integrable if
$\beta_{i}^{r=\infty}=0$. Therefore, Dirichlet boundary conditions must be
imposed.

With these considerations and for consistency in order to compare with the
Proca field normal modes, we still apply to the Maxwell field the Dirichlet
boundary conditions for both vector-type and scalar-type perturbations, for
every case, i.e., $q_{i}(r\to\infty)=0$.

###  V.2 Solutions of the Maxwell equations and normal mode frequencies

To obtain the normal modes, one can put the equations obeyed by $q_{12}$ and
$q_{3}$ in the form

$\displaystyle\partial_{r_{*}}^{2}q_{i}+\left(\omega_{i}^{2}-\frac{G_{i}}{\sin^{2}\left(\frac{r_{*}}{l}\right)}-\frac{H_{i}}{\cos^{2}\left(\frac{r_{*}}{l}\right)}\right)q_{i}=0\,,$
(79)

where $i\in\\{12,3\\}$, and the constants are
$G_{i}=\frac{4\ell(\ell+d-3)+(d-2)(d-4)}{4l^{2}}$,
$H_{12}=\frac{(d-4)(d-6)}{4l^{2}}$, and $H_{3}=\frac{(d-2)(d-4)}{4l^{2}}$.

Imposing the regularity and the Dirichlet boundary conditions for $d=4$ and
$d\geq 6$, and Dirichlet-Neumann boundary conditions for $d=5$, one obtains
that the solutions are described by the hypergeometric functions, see Appendix
B, and they yield

$\displaystyle q_{i}=$ $\displaystyle
a_{i}\left(\frac{r}{l}\right)^{\ell+\frac{d-2}{2}}\left(1+\frac{r^{2}}{l^{2}}\right)^{n-\frac{\omega_{i}l}{2}}$
$\displaystyle\times\prescript{}{2}{F}_{1}\left[-n+\omega_{i}l,-n,\ell+\frac{d-1}{2};\frac{\frac{r^{2}}{l^{2}}}{1+\frac{r^{2}}{l^{2}}}\right],$
(80)

for $i\in\\{12,3\\}$, with the normal mode frequencies

$\displaystyle\omega_{12}=$
$\displaystyle\begin{cases}2n+\ell+2&\mathrm{for}\,\,d=4\,,\\\
2n+\ell+2&\mathrm{for}\,\,d=5\,\,,\\\ 2n+\ell+d-3&\mathrm{for}\,\,d\geq
6\,,\end{cases}$ (81) $\displaystyle\omega_{3}=$ $\displaystyle\quad
2n+\ell+d-2\,\,\,\;\;\mathrm{for}\,\,d\geq 4\,,$ (82)

with $n\in\mathbb{N}_{0}$ and $\ell\in\mathbb{N}$, where $\mathbb{R}^{+}$ are
the positive real numbers.

It is interesting to reflect on the normal modes of the Maxwell
electromagnetic field as the massless limit of the Proca field. The scalar-
type perturbations need to be treated with some care as we have seen. For
$d=4$, the scalar-type perturbation has the same frequency as the vector-type.
Since the massless limit of the Proca field yields different frequencies for
the scalar and vector perturbations, this discrepancy indicates some
discontinuity in the massless limit, which indeed happens in the frequencies
of the scalar perturbations. For $d=5$, the normal mode frequencies of the
scalar-type perturbation of the electromagnetic field follow from the massless
limit of the Proca field. It must be noted however that the frequencies for
the scalar-type perturbation of the electromagnetic field were obtained using
Dirichlet-Neumann conditions, rather than Dirichlet boundary conditions that
were imposed to the original Proca field. Indeed, the Dirichlet boundary
conditions leave the eigenvalue problem ill-defined. For $d\geq 6$, the
electromagnetic scalar-type normal modes follow from the massless limit of the
modes in Eq. (61) for $k=2$ and $j_{2}=\ell-1$, indicating that $q_{2}$
describes the electromagnetic mode, but now extended to the massive case, and
that $q_{1}$ describes the scalar field degree of freedom of the Proca fields,
which in the massless case can be removed by the gauge freedom. On the other
hand, the vector-type perturbation of the electromagnetic follows directly
from the massless limit of the vector-type perturbation of the Proca field and
so one sets directly $\mu=0$ in Eq. (61) for $k=3$ and $j_{3}=\ell$, and
obtains the normal frequencies in Eq. (82).

We have obtained the normal modes for the Maxwell electromagnetic
perturbations by working in all detail the massless limit of the Proca
perturbations. Our results agree with those found in ishibashiwald , where a
direct analysis of the Maxwell field from a master equation in AdSd was
performed. The results also conform to the Maxwell electromagnetic
perturbations for AdSd found in natario ; ortega , and they recover the
$\mu=0$ limit of the Proca field for AdS4 tf , see also ckjpsl .

##  VI Conclusions

In this work, the normal modes of linear Proca perturbations in AdSd
background were obtained analytically, using the Ishibashi-Kodama formalism.
The Proca field was decomposed into different components according to its
tensorial behavior on the sphere, yielding scalar-type components, covering
two degrees of freedom of the field, and vector-type components, covering the
remaining $d-3$ degrees of freedom. In general, while in the scalar-type
sector the two degrees of freedom are coupled, due to the mass of the field,
in the vector-type sector the modes are completely decoupled. In AdSd, the
scalar-type perturbations can be indeed decoupled in the equations and they
are covered independently by two fields, $q_{1}$ and $q_{2}$. The vector-type
perturbations can be covered by one field only, $q_{3}$.

The usual regularity boundary conditions at the center and Dirichlet boundary
conditions at infinity were imposed to the Proca equations. We used a
Dirichlet condition because it is the condition that ensures that the usual
definition of the energy through the conservation of $\xi^{\mu}T_{\mu\nu}$,
where $\xi^{\mu}$ is the timelike Killing vector and $T_{\mu\nu}$ is the
stress-energy tensor, is finite and is independent of the $t$ constant
hypersurface. It would be interesting to investigate the eigenfrequency
problem for other possible boundary conditions. Using the regularity and the
Dirichlet conditions, the solutions for the Proca field were obtained and
found to be described by hypergeometric functions. The normal mode frequencies
were obtained for any value of the Proca mass $\mu$ and for $d\geq 4$. For the
$d=4$ case, the expression of the frequencies agrees with previous works. We
have also found an explicit expression for Breitenlohner-Freedman bound of the
Proca field in AdSd.

The normal modes of Maxwell electromagnetic perturbations in AdSd were also
obtained analytically through the $\mu=0$ limit of Proca perturbations. In
this case, by working with gauge-invariant variables, it was possible to
separate the physical modes from the non-physical ones. For consistency, the
regularity and Dirichlet boundary conditions were used, except for the case of
$d=5$ where the Dirichlet-Neumann boundary condition was used, in order to
analyze the massless limit of the Proca field. We have thus recovered the
normal mode frequencies of the Maxwell field found in previous works, with the
$d=4$ case regarding the scalar perturbations having to be treated with much
care, in particular, the massless limit of the Proca scalar perturbations in
$d=4$ does not lead directly to the Maxwell field scalar perturbation modes.

###### Acknowledgements.

We thank Antonino Flachi for conversations. We acknowledge financial support
from Fundação para a Ciência e Tecnologia - FCT through the project No.
UIDB/00099/2020 and project No. UIDP/00099/2020. TF acknowledges a grant from
FCT no. RD0970.

##  Appendix A Spherical harmonics on the ($d-2$)-sphere

###  A.1 Initial considerations

The decomposition of fields in spherical harmonics is important to study the
structure of their perturbations and to isolate the physical degrees of
freedom of the fields themselves. Here, we discuss in some detail the
properties of these special functions.

The approach we adopt here to construct spherical harmonics on the
$(d-2)$-sphere, $\mathcal{S}^{d-2}$, follows myers , where spherical harmonics
are constructed by embedding $\mathcal{S}^{d-2}$ in a ($d-1$)-Euclidean space,
$\mathbb{R}^{d-1}$. Firstly, we introduce some useful concepts: (a) a
polynomial $h_{\ell}:\mathbb{R}^{d-1}\rightarrow\mathbb{C}$ is homogeneous of
degree $\ell$ in $\mathbb{R}^{d-1}$ if $h_{\ell}(\lambda
x^{\mu})=\lambda^{\ell}h_{\ell}(x^{\mu})$, for any
$\lambda\in\mathbb{R},{x^{\mu}}\in\mathbb{R}^{d-1}$; (b) a polynomial
$h_{\ell}:\mathbb{R}^{d-1}\rightarrow\mathbb{C}$ is harmonic in
$\mathbb{R}^{d-1}$ if $\Box_{E}h_{\ell}=0$, where $\Box_{E}$ is the Laplacian
on $\mathbb{R}^{d-1}$; (c) a spherical harmonic of degree $\ell$ on
$\mathcal{S}^{d-2}$ is a function
$Y_{\ell}:\mathcal{S}^{d-2}\rightarrow\mathbb{C}$ such that, for some
homogeneous and harmonic polynomial in $\mathbb{R}^{d-1}$, $h_{\ell}$,
$Y_{\ell}(\theta)=h_{\ell}(\theta)$ for all
$\theta=(\theta^{2},...\theta^{d-1})\in\mathcal{S}^{d-2}$. Spherical harmonics
on the $(d-2)$-sphere can also be constructed recursively by dimensional
reduction, see, e.g., higuchi , but the approach becomes cumbersome when
constructing vector spherical harmonics. For more details on the matter we
refer to lovenumbers ; myers ; higuchi ; abramowitz , see also
ishibashi_kodama2 .

###  A.2 Scalar spherical harmonics

In spherical coordinates, the $\mathbb{R}^{d-1}$ line element, $ds^{2}$, is
related to the $\mathcal{S}^{d-2}$ line element,
$(d\Omega^{d-2})^{2}={\hat{g}}_{ij}d\theta^{i}d\theta^{j}$, by
$ds^{2}=dr^{2}+r^{2}(d\Omega^{d-2})^{2}$. Using Eq. (III.1), the non-vanishing
Christoffel symbols associated to $g_{\mu\nu}$ in these coordinates are

$\Gamma^{r}_{\ ij}=-r{\hat{g}}_{ij}\,,\quad\Gamma^{i}_{\
rj}=\frac{1}{r}\delta^{i}_{j}\,,\quad\Gamma^{i}_{\ jk}=\hat{\Gamma}^{i}_{\
jk},$ (83)

and the condition for a homogeneous polynomial of degree $\ell$, $h_{\ell}$,
to be harmonic in $\mathbb{R}^{d-1}$, becomes

$\Box_{E}h_{\ell}=\frac{1}{r^{d-2}}\partial_{r}\left(r^{d-2}\partial_{r}h_{\ell}\right)+\frac{1}{r^{2}}\hat{\Box}h_{\ell}=0\,,$
(84)

where $\hat{\Box}={\hat{g}}^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}$. Since
$h_{\ell}$ is homogeneous, it follows that
$h_{\ell}(x)|_{S}=r^{\ell}h_{\ell}({\hat{x}})|_{S}=r^{\ell}Y_{\ell}(\theta)$,
where $x^{\mu}=r\hat{x}^{\mu}$, so that, substituting in Eq. (84), one has

$\hat{\Box}Y_{\ell}=-\ell(\ell+d-3)Y_{\ell}\,\,,\,\,\ell=0,1,2,...$ (85)

$Y_{\ell}$ are called scalar spherical harmonics. Besides being eigenfunctions
of $\hat{\Box}$, it can also be shown that they form a complete and orthogonal
set on $\mathcal{S}^{d-2}$.

###  A.3 Vector spherical harmonics

Vector spherical harmonics can be constructed in the same way as scalar
spherical harmonics, only this time one starts with vector functions
$V_{\nu}^{\ell}:\mathbb{R}^{d-1}\rightarrow\mathbb{C}^{d-1}$. Using the
Helmholtz-Hodge theorem, $V_{\nu}^{\ell}$ can be written as, in analogy to Eq.
(15),

$V_{\nu}^{\ell}dx^{\nu}=V_{r}^{\ell}dr+(W_{i}^{\ell}+\hat{\nabla}_{i}\sigma^{\ell})d\theta^{i}\,\,,\,\,\hat{\nabla}^{i}W_{i}^{\ell}=0\,\,,$
(86)

where $W^{i}_{\ell}$ is a vector on the $(d-2)$-sphere and $V_{r}^{\ell}$ and
$\sigma^{\ell}$ are scalars. Expanding $\Box_{E}V_{\nu}^{\ell}$ in spherical
coordinates, and assuming that $V_{\nu}^{\ell}$ is harmonic, one has

$\Box_{E}W_{i}^{\ell}=\partial^{2}_{r}W_{i}^{\ell}+\frac{d-4}{r}\partial_{r}W_{i}^{\ell}-\frac{d-3}{r^{2}}W_{i}^{\ell}+\frac{1}{r^{2}}\hat{\Box}W_{i}^{\ell}=0\,,$
(87)

as well as two coupled equations for the scalars $V_{r}^{\ell}$ and
$\sigma^{\ell}$ myers . Here, we are only interested in the equations for
$W_{i}^{\ell}$. Since $V_{\nu}^{\ell}$ is homogeneous,
$V_{\mu}^{\ell}(x)=r^{\ell}V_{\mu}^{\ell}(\theta)$. This means that
${V}_{\nu}^{\ell}(x)dx^{\nu}=\frac{\partial x^{\mu}}{\partial
r}r^{\ell}V_{\mu}^{\ell}(\theta)dr+\frac{\partial
x^{\mu}}{\partial\theta^{i}}r^{\ell}V_{\mu}^{\ell}(\theta)d\theta^{i}$ and so
$W^{\ell}_{i}+\hat{\nabla}_{i}\sigma^{\ell}=r^{\ell+1}\frac{\partial\hat{x}^{\mu}}{\partial\theta^{i}}V_{\mu}^{\ell}(\theta)$,
where $x^{\mu}=r\hat{x}^{\mu}$. Hence, we define $Y_{\ell i}(\theta)$ as the
angular dependence of $W^{\ell}_{i}(x)$, i.e.
$W^{\ell}_{i}(x)|_{S}=r^{\ell+1}Y_{\ell\,i}(\theta)$, which put in Eq. (87),
it follows

$\hat{\Box}Y_{\ell\,i}=-\left[\ell(\ell+d-3)-1\right]Y_{\ell\,i}\,,\,\,\,\,\ell=1,2,...$
(88)

The vectors $Y_{\ell\,i}$ are called transverse vector spherical harmonics, as
they verify $\hat{\nabla}^{i}Y_{\ell\,i}=0$. One can also construct
longitudinal vector spherical harmonics, $Y_{\ell\,i}^{L}$, by taking the
gradient of scalar fields on $\mathcal{S}^{d-2}$. These are defined as
lovenumbers

$Y_{\ell\,i}^{L}\equiv\frac{1}{\sqrt{\ell(\ell+d-3)}}\hat{\nabla}_{i}Y_{\ell}\,,$
(89)

and have eigenmodes

$\hat{\Box}{\hat{\nabla}_{i}}Y_{\ell}=-\left[\ell(\ell+d-3)-(d-3)\right]{\hat{\nabla}_{i}}Y_{\ell}\,\,,\,\,\ell=1,2,...$
(90)

###  A.4 Properties under rotation and parity transformations

Now, let $\varphi$ and $u^{i}$ be, respectively, a scalar field and a vector
field on $\mathcal{S}^{d-2}$. The action of the SO($d-1$) Casimir operator,
$\hat{J}^{2}$, on these two entities is $\hat{J}^{2}\varphi=-\Box_{S}\varphi$
and $\hat{J}^{2}u^{i}=-\left(\Box_{S}-(d-3)\right)u^{i}$, see ishibashiwald .
Using Eqs. (85), (88) and (90), one gets the following

$\displaystyle\hat{J}^{2}Y_{\ell}=\ell(\ell+d-3)Y_{\ell}\,\,,$ (91)
$\displaystyle\hat{J}^{2}\hat{\nabla}_{i}Y_{\ell}=\ell(\ell+d-3)\hat{\nabla}_{i}Y_{\ell}\,\,,$
(92)
$\displaystyle\hat{J}^{2}Y_{\ell\,i}=\left[\ell(\ell+d-3)+d-4\right]Y_{\ell\,i}\,\,.$
(93)

One sees that, for $\ell\geq 1$ and $d>4$, the Casimir eigenvalues of the
scalar spherical harmonics and of the longitudinal vector spherical harmonics
are the same. One then expects these modes to mix. On the contrary, the
Casimir eigenvalues of the transverse vector spherical harmonics are never
equal for $d>4$ to the eigenvalues of the longitudinal vector spherical
harmonics, so that these completely decouple. For $d=4$, the last are equal
and one might expect to have mixed modes. However, in this case, the modes are
decoupled due to their different parity eigenvalues. Indeed, under parity
transformations $\theta_{i=2}\rightarrow\pi-\theta_{i=2}$ and
$\theta_{i=3}\rightarrow\pi+\theta_{i=3}$, one has
$\hat{\mathcal{P}}Y_{\ell}=\left(-1\right)^{\ell}Y_{\ell}$,
$\hat{\mathcal{P}}\hat{\nabla}_{i}Y_{\ell}=\left(-1\right)^{\ell}\hat{\nabla}_{i}Y_{\ell}$
and $\hat{\mathcal{P}}Y_{\ell\,i}=\left(-1\right)^{\ell+1}Y_{\ell\,i}$. Note
that a vector, $A_{\theta_{i}}$, on the $(d-2)$-sphere transforms under parity
as $A_{\theta_{i=1}}\rightarrow A_{\theta_{i=1}}$ and $A_{\theta_{i\neq
1}}\rightarrow-A_{\theta_{i\neq 1}}$.

##  Appendix B Hypergeometric differential equation

###  B.1 General equation

We study now the solutions of the hypergeometric differential equation, based
on abramowitz , see also ishibashiwald .

The Proca and electromagnetic field equations can be separated and decoupled
to assume the reduced form

$\partial_{r_{*}}^{2}q+\left(\omega^{2}-\frac{G}{\sin^{2}\left(\frac{r_{*}}{l}\right)}-\frac{H}{\cos^{2}\left(\frac{r_{*}}{l}\right)}\right)q=0\,,$
(94)

as it appears in Eqs. (59) and (79), where
$r_{*}=l\arctan\left(\frac{r}{l}\right)$,
$G=\frac{4j(j+d-3)+(d-2)(d-4)}{4l^{2}}$ for $j\in\mathbb{N}_{0}$, and $H$ for
now is some constant that depends on $d$, $l$, $\mu$, and $j$ with units of
$\frac{1}{l^{2}}$. By making the change

$q(z)=z^{\alpha}(1-z)^{\beta}\Theta(z)\,,$ (95)

with

$z=\sin^{2}\left(\frac{r_{*}}{l}\right)\,,$ (96)

and recalling $r_{*}=l\arctan\frac{r}{l}$, one can transform Eq. (94) into the
hypergeometric differential equation

$z(1-z)\frac{d^{2}\Theta}{dz^{2}}+\left[c-(a+b+1)z\right]\frac{d\Theta}{dz}-ab\Theta=0\,,$
(97)

with $a=\alpha+\beta+\frac{\omega l}{2}$, $b=\alpha+\beta-\frac{\omega l}{2}$,
$c=2\alpha+\frac{1}{2}$, and $\alpha=\frac{2j+d-2}{4}$,
$\beta=\frac{1}{4}\left(1+\sqrt{1+4Hl^{2}}\right)$. For $d$ even, the
solutions of (97) are given in terms of the hypergeometric function,
$\prescript{}{2}{F}_{1}$, as abramowitz

$\displaystyle
q(z)=Az^{\alpha}(1-z)^{\beta}\prescript{}{2}{F}_{1}\left[a,b,c;z\right]$
$\displaystyle+Bz^{1/2-\alpha}(1-z)^{\beta}\prescript{}{2}{F}_{1}\left[a-c+1,b-c+1,2-c;z\right]$
$\displaystyle\hskip 158.00026pt\mathrm{for}\,\,d\,\,\mathrm{even}\,,$ (98)

where $A$ and $B$ are constants of integration. For $d$ odd, the indicial
roots of Eq. (97) are separated by an integer, making the solutions presented
linearly dependent. In this case, since $c\geq 2$ and it is an integer, $q(z)$
can be written as abramowitz

$\displaystyle
q(z)=Az^{\alpha}(1-z)^{\beta}\prescript{}{2}{F}_{1}\left[a,b,c;z\right]+Bz^{\alpha}(1-z)^{\beta}$
$\displaystyle\times\Bigg{[}\prescript{}{2}{F}_{1}\left[a,b,c;z\right]\ln{z}+\sum^{\infty}_{i=1}v_{i}z^{i}-\sum^{c-1}_{i=1}w_{i}z^{-i}\Bigg{]}$
$\displaystyle\hskip 155.00023pt\mathrm{for}\,\,d\,\,\mathrm{odd}\,,$ (99)

where

$\displaystyle
v_{i}=\frac{(a)_{i}(b)_{i}}{(c)_{i}(i!)}[\Psi(a+i)-\Psi(a)+\Psi(b+i)-\Psi(b)$
$\displaystyle-\Psi(c+i)+\Psi(c)-\Psi(1+i)+\Psi(1)]\,\,,$ (100) $\displaystyle
w_{i}=\frac{(i-1)!(1-c)_{i}}{(1-a)_{i}(1-b)_{i}}\,\,,$ (101)

$(a)_{i}=\frac{\Gamma(a+i)}{\Gamma(a)}$ if $a>0$,
$(a)_{i}=(-1)^{i}(-a-i+1)_{i}$ if $a<0$, and the same for $(b)_{i}$ and
$(c)_{i}$, and $\Psi(a)=\frac{\Gamma^{\prime}(a)}{\Gamma(a)}$ is the digamma
function, see abramowitz .

###  B.2 Imposing regularity conditions near $r=0$, i.e., $z=0$

To obtain the normal modes, one needs to impose boundary conditions that
characterize the system. At $r=0$ or $z=0$, one imposes regularity conditions
so that the solution does not diverge there.

For even spacetimes, it is seen from Eq. (B.1) that for $\alpha>\frac{1}{2}$
one needs to set $B=0$, considering that $\lim_{z\to
0}\prescript{}{2}{F}_{1}\left[a,b,c;z\right]=1$. For $\alpha=\frac{1}{2}$,
which corresponds to the case $q_{2}$ with $\ell=1$ and $d=4$, the exponent in
$z$ vanishes and the solution seems to be finite near $z=0$. However, this is
just an artifact of having removed the origin when separating the field in
spherical harmonics, see ishibashiwald . Indeed, if one writes the components
of the four-dimensional Proca field $A_{r}$, $A_{\theta_{2}}$ and
$A_{\theta_{3}}$ in terms of $q_{2}$ with $\ell=1$ and the other fields put to
zero, one arrives at $A_{r}=\frac{1}{rf}\sum_{m=-1}^{1}q_{2\,(0,m)}Y_{(1,m)}$,
$A_{\theta_{2}}=\sum_{m=-1}^{1}q_{2\,(0,m)}\partial_{\theta_{2}}Y_{(1,m)}$,
and $A_{\theta_{3}}=-\sum_{m=-1}^{1}$
$\frac{q_{2\,(0,m)}}{d-3}\partial_{\theta_{3}}Y_{(1,m)}$, where $q_{2\,(0,m)}$
is $q_{2}$ with $j_{2}=0$, $\ell=1$ and azimuthal number $m$. Since near the
origin, the spacetime is flat, i.e., $f(r)\simeq 1$, and
$q_{2\,(0,m)}=B_{(1,m)}$ where $B_{(1,1)}$, $B_{(1,0)}$ and $B_{(1,-1)}$ are
constants, the behavior of the Proca field near the origin becomes
$A_{r}\simeq\frac{1}{r}\left(B_{(1,1)}Y_{(1,1)}+B_{(1,0)}Y_{(1,0)}+B_{(1,-1)}Y_{(1,-1)}\right)$,
$A_{\theta_{2}}\simeq
B_{(1,1)}\partial_{\theta_{2}}Y_{(1,1)}+B_{(1,0)}\partial_{\theta_{2}}Y_{(1,0)}+B_{(1,-1)}\partial_{\theta_{2}}Y_{(1,-1)}$,
$A_{\theta_{3}}\simeq
B_{(1,1)}\partial_{\theta_{3}}Y_{(1,1)}+B_{(1,0)}\partial_{\theta_{3}}Y_{(1,0)}+B_{(1,-1)}\partial_{\theta_{3}}Y_{(1,-1)}$.
The components of a vector field transform as $A_{\mu^{\prime}}=\frac{\partial
x^{\mu}}{\partial x^{\mu^{\prime}}}A_{\mu}$ to give
$A_{x},A_{y},A_{z}\sim\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}$, in Cartesian
coordinates. Since the Proca field equations Eq. (6) can be written near the
origin in Cartesian coordinates as
$\partial^{\nu}\partial_{\nu}A_{\rho}-\mu^{2}A_{\rho}=0$, where
$\alpha=\\{t,x,y,z\\}$, one has
$\left(\partial^{2}_{x}+\partial^{2}_{y}+\partial^{2}_{z}\right)A_{x,y,z}\sim\delta(x)\delta(y)\delta(z)$.
Due to this additional delta term, $q_{2\,(0,m)}=B_{(1,m)}$, with
$B_{(1,m)}\neq 0$, cannot be a solution near the origin and one needs to set
$B=0$ in Eq. (B.1), even in this special case. Moreover, if one takes the
asymptotic expansion of the integrand of the usual energy,
$T_{tt}\frac{r^{d-2}}{f}$, near $r=0$, then one gets divergent terms $r^{-1}$,
$r^{-2}$ and $r^{-4}$ which only vanish if $B=0$. Thus, in this particular
case of $\alpha=\frac{1}{2}$, which corresponds to the case $q_{2}$ with
$\ell=1$ and $d=4$, one has also $B=0$.

For odd spacetimes, all the terms of Eq. (B.1) except the last one vanish in
the limit $z\rightarrow 0$. For the last term, as $c\geq 2$ and it is an
integer, it can be seen that $\sum^{c-1}_{i=1}w_{i}z^{\alpha-i}$ contains a
power $r^{s}$ with $s\leq-\frac{1}{2}$ always and so the field $q$ would not
be square integrable. Thus, one also needs to set $B=0$ in this case.

###  B.3 Imposing Dirichlet boundary conditions at spatial infinity,
$r\to\infty$, i.e., $z\to 1$

####  B.3.1 Expansion at spatial infinity for general case

To impose the remaining boundary condition, one uses the transformation law
$z\rightarrow 1-z$ of $\prescript{}{2}{F}_{1}$, so that, if
$c-a-b=\frac{1}{2}-2\beta\neq-m^{\prime}$ with $m^{\prime}\in\mathbb{N}_{0}$
one has abramowitz

$\displaystyle q(z)=Az^{\alpha}(1-z)^{\beta}$
$\displaystyle\times\Bigg{[}\frac{\Gamma(c)\Gamma(1-e)}{\Gamma(\bar{a})\Gamma(\bar{b})}\prescript{}{2}{F}_{1}\left[a,b,e;1-z\right]$
$\displaystyle+(1-z)^{\frac{1}{2}-2\beta}\frac{\Gamma(c)\Gamma(e-1)}{\Gamma(a)\Gamma(b)}\prescript{}{2}{F}_{1}\left[\bar{a},\bar{b},2-e;1-z\right]\Bigg{]}\,,$
(102)

where $e=1-c+a+b$, $\bar{a}=c-a$ and $\bar{b}=c-b$. If
$c-a-b=\frac{1}{2}-2\beta=0$, one has

$\displaystyle
q(z)=Az^{\alpha}(1-z)^{\beta}\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\sum_{i=0}^{\infty}\frac{(a)_{i}(b)_{i}}{(i!)^{2}}(1-z)^{i}$
$\displaystyle\times\left[2\Psi(i+1)-\Psi(a+i)-\Psi(b+i)-\log(1-z)\right]\,.$
(103)

If $c-a-b=\frac{1}{2}-2\beta=-m^{\prime}$ with $m^{\prime}\in\mathbb{N}$, one
has

$\displaystyle
q(z)=Az^{\alpha}(1-z)^{\beta}\Bigg{(}\frac{\Gamma(m^{\prime})\Gamma(c)}{\Gamma(a)\Gamma(b)}\sum^{m^{\prime}-1}_{i=0}v^{\prime}_{i}(1-z)^{i-m^{\prime}}$
$\displaystyle-(-1)^{m^{\prime}}\sum^{\infty}_{i=0}\frac{\Gamma(c)(1-z)^{i}}{\Gamma(a-m^{\prime})\Gamma(b-m^{\prime})}\left[w^{\prime}_{i}\ln{(1-z)}+t_{i}\right]\Bigg{)}\,\,,$
(104)

with the coefficients defined by

$\displaystyle
v^{\prime}_{i}=\frac{(a-m^{\prime})_{i}(b-m^{\prime})_{i}}{i!(1-m^{\prime})_{i}}\,\,,$
(105) $\displaystyle
w^{\prime}_{i}=\frac{(a)_{i}(b)_{i}}{i!(i+m^{\prime})!}\,\,,$ (106)
$\displaystyle t_{i}=w^{\prime}_{i}[\Psi(a+i)+\Psi(b+i)$
$\displaystyle-\Psi(i+1)-\Psi(i+m^{\prime}+1)]\,\,.$ (107)

####  B.3.2 Proca field case

In the Proca field case, we have three fields $q_{1}$, $q_{2}$ and $q_{3}$,
where the value of $j$ that appears in Eq. (B.1) for each field is
$j_{1}=\ell+1$, $j_{2}=\ell-1$ and $j_{3}=\ell$. The constant $H$ in this case
is $H=\frac{(d-2)(d-4)+4\mu^{2}l^{2}}{4l^{2}}$, which leads to the expression
of $\beta$ in Eq. (B.1) to become
$\beta=\frac{1}{4}(1+\sqrt{(d-3)^{2}+4\mu^{2}l^{2}})$. Moreover, we have
$c-a-b\neq-m^{\prime}$ with $m^{\prime}\in\mathbb{N}_{0}$. Therefore the
correct expansion at spatial infinity of the Proca field is described by Eq.
(B.3.1). Since $\beta>\frac{1}{2}$, the first term of Eq. (B.3.1) vanishes as
the gamma function in the numerator is finite. The remaining term must be zero
to satisfy the Dirichlet boundary condition, which only occurs if either $a$
or $b$ are non-positive integers. By requiring that $\omega>0$, the Dirichlet
boundary condition leads to $b=-n$, with $n\in\mathbb{N}_{0}$, and so the
normal mode eigenfrequencies $\omega_{k}l$ for each field $q_{k}$ are

$\displaystyle\omega_{k}L=2n+j_{k}+\frac{d-1}{2}+\frac{1}{2}\sqrt{\left(d-3\right)^{2}+4\mu^{2}L^{2}}\,\,,$
(108)

where the normal mode eigenfunctions are given by

$\displaystyle
q_{k}(r)=A_{k}\left(\frac{r}{l}\right)^{\frac{2j_{k}+d-2}{2}}\left(1+\frac{r^{2}}{l^{2}}\right)^{n-\frac{\omega_{k}l}{2}}$
$\displaystyle\times\prescript{}{2}{F}_{1}\left[-n+\omega_{k}l,-n,j_{k}+\frac{d-1}{2};\frac{r^{2}/l^{2}}{1+r^{2}/l^{2}}\right]\,\,.$
(109)

####  B.3.3 Maxwell electromagnetic field case

Scalar-type perturbation:

The scalar-type perturbations for the Maxwell field are described by the
function $q_{12}$, where the value of $j$ that appears in Eq. (B.1) is
$j=\ell$. The constant $H$ in this case is $H=\frac{(d-4)(d-6)}{4l^{2}}$,
which leads to the expression of $\beta$ in Eq. (B.1) to become
$\beta=\frac{1}{4}(1+\absolutevalue{d-5})$.

For even dimensions, the expansion at spatial infinity of $q_{12}$ is
described by Eq. (B.3.1), since $c-a-b\neq-m^{\prime}$, with
$m^{\prime}\in\mathbb{N}_{0}$. We distinguish two cases here, $d=4$ and $d\geq
6$. At spatial infinity for $d=4$, since $\beta=\frac{1}{2}$, the first term
in Eq. (B.3.1) vanishes while the second term is finite. We still impose the
Dirichlet boundary conditions and the first term vanishes if $b=-n$, with
$n\in\mathbb{N}_{0}$ and $\omega l>0$. Therefore, the normal mode frequencies
are

$\displaystyle\omega_{12}=2n+\ell+2\,,\quad\quad\mathrm{for}\,\,d=4\,.$ (110)

For $d\geq 6$ and even, at spatial infinity, since $\beta\geq\frac{1}{2}$, the
first term in Eq. (B.3.1) vanishes while the second term seems to diverge for
$d>6$ and seems to assume a finite value for $d=6$. Despite the $d=6$ case, we
impose the Dirichlet boundary conditions and the second term vanishes if
$b=-n$, with $n\in\mathbb{N}_{0}$ and $\omega l>0$. Therefore, the normal mode
frequencies are

$\displaystyle\omega_{12}=2n+\ell+d-3\,,\quad\quad\mathrm{for}\,\,d\geq 6\,.$
(111)

For odd dimensions, we split the analysis for $d=5$ and $d\geq 7$. For $d=5$,
the expansion at spatial infinity of $q_{12}$ is described by Eq. (B.3.1),
since $c-a-b=0$. Therefore, at spatial infinity, all the terms vanish in Eq.
(B.3.1). The Dirichlet boundary conditions, which impose the field to vanish
at $r\rightarrow+\infty$, makes the eigenvalue problem ill-defined. One can
instead impose a one parameter family boundary condition. For example, one can
choose a boundary condition such that the logarithmic term vanishes called
Dirichlet-Neumann, which is satisfied if $b=-n$, with $n\in\mathbb{N}_{0}$,
and so the frequency is given by

$\displaystyle\omega_{12}=2n+\ell+2\,,\quad\quad\mathrm{for}\,\,d=5\,.$ (112)

For $d\geq 7$, the expansion at spatial infinity of $q_{12}$ is described by
Eq. (B.3.1), with $\beta>\frac{1}{2}$ and $c-a-b=-m^{\prime}$ with
$m^{\prime}\in\mathbb{N}$. In this case, the second term vanishes while the
first term seems to diverge. We impose the Dirichlet boundary conditions and
so the first term vanishes if $b=-n$ with $n\in\mathbb{N}_{0}$. The normal
mode eigenfrequencies are then also described by Eq. (111).

The eigenfunctions for all the even and odd dimension cases are

$\displaystyle
q_{12}(r)=A_{12}\left(\frac{r}{l}\right)^{\frac{2\ell+d-2}{2}}\left(1+\frac{r^{2}}{l^{2}}\right)^{n-\frac{\omega_{12}l}{2}}$
$\displaystyle\times\prescript{}{2}{F}_{1}\left[-n+\omega_{12}l,-n,\ell+\frac{d-1}{2};\frac{\frac{r^{2}}{l^{2}}}{1+\frac{r^{2}}{l^{2}}}\right]\,.$
(113)

Vector-type perturbation:

The vector-type perturbations are described by the function $q_{3}$, where the
value of $j$ that appears in Eq. (B.1) is $j=\ell$. The constant $H$ in this
case is $H=\frac{(d-2)(d-4)}{4l^{2}}$, which leads to the expression of
$\beta$ in Eq. (B.1) to become $\beta=\frac{1}{4}(d-2)$.

For even dimensions, the expansion at spatial infinity of $q_{3}$ is described
by Eq. (B.3.1), since $\beta\geq\frac{1}{2}$ and $c-a-b\neq-m^{\prime}$, with
$m^{\prime}\in\mathbb{N}_{0}$. At spatial infinity, it turns out that the
first term in Eq. (B.3.1) vanishes while the second term seems to diverge for
$d>4$ and assumes a finite value for $d=4$. Despite the $d=4$ case, we impose
Dirichlet boundary conditions and the second term only vanishes if $b=-n$,
with $n\in\mathbb{N}_{0}$. Therefore, the normal mode eigenfrequencies are
given by

$\displaystyle\omega_{3}l=2n+\ell+d-2\,\,.$ (114)

For odd dimensions, the expansion at spatial infinity of $q_{3}$ is described
by Eq. (B.3.1), since $\beta>\frac{1}{2}$ and $c-a-b=-m^{\prime}$, with
$m^{\prime}\in\mathbb{N}$. At spatial infinity, the first term in Eq. (B.3.1)
seems to diverge. Imposing the Dirichlet boundary conditions, these terms
vanish if again $b=-n$ with $n\in\mathbb{N}_{0}$. The normal mode
eigenfrequencies for odd dimensions have then the same expression as Eq.
(114).

The eigenfunctions for all the even and odd dimension cases are

$\displaystyle
q_{3}=A_{3}\left(\frac{r}{l}\right)^{\frac{2\ell+d-2}{2}}\left(1+\frac{r^{2}}{l^{2}}\right)^{n-\frac{\omega_{3}l}{2}}$
$\displaystyle\times\prescript{}{2}{F}_{1}\left[-n+\omega_{3}l,-n,\ell+\frac{d-1}{2};\frac{\frac{r^{2}}{l^{2}}}{1+\frac{r^{2}}{l^{2}}}\right]\,\,.$
(115)

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