# Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

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  David Hilditch  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  Vítor
Cardoso  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

We present new results concerning the Proca massive vector field in a
Schwarzschild-anti-de Sitter (Schwarzschild-AdS) black hole geometry. We
provide a first principles analysis of Proca vector fields in this geometry
using both the vector spherical harmonic (VSH) decomposition and separation
method and the Frolov-Krtouš-Kubizňák-Santos (FKKS) method that separates the
relevant equations in spinning geometries. The analysis in the VSH method
shows, on one hand, that it is arduous to separate the scalar-type from the
vector-type polarizations of the electric sector of the Proca field, and on
the other hand, it displays clearly the electric and the magnetic mode
sectors. The analysis in the FKKS method is performed by taking the
nonrotating limit of the Kerr-AdS spacetime, and shows that the Ansatz
decouples the polarizations in the electric mode sector even in the
nonrotating limit. On the other hand, it captures only two of the three
possible polarizations; indeed, the magnetic mode sector, which is of vector-
type, is missing. The reason for the absence of the remaining polarization is
related to the degeneracy of the principal tensor in static spherical
symmetric spacetimes. The degrees of freedom and quasinormal modes in both
separation methods of the Proca field are found. The frequencies of the
quasinormal modes are also carefully computed. For the electric mode sector in
the VSH method the frequencies are found through an extension, which
substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny
numerical procedure, whereas for the magnetic mode sector in the VSH method
and the electric sector of the FKKS method it is shown that a direct use of
the procedure can be made. The values of the quasinormal mode frequencies
obtained for each method are compared and shown to be in good agreement with
each other. This further supports the analytical approaches presented here for
the behavior of the Proca field in a Schwarzschild-AdS black hole background.

##  I Introduction

Vector fields are known to describe the electroweak and the strong
interactions of the standard model of particle physics. In particular, the
mediator of electromagnetic interactions is the photon and it can be
identified as an excitation of an Abelian massless vector field. Since the
standard model does not explain the existence of dark matter, new kinds of
fields have been proposed as candidates. Such fields would have very feeble
interactions with ordinary matter, but would interact gravitationally. Thus,
bodies with strong gravitational effects serve as devices to probe the
existence of such new fields. General relativistic black holes and their
dynamical description are thus important to understand the behavior of
possible new matter, such as massive vector fields that obey the Proca
equations.

The analysis of the Maxwell massless vector field in static spacetimes has
been performed using the vector spherical harmonics method, or VSH method for
short. In Schwarzschild spacetimes, the Maxwell equations were separated,
decoupled, and reduced to a single master equation in Ruffini:1973 . In
Schwarzschild-anti-de Sitter (Schwarzschild-AdS) spacetimes, the Maxwell
equations and their quasinormal mode content were studied using reflective
boundary conditions at infinity with VSH techniques in Cardoso:2001 ; ckl ,
and using vanishing energy flux boundary conditions at infinity, the
quasinormal modes were found in WangHerdeiroSampaio ; ChenChoCornell . The
massive Proca field in Schwarzschild-type spacetimes with an interest on its
late time behavior was studied in KZM , and a quasinormal mode analysis for
Schwarzschild spacetimes was performed in Rosa:2012 . Quasinormal modes in
Schwarzschild-AdS spacetimes with a focus on their monopole term were analyzed
in Konoplya:2006 .

The separation of massless vector fields, such as the Maxwell field, in the
Kerr spacetime used a completely different approach, that of the Newman-
Penrose formalism, despite the reduced group of explicit symmetries
Teukolsky:1972 . This already hinted at the existence of another kind of
symmetry. Massive vector fields were considered not only under a small
rotation approximation with the equations yielding a ladder of coupled
multipoles Pani:2012vp , but also without approximations using a fully
numerical approach Witek:2012tr ; Cardoso:2018tly . In a different
development, and following previous work Frolov:2008jr ; Houri:2008ng ;
Frolov:2017whj ; Frolov:2017 ; Lunin:2017  that takes into account the
presence of the principal tensor in the Kerr-NUT-AdS and Kerr-NUT-dS
spacetimes, i.e., spacetimes describing a rotating black hole in four and
higher dimensions and that include the NUT parameters and a cosmological
constant, Frolov, Krtouš, Kubizňák, and Santos Frolov:2018 , or FKKS for
short, were able to extend the perturbation analysis to the case of a Proca
massive vector field in spinning geometries. It was further shown that with
this formalism the Proca equations can be separated. It is still unclear
whether the FKKS method covers all the degrees of freedom, including
polarizations and quasinormal modes, of the Proca field in the whole Kerr-NUT-
AdS and Kerr-NUT-dS family. However, with the help of an analysis for the
marginally bound state case Dolan:2018dqv ; Baumann:2019eav , it was found
that for the Kerr spacetime the FKKS Ansatz describes all possible modes. The
quasinormal modes for the FKKS Ansatz in Kerr were then recalculated and found
to yield excellent agreement with previous approaches Percival:2020skc . The
investigation of the degrees of freedom covered by the FKKS Ansatz in spinning
geometries can be performed either in a small spin approximation or
numerically, whereas analytically, this investigation is arduous due to the
complexity of the system comprised of partial differential equations and a
polynomial equation.

In this paper, we analyze Proca fields in the Schwarzschild-AdS spacetime
through both the VSH and the FKKS techniques, i.e., we investigate the degrees
of freedom and quasinormal modes through both methods. In the computation of
the quasinormal modes, we use reflective boundary conditions at infinity and
pure incoming wave boundary conditions at the event horizon. The analysis in
the VSH method has the advantage that both the electric and the magnetic
sectors appear in a natural way. The analysis in the FKKS method has the
advantage that the polarizations in the electric mode sector decouple
promptly. To do this we use a useful numerical procedure set up by Horowitz
and Hubeny Horowitz:2000  to study perturbations for scalar fields in
Schwarzschild-AdS spacetimes and to find their quasinormal mode frequencies,
see also cl . An extension of this method which substitutes the number
coefficients by matrix coefficients will be performed by us here, see also
Delsate:2011qp . For a review of all these treatments and an overview of the
FKKS Ansatz see the thesis thesis .

The paper is organized as follows. In Sec. II, we present the general
Einstein-Proca field equations and specialize them for a fixed background
geometry with a negative cosmological constant. In Sec. III, we separate the
Proca equations for a Schwarzschild-AdS spacetime using the VSH method and
compute its quasinormal modes. In Sec. IV, we present the FKKS method adapted
to the Kerr-AdS spacetime, take the nonrotating limit to the Schwarzschild-AdS
spacetime and compute its quasinormal modes. In Sec. V, we compare the
quasinormal modes given by the VSH and FKKS methods. In Sec. VI, we conclude.
Geometric units $G=c=1$ are used throughout the paper.

##  II Einstein-Proca field equations with a negative cosmological constant:
Fixed background

The action for the minimally coupled Einstein-Proca system is given by

$S=\int\differential[4]x\sqrt{-g}(\mathcal{L}_{EH}-\mathcal{L}_{A})\,,$ (1)

where $g$ is the determinant of the metric $g_{ab}$,

$\mathcal{L}_{EH}=\frac{R-2\Lambda}{16\pi}$ (2)

is the Einstein-Hilbert Lagrangian density with $R=R_{ab}g^{ab}$, and $R_{ab}$
the Ricci tensor, both built out of the Riemann tensor ${R^{a}}_{bcd}$ made of
the metric and its first and second derivatives. $\Lambda$ is the cosmological
constant, with which we can define a characteristic length
$l=\sqrt{\frac{3}{\absolutevalue{\Lambda}}}$, and

$\mathcal{L}_{A}=\frac{F_{ab}F^{ab}}{4}+\frac{\mu^{2}}{2}A_{a}A^{a}\,,$ (3)

is the Proca Lagrangian density with $F_{ab}$, given by
$F_{ab}=\nabla_{a}A_{b}-\nabla_{b}A_{a}$ being the Proca field strength,
$A_{a}$ being the Proca vector potential, and $\mu$ the field mass. It follows
from the Euler-Lagrange equations for the metric field $g_{ab}$ that the
Einstein equation must be obeyed, i.e.,

$\displaystyle G_{ab}+\Lambda g_{ab}=8\pi T_{ab}\,,$ (4)

where $G_{ab}$ is the Einstein tensor defined as
$G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$, and $T_{ab}$ is the energy-momentum tensor
given by

$\displaystyle
T_{ab}=F_{ac}F_{bd}g^{cd}-\frac{1}{4}g_{ab}F_{cd}F^{cd}+\mu^{2}\left(A_{a}A_{b}-\frac{1}{2}g_{ab}A_{c}A^{c}\right)\,.$
(5)

It follows from the Euler-Lagrange equations for the Proca field $A_{a}$ that
the Proca equation must be obeyed, i.e.,

$\displaystyle\nabla_{b}F^{ab}+\mu^{2}A^{a}=0\,.$ (6)

The tensor field $F_{ab}$ also obeys the internal equations
$\nabla_{[a}F_{bc]}=0$.

The Einstein-Proca system consists of nonlinear coupled partial differential
equations, Eqs. (4)-(6). We now want to study the perturbations of the $A_{a}$
field in a fixed background with negative cosmological constant, $\Lambda<0$.
In other words, we consider a perturbative expansion of the field equations
when the vector field and its derivatives are small. In this case, from Eq.
(4) one has the vacuum Einstein equation, namely,

$\displaystyle G_{ab}-\frac{3}{l^{2}}g_{ab}=0\,.$ (7)

Now, Eq. (7) obeys the field Bianchi identities,
$\nabla_{a}\left(G^{ab}-\frac{3}{l^{2}}g^{ab}\right)=0$. Although to first
order the stress-energy tensor $T_{ab}$ is zero, in second order it is not; it
must obey the conservation equation

$\displaystyle\nabla_{a}T^{ab}=0\,,$ (8)

where $T_{ab}$ is the Proca energy-momentum tensor given in Eq. (5). For the
Proca equation given in Eq. (6), one can make use of the commutation relations
for covariant derivatives that involve the Riemann tensor to yield
$g^{cd}\nabla_{c}\nabla_{d}A^{a}-\mu^{2}A^{a}-R^{a}_{\,\,d}A^{d}=0$, where use
of the Bianchi identity for the Proca field was also made. In a background
spacetime with a negative cosmological constant, we have from Eq. (7) that
$R_{ab}=-\frac{3}{l^{2}}g_{ab}$. So the Proca equation in a fixed negative
cosmological constant background becomes

$\displaystyle
g^{cd}\nabla_{c}\nabla_{d}A^{a}-\left(\mu^{2}-\frac{3}{l^{2}}\right)A^{a}=0\,.$
(9)

The Bianchi identity for the Proca $A^{a}$ field is clearly

$\displaystyle\nabla_{a}A^{a}=0\,,$ (10)

i.e., its divergence is zero. Note that for the massless case, $\mu=0$, the
Bianchi identity given in Eq. (10) is the Lorenz gauge condition that may or
may not be imposed, indeed in this case the field equations are invariant
under the gauge transformation $A_{a}\rightarrow A_{a}+\nabla_{a}\chi$, where
$\chi$ is a scalar obeying the Klein-Gordon equation, i.e.
$\nabla_{a}\nabla^{a}\chi=0$.

Thus, to study the perturbations of the Proca field $A_{a}$ with a fixed
spacetime background, and thus a fixed metric, one uses Eq. (9) with the help
of Eq. (10). Of interest is the case in which the cosmological constant is
negative and the background is a Schwarzschild-AdS black hole. We apply two
methods to solve the Proca set of equations given in Eqs. (9) and (10), the
VSH and the FKKS methods.

##  III Proca field perturbations in Schwarzschild-AdS: The VSH method

###  III.1 Schwarzschild-AdS metric

The spacetime considered here is the Schwarzschild-AdS spacetime, whose line
element can be written in spherical $(t,r,\theta,\phi)$ coordinates as

$ds^{2}=-f(r)dt^{2}+\dfrac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\,,$ (11)

where the function $f(r)$ is

$f(r)=1-\frac{2M}{r}+\frac{r^{2}}{l^{2}}\,,$ (12)

and $M$ is the spacetime mass. The Schwarzschild-AdS metric describes a vacuum
spherically symmetric spacetime with a negative cosmological constant. It is a
static black hole solution with an event horizon located at $r_{+}$ given by
the positive root of the equation $f(r)=0$, i.e.,

$r_{+}^{3}+r_{+}l^{2}-2Ml^{2}=0\,.$ (13)

###  III.2 Perturbations in the Proca field, separation of variables, and the
VSH method

####  III.2.1 General case

In the Schwarzschild-AdS spacetime, the Proca equations (9) can be separated
using vector spherical harmonics which can be obtained by the sum of a spin
$s=1$ with angular momentum $\ell$ as it is well known from any modern quantum
mechanics textbook. This vector spherical harmonics method, or VSH method, was
first used in Ruffini:1973  to separate the Maxwell equations in static
geometries. In the VSH method the following Ansatz is assumed for the field
$A_{a}$,

$\displaystyle
A_{a}=\frac{1}{r}\sum_{i=0}^{3}\sum_{lm}c_{i}u_{(i)}(t,r)Z_{a}^{(i)\ell
m}(\theta,\phi)\,,$ (14)

where $c_{0}=1$, $c_{1}=1$,
$c_{2}=\frac{1}{\left[\ell(\ell+1)\right]^{\frac{1}{2}}}$,
$c_{3}=\frac{1}{\left[\ell(\ell+1)\right]^{\frac{1}{2}}}$, the $u_{(i)}$, with
$i=0,1,2,3$, are functions of $t$ and $r$ that should be written as
$u_{(i)}(t,r)\equiv u_{(i)}^{\ell m}(t,r)$, but the indices $\ell$ and $m$
have been suppressed to not overcrowd the notation, and the $Z_{a}^{(i)\ell
m}$ are given by

$\displaystyle Z_{a}^{(0)\ell m}$ $\displaystyle=(1,0,0,0)Y^{\ell m}\,,$ (15)
$\displaystyle Z_{a}^{(1)\ell m}$ $\displaystyle=(0,\dfrac{1}{f},0,0)Y^{\ell
m}\,,$ (16) $\displaystyle Z_{a}^{(2)\ell m}$
$\displaystyle=\frac{r}{\sqrt{\ell(\ell+1)}}(0,0,\partial_{\theta},\partial_{\phi})Y^{\ell
m}\,,$ (17) $\displaystyle Z_{a}^{(3)\ell m}$
$\displaystyle=\frac{r}{\sqrt{\ell(\ell+1)}}\Bigg{(}0,0,\frac{\partial_{\phi}}{\sin\theta},-\sin\theta\partial_{\theta}\Bigg{)}Y^{\ell
m}\,,$ (18)

where $Y^{\ell m}$ are the spherical harmonics, with $\ell$ being the
principal number and $m$ the azimuthal number.

Indeed, by inserting the Ansatz given in Eq. (14) into the Proca equations
(9), one finds after rearrangement the following system of equations for the
functions $u_{(i)}$,

$\displaystyle{\hat{\mathcal{D}}}^{2}u_{(0)}+(\partial_{r}f)(\dot{u}_{(1)}-u^{\prime}_{(0)})=0\,,$
(19)
$\displaystyle{\hat{\mathcal{D}}}^{2}u_{(1)}+\frac{2f}{r^{2}}\Big{(}1-\frac{3M}{r}\Big{)}(u_{(2)}-u_{(1)})=0\,,$
(20)
$\displaystyle{\hat{\mathcal{D}}}^{2}u_{(2)}+\Bigg{[}\frac{2f\ell(\ell+1)}{r^{2}}u_{(1)}\Bigg{]}=0\,,$
(21) $\displaystyle{\hat{\mathcal{D}}}^{2}u_{(3)}=0\,,$ (22)

where $\dot{u}_{(i)}=\dfrac{\partial{u_{(i)}}}{\partial t}$,
$u^{\prime}_{(i)}=\dfrac{\partial{u_{(i)}}}{\partial r_{*}}$ with $r_{*}$
being defined by $\dfrac{dr_{*}}{dr}=\dfrac{1}{f}$, and where
${\hat{\mathcal{D}}}^{2}$ is shorthand for

${\hat{\mathcal{D}}}^{2}=-\partial_{t}^{2}+\partial_{r_{*}}^{2}-f\Big{[}\frac{\ell(\ell+1)}{r^{2}}+\mu^{2}\Big{]}\,.$
(23)

It must be noted that $u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ describe the
electric modes since, under parity transformations, $Z^{(0)}_{a}$,
$Z^{(1)}_{a}$, and $Z^{(2)}_{a}$ gain a factor of $(-1)^{\ell}$, whereas
$u_{(3)}$ describes the magnetic modes since, under parity transformations,
$Z^{(3)}_{a}$ gains a factor of $(-1)^{\ell+1}$. All this follows the notation
given in Rosa:2012 , where the Proca equations for a pure Schwarzschild
background have been presented.

Furthermore, inserting the Ansatz in Eq. (14) into the Bianchi identity given
in Eq. (10), $\nabla^{a}A_{a}=0$, one obtains

$\frac{1}{rf}\Bigg{[}u^{\prime}_{(1)}-\dot{u}_{(0)}+\frac{f}{r}(u_{(1)}-u_{(2)})\Bigg{]}=0\,.$
(24)

Equation (24) was used to find Eq. (20). Indeed, the Ansatz given in Eq. (14)
when put into Eq. (9) leads directly to
$\hat{\mathcal{D}}u_{(1)}+(\partial_{r}f)(\dot{u}_{(0)}-u^{\prime}_{(1)})+\frac{2f^{2}}{r^{2}}(u_{(2)}-u_{(1)})=0$,
which upon using Eq. (24) yields Eq. (20).

Therefore, the system consisting of Eqs. (19)-(22), taking into account the
definition given in Eq. (23), determines the solution. The Bianchi identity,
Eq. (24), also helps in the determination of the solution. For example, the
static part of $u_{(0)}$ must be obtained from Eq. (19), but the dynamical
part of $u_{(0)}$ can be described by the Bianchi identity Eq. (24). Equations
(20) and (21), for $u_{(1)}$ and $u_{(2)}$, are coupled together, whereas Eq.
(22), for $u_{(3)}$, is decoupled.

We shall assume that the time dependence of the functions $u_{(0)}$,
$u_{(1)}$, $u_{(2)}$, and $u_{(3)}$ goes as $e^{-i\omega t}$. In this case,
the system given by Eqs. (19)-(22) can be treated as an eigenvalue problem to
$\omega$. We classify the eigenvectors of the system according to the three
degrees of freedom, i.e., the three polarizations, of the Proca vector
$A_{a}$. These three polarizations consist of one scalar-type polarization and
two vector-type polarizations. The electric modes of $A_{a}$, characterized by
$u_{(0)}$, $u_{(1)}$, and $u_{(2)}$, possess one scalar-type polarization and
one vector-type polarization. The scalar-type polarization of $A_{a}$ has a
behavior similar to a scalar field, and can be picked up to the higher $\ell$
modes from the $\ell=0$ mode, which has solely scalar-type polarization.
Moreover, setting the Proca mass $\mu$ to zero, $\mu=0$, the scalar-type
polarization becomes nonphysical, more precisely, at the massless limit it can
be removed by the gauge freedom. The vector-type polarization of $A_{a}$ of
the electric modes are then picked up by exclusion, i.e., they are the ones
that are not scalar-type. Since the system in the electric mode sector is
coupled, it is not trivial to obtain the scalar-type and the vector-type
polarization of $A_{a}$ in terms of the functions $u_{(1)}$ and $u_{(2)}$. The
magnetic modes of $A_{a}$, characterized by $u_{(3)}$, possess the remaining
vector-type polarization.

As an addendum, we note that it is possible to decouple the pair of equations
for $u_{(1)}$ and $u_{(2)}$ which are also present in the pure Schwarzschild
case, see Rosa:2012 . Equation (21) can be used to find $u_{(1)}$ as a
function of $u_{(2)}$, i.e.,
$u_{(1)}=-\frac{r^{2}\hat{\mathcal{D}}^{2}u_{(2)}}{2f\ell(\ell+1)}$.
Substituting this expression in Eq. (20), a decoupled equation for $u_{(2)}$
can be found, $\mathcal{H}\left(ru_{(2)}\right)=0$, where
$\mathcal{H}=\hat{\mathcal{D}}^{2}\Big{[}\frac{1}{f}\hat{\mathcal{D}}_{1}^{\,2}\Big{]}-2f(\partial_{r}f)\mu^{2}$
and
${\hat{\mathcal{D}}_{1}}^{\,2}=-\partial_{t}^{2}+\partial_{r_{*}}^{2}-f\Big{[}\frac{\ell(\ell+1)}{r^{2}}+\mu^{2}+\frac{\partial_{r}f}{r}\Big{]}$.
Thus, one can decouple the pair of equations in this way by paying the price
of increasing the order of the partial differential equations. This happens
since $u_{(2)}$ must contain the scalar-type and the vector-type
polarizations. In the massless limit, $\mu=0$, the operator $\mathcal{H}$
factorizes, becoming the product between the operators $\hat{\mathcal{D}}^{2}$
and $f^{-1}\hat{\mathcal{D}}_{1}^{\,2}$. One must notice that in this limit,
the scalar-type polarization can be removed by the gauge freedom, meaning
$u_{(2)}$ will contain both a spurious degree of freedom and the physical
vector-type polarization associated with the electric modes. It can be shown
that the scalar $\Psi=f^{-1}\mathcal{D}_{1}^{\,2}(ru_{(2)})$ is related to the
field strength tensor $F_{ab}$, thus in the massless case it describes
appropriately the vector-type polarization related to the electric modes
without the spurious degree of freedom. Moreover, the equation
$\mathcal{H}(ru_{(2)})=0$ indicates that $\Psi$ will satisfy the same equation
as $u_{(3)}$, which means the two vector-type polarizations of the massless
vector field become degenerate. For the massive case, the factorization of
$\mathcal{H}$ does not appear to be possible, which makes the analytical
decoupling difficult.

####  III.2.2 Monopole case

The monopole case $\ell=0$ for the massive vector field simplifies the system
considerably. Only the functions $u_{(0)}$ and $u_{(1)}$ survive, the
functions $u_{(2)}$ and $u_{(3)}$ vanish since $Y^{00}$ is a constant. The
equation for the function $u_{(0)}$ can be obtained from Eq. (19) together
with Eq. (23), to give

$\displaystyle-\ddot{u}_{(0)}+u_{(0)}^{\prime\prime}+(\partial_{r}f)(\dot{u}_{(1)}-u_{(0)}^{\prime})-f\mu^{2}u_{(0)}=0\,.$
(25)

The equation for $u_{(1)}$ can be obtained from Eq. (20) together with Eq.
(23), i.e.,

$\displaystyle
u^{\prime\prime}_{(1)}-\ddot{u}_{(1)}-f\Big{[}\mu^{2}+\frac{2}{r^{2}}\Big{(}1-\frac{3M}{r}\Big{)}\Big{]}u_{(1)}=0\,.$
(26)

The Bianchi identity given in Eq. (24) is now

$\displaystyle{\dot{u}}_{(0)}=u^{\prime}_{(1)}+\frac{f}{r}u_{(1)}\,.$ (27)

Note that for the $\ell=0$ case the function $u_{(0)}$ could be written as
$u_{(0)}=u_{(0)\text{s}}(r)+u_{(0)\text{t}}(t,r)$, where $u_{(0)\text{s}}(r)$
is the static part of $u_{(0)}$ and can be obtained directly from Eq. (25),
and $u_{(0)\text{t}}(t,r)$ is the dynamic part of $u_{(0)}$ and can be
obtained directly from the Bianchi identity given in Eq. (27). The function
$u_{(1)}$ is taken from Eq. (26). We are not considering the static part which
would give a static spherically symmetric second-order perturbed
Schwarzschild-AdS geometry with a new Proca gravitational term with the
corresponding first-order Proca field term, rather than the Schwarzschild-AdS
background geometry we are working with. By taking the zero-mass field limit,
the static part would give a static spherically symmetric second-order
perturbed Schwarzschild-AdS geometry with a new Maxwell gravitational term,
i.e., a second-order Reissner-Nordström-AdS geometry, with the corresponding
first-order Maxwell field.

###  III.3 Quasinormal modes of Proca in Schwarzschild-AdS in the VSH method

Having found the equations obeyed by the Proca field in a Schwarzschild-AdS
background, namely Eqs. (19)-(22) for $u_{(i)}$ with $i=0,1,2,3$, we can now
study the quasinormal modes of a Schwarzschild-AdS black hole for a Proca
field. The quasinormal modes are defined as solutions that solve the equations
of motion given in Eqs. (19)-(22) with boundary conditions such that at the
event horizon there are only purely incoming waves and at infinity the Proca
field is zero, i.e., $u_{(i)}\rightarrow 0$ at infinity, with $i=0,1,2,3$. The
analysis of the system is concluded by integrating the equations. To find the
quasinormal modes and the quasinormal frequencies in this spacetime, we
implement the numerical procedure used in Horowitz:2000 , see also
Cardoso:2001 . One can write $u_{(i)}$, with $i=0,1,2,3$, as

$\displaystyle u_{(i)}=U_{(i)}e^{-i(t+r_{*})\omega}\,,$ (28)

where $\omega$ is a frequency and the $U_{(i)}$ are functions of $r$. This
transformation is useful since it expresses explicitly the behavior of
$u_{(i)}$ as an incoming wave at the event horizon. Assuming analyticity, it
is possible to write every $U_{(i)}$ as an expansion series around the horizon
$r_{+}$,

$\displaystyle U_{(i)}=\sum_{n=0}^{\infty}a_{(i)n}(x-x_{+})^{n}\,,$ (29)

where the $a_{(i)n}$ are expansion coefficients, $x=\frac{1}{r}$, and
$x_{+}=\frac{1}{r_{+}}$.

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (VSH) |  $\omega l$ (VSH)
$0.01$ | $3.331-2.489\,\,i$ | $184.968-266.394\,\,i$
$0.10$ | $3.339-2.500\,\,i$ | $185.604-267.461\,\,i$
$0.20$ | $3.362-2.531\,\,i$ | $187.452-270.612\,\,i$
$0.40$ | $3.446-2.645\,\,i$ | $193.925-282.119\,\,i$
$0.50$ | $3.501-2.722\,\,i$ | $198.077-289.799\,\,i$
(a) Scalar-type polarization.

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (VSH) |  $\omega l$ (VSH)
$0.01$ | $1.554-0.542\,\,i$ | $0-149.984\,\,i$
$0.10$ | $1.557-0.552\,\,i$ | $0-152.099\,\,i$
$0.20$ | $1.568-0.583\,\,i$ | $0-158.432\,\,i$
$0.30$ | $1.585-0.633\,\,i$ | $0-168.817\,\,i$
$0.40$ | $1.607-0.699\,\,i$ | $0-183.291\,\,i$
$0.50$ | $1.634-0.777\,\,i$ | $0-202.684\,\,i$
(b) Vector-type polarization.

Table 1:  Quasinormal mode frequencies $\omega l$ of the Proca field electric modes $u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ with $\ell=1$, using the VSH method in Schwarzschild-AdS for $r_{+}=l$ and $r_{+}=100l$ and for several values of the Proca field mass $\mu l$. (a) The frequencies of the scalar-type polarization of the electric modes are displayed. (b) The frequencies of the vector-type polarization of the electric modes are displayed.  | $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (VSH) |  $\omega l$ (VSH)
$0.01$ | $2.163-1.699\,\,i$ | $0-150.069\,\,i$
$0.10$ | $2.171-1.710\,\,i$ | $0-152.187\,\,i$
$0.20$ | $2.193-1.743\,\,i$ | $0-158.526\,\,i$
$0.30$ | $2.228-1.795\,\,i$ | $0-168.922\,\,i$
$0.40$ | $2.273-1.863\,\,i$ | $0-183.419\,\,i$
$0.50$ | $2.327-1.944\,\,i$ | $0-202.860\,\,i$
Table 2:  Quasinormal mode frequencies $\omega l$ of the Proca field magnetic
modes $u_{(3)}$ with $\ell=1$, using the VSH method in Schwarzschild-AdS for
$r_{+}=l$ and $r_{+}=100l$ and for several values of the Proca field mass $\mu
l$. Magnetic modes only have vector-type polarization.

The functions $u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ give the electric modes. For
$u_{(0)}$, the equation to be solved is given in Eq. (19), and one sees it is
coupled to the equation for $u_{(1)}$, Eq. (20). The variables $u_{(1)}$ and
$u_{(2)}$ are also coupled, see Eqs. (20) and (21), and this means that
careful treatment is required to solve them. The strategy that we follow here
is to determine $u_{(1)}$ and $u_{(2)}$ from Eqs. (20) and (21) and then use
the Bianchi identity to determine Eq. (24) to determine $u_{(0)}$. Since the
Horowitz-Hubeny numerical procedure Horowitz:2000  was designed for decoupled
equations, an extension is needed for this case. Hence we substitute the
number coefficients by matrix coefficients, see also Delsate:2011qp ; thesis .
Let us first define the following polynomials by

$\displaystyle s(x)=\frac{x^{4}f(x)}{x-x_{+}}\,\,,$ (30) $\displaystyle
t(x)=x^{2}\partial_{x}\Big{(}x^{2}f(x)\Big{)}+2i\omega x^{2}\,,$ (31)
$\displaystyle u(x)=-(x-x_{+})\Big{[}x^{2}\ell(\ell+1)+\mu^{2}\Big{]}\,,$ (32)

where $f(x)$ is $f(r)$ of Eq.(12) transformed to the variable $x$, i.e.,

$\displaystyle
f(x)=\frac{1}{x^{2}}\left(\frac{1}{l^{2}}+x^{2}-2Mx^{3}\right)\,,$ (33)

and the matrix $\bm{K}$ by

$\displaystyle\bm{K}=(x-x_{+})\begin{bmatrix}-2x^{2}(1-3Mx)&2x^{2}(1-3Mx)\\\
2x^{2}\ell(\ell+1)&0\\\ \end{bmatrix}\,.$ (34)

We then substitute Eq. (28) into Eqs. (20) and (21), finding a matrix equation
for the $U_{(i)}$ given by

$\displaystyle(x-x_{+})s(x)\partial_{x}^{2}\bm{U}+t(x)\partial_{x}\bm{U}+\frac{u(x)}{x-x_{+}}\bm{U}$
$\displaystyle+\frac{1}{x-x_{+}}\bm{K}\cdot\bm{U}=0\,,$ (35)

where the matrix $\bm{U}$ is defined in a natural way by

$\displaystyle\bm{U}=\begin{bmatrix}U_{(1)}\\\ U_{(2)}\end{bmatrix}\,.$ (36)

Notice that in the polynomials above, there is only a linear dependence in
$\omega$. This happens because the second time derivative and the second
$r_{*}$ derivative have different signs in Eq. (23), and when applying each of
the derivatives to the exponential in Eq. (28) the term in $\omega^{2}$
cancels. Now, to solve the problem we have to expand the matrix $\bm{U}$ as it
was done in Eq. (29), i.e.,

$\displaystyle\bm{U}(x)=\sum_{n=0}^{\infty}{\bm{a}_{n}}(x-x_{+})^{n}\,,$ (37)

where $\bm{a}_{n}$ for the $a_{(i)n}$, $i=1,2$, is given by

$\displaystyle\bm{a}_{n}=\begin{bmatrix}a_{(1)n}\\\ a_{(2)n}\,\end{bmatrix}.$
(38)

It is helpful to write

$\displaystyle\bm{a}_{n}=\bm{M}_{n}\bm{a}_{0}\,\,$ (39)

for some $2\times 2$ matrix $\bm{M}_{n}$ that has to be found, with the
obvious definition that $\bm{M}_{0}$ is the identity matrix $\bm{I}$, i.e.,
$\bm{M}_{0}=\bm{I}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix}$. Note now that the
polynomials defined in Eqs. (30)-(32) can be expanded around $x_{+}$ so that
it is possible to write

$s(x)=\sum_{j=0}^{\infty}s_{j}(x-x_{+})^{j}\,,$ (40)
$t(x)=\sum_{j=0}^{\infty}t_{j}(x-x_{+})^{j}\,,$ (41)
$u(x)=\sum_{j=0}^{\infty}u_{j}(x-x_{+})^{j}\,,$ (42)

where $s_{j}$, $t_{j}$, and $u_{j}$ are expansion coefficients that vanish for
$j$ higher than some value, since $s(x)$, $t(x)$, and $u(x)$ are finite
polynomials. We also can expand $\bm{K}$ in Eq. (34) as

$\bm{K}(x)=\sum_{j=0}^{\infty}{\bm{K}}_{j}(x-x_{+})^{j}\,,$ (43)

where ${\bm{K}}_{j}$ are the expansion coefficients which also vanish for $j$
higher than some value, since the components of the matrix $\bm{K}$ are finite
polynomials. Equation (35), which is equivalent to the Proca equations for
$u_{(1)}$ and $u_{(2)}$, is then reduced to the recurrence relation

$\displaystyle\bm{M}_{n}=-\frac{1}{P_{n}}\sum_{j=0}^{n-1}\bm{V}_{nj}\cdot\bm{M}_{j}\,,$
(44)

where

$\displaystyle\bm{V}_{nj}=\big{[}j(j-1)s_{n-j}+jt_{n-j}+u_{n-j}\big{]}\bm{I}+\bm{K}_{n-j}\,$
(45)

and

$P_{n}=n(n-1)s_{0}+nt_{0}\,.$ (46)

The quasinormal mode frequencies for the functions $U_{(1)}$ and $U_{(2)}$
that appear in Eq. (29) and are put in matrix form in Eq. (36) can be obtained
by imposing that the series appearing in Eq. (37) vanishes at $x\rightarrow
0$, i.e., $r\rightarrow+\infty$. More specifically, from Eqs. (36)-(39), the
series given in Eq. (37) vanishes at $x\rightarrow 0$ if either
$\bm{a}_{0}=0$, which means the series vanishes everywhere trivially, or
$\sum_{j=0}^{\infty}\bm{M}_{j}(-x_{+})^{j}$ is singular, which means the
determinant of the matrix resulting from the sum is zero. Thus, discarding the
trivial solution, the boundary condition is satisfied when

$\displaystyle\det\Bigg(\sum_{j=0}^{N}\bm{M}_{j}(-x_{+})^{j}\Bigg{missing})=0\,,$
(47)

where $N$ is in principle infinite. Then, for $u_{(0)}$ the quasinormal mode
frequencies are directly determined through the Bianchi identity, Eq. (24). Of
course, the quasinormal frequencies for $u_{(0)}$, $u_{(1)}$, and $u_{(2)}$
are the same, they are the electric modes. We do not present the modes for
$\ell=0$, but they were calculated and agree with Konoplya:2006 . The modes
for $\ell=1$ are shown in the Table 1, in particular the scalar-type
polarization is shown in Table 1(a) and the vector-type polarization is shown
in Table 1(b), where $N=40$ was taken, and we set $r_{+}$ and $\omega$ in
units of $l$ and $l^{-1}$, respectively. In the tables we wrote the number 0
which in the numerical procedure we use means a real number that is very close
to zero. This hints that the quasinormal modes are purely imaginary.

The function $u_{(3)}$ gives the magnetic modes. For $u_{(3)}$, Eq. (22) can
be written with the help of Eq. (28) and with $x=\frac{1}{r}$ and
$x_{+}=\frac{1}{r_{+}}$ as

$\displaystyle\Bigg{[}(x-x_{+})s(x)\partial_{x}^{2}+t(x)\partial_{x}+\frac{u(x)}{x-x_{+}}\Bigg{]}U_{(3)}=0\,,$
(48)

where the polynomials $s(x)$, $t(x)$, and $u(x)$ are in Eqs. (30)-(32). Now to
solve the problem we have to expand $U_{(3)}$ as in Eq. (29), i.e.,

$\displaystyle U_{(3)}=\sum_{n=0}^{\infty}a_{(3)n}(x-x_{+})^{n}\,,$ (49)

where the $a_{(3)n}$ are expansion coefficients. Using the expansions for
$s(x)$, $t(x)$, and $u(x)$ given in Eqs. (40)-(42), one finds that Eq. (48)
can then be reduced to the recurrence relation,

$\displaystyle
a_{(3)n}=-\frac{1}{P_{n}}\sum_{j=0}^{n-1}\Big{(}j(j-1)s_{n-j}+jt_{n-j}+u_{n-j}\Big{)}a_{(3)j}\,.$
(50)

The quasinormal mode frequencies $\omega$ can be found by imposing that the
series vanishes at $x\rightarrow 0$, i.e., $r\rightarrow+\infty$,

$\displaystyle\sum_{j=0}^{N}a_{(3)j}(-x_{+})^{j}=0\,,$ (51)

with $N$ being again, in principle, infinite. The quasinormal frequencies for
$u_{(3)}$, i.e., the magnetic mode frequencies, are calculated numerically for
$\ell=1$ where $N=40$ was taken, and are displayed in the Table 2. We set
$r_{+}$ and $\omega$ in units of $l$ and $l^{-1}$, respectively.

A comment on the distinction of the polarizations of the electric modes
$u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ is in order. Since Eqs. (20) and (21)
cannot be decoupled trivially, the distinction of the modes for each
polarization is made by inference. From Table 1, the electric mode
polarizations, namely the scalar-type and vector-type, are difficult to
distinguish at small $\frac{r_{+}}{l}$, e.g., $r_{+}=l$. A possible method to
distinguish them is to compare the frequencies at $\ell=0$, where only the
modes of the scalar-type polarization are present, with the frequencies at
$\ell=1$, as it is expected they have a higher modulus for higher $\ell$. For
large $r_{+}$, e.g., $r_{+}=100l$, the distinction is easier since we can
compare the frequencies in Table 1 with the frequencies in Table 2, because
the modes in the vector-type polarizations have the same behavior, namely,
negligible real frequency. The reason for this is that the mass of the field
is very small compared with the mass of the black hole, thus the effect of the
field mass on the vector-type modes is almost negligible. They also approach
the values computed for the massless case, done in Cardoso:2001 , which were
confirmed by our numerics.

##  IV Proca field perturbations in Schwarzschild-AdS: The FKKS method

###  IV.1 Kerr-AdS metric and the principal tensor in Kerr-AdS

We now employ another, very interesting, method to find the quasinormal modes
of the Proca field in a Schwarzschild-AdS background. The method relies on
symmetries for a rotating body in general relativity, specifically on the
symmetries of the Kerr-NUT-AdS and of the Kerr-NUT-dS. To use this formalism
in the Schwarzschild-AdS background we display it for the Kerr-AdS putting
zero NUT charge from the start and then take the limit $a=0$ of the Kerr-AdS.

Symmetries are important in the analysis of physical systems as they can allow
for separability of field equations or the integrability of equations of
motion for test particles. There are explicit symmetries and hidden
symmetries. The quantity of interest related to hidden symmetries is the
principal tensor $h_{ab}$, a nondegenerate closed conformal Killing-Yano
2-form, i.e., an antisymmetric tensor. The principal tensor $h_{ab}$ is
nondegenerate when its matrix representation in any coordinate system has
maximal rank. Denoting $i$ as the imaginary unit, and denominating for
convenience the eigenvalues of $h^{a}_{\,\,b}$ in four-dimensional spacetime
as $\pm ix_{A}$, where $A=1,2$, nondegeneracy implies that the $x_{A}$ are
functionally independent and nonvanishing, i.e., the Jacobian matrix of
$X=(x_{1},x_{2})$ is nonsingular.

The principal tensor is able to generate the Killing tower Frolov:2008jr , a
set of symmetries that allows the integration of the Hamilton-Jacobi and the
Klein-Gordon equations in spinning geometries. It was shown in Houri:2008ng
that Kerr-NUT with a cosmological constant, a family of solutions of the
Einstein vacuum equations valid in four and higher dimensions, is the unique
family of spacetimes with a principal tensor, under the condition that the
gradient of the eigenvalues of the principal tensor are spacelike vectors, or
timelike via Wick rotation. Principal tensors are very interesting quantities.
As reviewed in Frolov:2017 , they have applications in the Kerr-NUT-AdS and
Kerr-NUT-dS families, as well as in another set of spacetimes with Lorentzian
signature with a principal tensor built from eigenvalues with null gradient,
an issue that it is far from being fully understood, and is currently an open
problem Frolov:2017whj .

Here we are interested in a particular case of the four-dimensional Kerr-NUT-
AdS spacetime which is the four-dimensional Kerr-AdS spacetime. In Boyer-
Lindquist coordinates $(t,r,\theta,\phi)$, the Kerr-AdS spacetime has line
element given by

$\displaystyle ds^{2}=$
$\displaystyle-\frac{\Delta_{\Lambda}}{\Sigma}\Big{[}dt-a\sin^{2}\theta
d\phi\Big{]}^{2}$
$\displaystyle+\frac{\Delta_{\theta}\sin^{2}\theta}{\Sigma}\Big{[}adt-(a^{2}+r^{2})d\phi\Big{]}^{2}$
$\displaystyle+\frac{\Sigma}{\Delta_{\Lambda}}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}\,,$
(52)

where $\Delta_{\Lambda}=\,\,r^{2}-2Mr+a^{2}+\frac{r^{2}}{l^{2}}(r^{2}+a^{2})$,
$\Sigma=\,\,r^{2}+a^{2}\cos^{2}\theta$,
$\Delta_{\theta}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta$, and $a$ is related to
the angular momentum of the black hole, $J$, by $J=aM$, with $M$ being the
spacetime mass. The principal tensor $h_{bc}$ obeys the following equation in
this spacetime $\nabla_{a}h_{bc}=2g_{a[b}\xi_{c]}$, where $\xi_{b}$ is defined
by $\xi_{b}=\frac{1}{3}\nabla^{c}h_{cb}$, and square brackets mean
antisymmetrization on the indices. In addition, $h_{bc}$ obeys the following
integrability conditions,
$\nabla^{a}\nabla^{b}h_{cd}=-\Big{(}R^{a}_{\,\,e}\delta^{b}_{[c}h^{e}_{\,\,d]}+\frac{1}{2}R_{fe\,\,[c}^{\,\,\,\,a}\delta^{b}_{d]}h^{fe}\Big{)}$,
$R^{[a}_{\,\,e}\delta^{b]}_{[c}h^{e}_{\,\,d]}-R^{ab}_{\,\,\,\,e[c}h^{e}_{\,\,d]}+R_{fe\,\,\,[c}^{\,\,\,\,[a}\delta^{b]}_{d]}h^{fe}=0$,
with $R_{abcd}$ being the spacetime Riemann tensor. From these conditions it
follows both that the principal tensor commutes with the Ricci tensor and that
$\xi_{c}$ is a Killing vector field. The principal tensor having these
properties is given by

$\displaystyle\bm{h}=-(rdr+a^{2}\cos\theta\sin\theta d\theta)\wedge dt$
$\displaystyle+a\sin\theta\left(r\sin\theta dr+(r^{2}+a^{2})\cos\theta
d\theta\right)\wedge d\phi\,.$ (53)

The components $h_{ab}$ of the principal tensor can be extracted directly from
Eq. (53). From $\xi_{b}=\frac{1}{3}\nabla^{c}h_{cb}$ one then finds
$\xi^{a}\partial_{a}=\partial_{t}$. The eigenvalues of $h^{a}_{\,\,b}$ given
by $\pm ix_{A}$ as referred to above have the following functions $x_{1}=ir$
and $x_{2}=a\cos\theta$.

###  IV.2 Proca field and FKKS Ansatz

####  IV.2.1 Kerr-AdS

The Kerr-AdS spacetime is axisymmetric which by itself is not enough to
separate the equations for the Proca field. For instance, for the massless
vector field in the Kerr spacetime one needs to use a null frame to have a
neat separation of the equations Teukolsky:1972 . Extending this result to the
massive case has been a hard task. Nevertheless, an Ansatz, called the FKKS
Ansatz Frolov:2018 , has been discovered and it is able to separate the Proca
equations in Kerr-AdS. This approach uses the principal tensor, i.e., a
nondegenerate closed conformal Killing-Yano antisymmetric two tensor. For
Kerr-AdS the Ansatz is given by

$\displaystyle A^{a}=B^{ab}\nabla_{b}Z\,,$ (54)

with $B^{ab}$ being given implicitly in terms of the metric $g_{bc}$ and the
principal tensor $h_{bc}$ by

$\displaystyle B^{ab}(g_{bc}-\beta h_{bc})=\delta^{a}_{c}\,,$ (55)

where $\beta$ is a complex constant with discrete values to be found, and $Z$
is a function given by

$\displaystyle Z=R(r)S(\theta)\exp\left(-i\omega t+im\phi\right)\,.$ (56)

The polarization tensor defined in Eq. (55) in Kerr-AdS, where the metric
$g_{ab}$ is taken from the line element given in Eq. (52) and the principal
tensor $h_{ab}$ is taken from Eq. (53), can be written as a symmetric part
$\bm{B}_{\text{S}}$ and an antisymmetric part $\bm{B}_{\text{A}}$, namely,

$\displaystyle\bm{B}_{\text{S}}=\frac{\Delta_{\Lambda}}{q_{r}\Sigma}\partial_{r}^{2}+\frac{q_{\Lambda}}{q_{\theta}\Sigma}\partial_{\theta}^{2}-\frac{1}{q_{r}\Delta_{\Lambda}\Sigma}\Big{[}(r^{2}+a^{2})\partial_{t}+a\partial_{\phi}\Big{]}^{2}$
$\displaystyle+\frac{1}{\Sigma
q_{\theta}q_{\Lambda}\sin^{2}\theta}\Big{[}a\sin^{2}\theta\partial_{t}+\partial_{\phi}\Big{]}^{2}\,,$
(57) $\displaystyle\bm{B}_{\text{A}}=\frac{\beta
r}{q_{r}\Sigma}\Big{[}(r^{2}+a^{2})(\partial_{r}\partial_{t}-\partial_{t}\partial_{r})+a(\partial_{r}\partial_{\phi}-\partial_{\phi}\partial_{r})\Big{]}$
$\displaystyle-\frac{\beta a\sin 2\theta}{2\Sigma
q_{\theta}}\Big{[}a(\partial_{t}\partial_{\theta}-\partial_{\theta}\partial_{t})+\frac{1}{\sin^{2}\theta}(\partial_{\phi}\partial_{\theta}-\partial_{\theta}\partial_{\phi})\Big{]}\,,$
(58)

with $q_{\Lambda}=1-\frac{a^{2}}{l^{2}}\cos^{2}\theta$,
$q_{r}=1-\beta^{2}r^{2}$, and $q_{\theta}=1+\beta^{2}a^{2}\cos^{2}\theta$. The
components ${\bm{B}_{\text{S}}}_{ab}$ and ${\bm{B}_{\text{A}}}_{ab}$ can be
extracted directly from Eqs. (57) and (58), respectively.

One can now put the $B_{ab}$ found in Eqs. (57) and (58) and the Ansatz Eq.
(56) for $Z$ into the Proca field equation given in Eq. (54), and then into
the Proca equations given in Eq. (9), to get the following equations that
$R(r)$ and $S(\theta)$ of $Z$ must obey,

$\displaystyle\partial_{r}\Bigg{[}\frac{\Delta_{\Lambda}}{q_{r}}\partial_{r}R(r)\Bigg{]}+\Bigg{[}\frac{K_{r}^{2}}{q_{r}\Delta_{\Lambda}}+i\frac{2-q_{r}}{q_{r}^{2}\beta}\sigma+\frac{\mu^{2}}{\beta^{2}}\Bigg{]}R(r)=0\,,$
(59)
$\displaystyle\frac{1}{\sin\theta}\partial_{\theta}\Bigg{[}\frac{q_{\Lambda}\sin\theta}{q_{\theta}}\partial_{\theta}S(\theta)\Bigg{]}-$
$\displaystyle\Bigg{[}\frac{K_{\theta}^{2}}{q_{\theta}q_{\Lambda}\sin^{2}\theta}+i\frac{2-q_{\theta}}{q_{\theta}^{2}\beta}\sigma+\frac{\mu^{2}}{\beta^{2}}\Bigg{]}S(\theta)=0\,,$
(60)

with $K_{r}=am-(a^{2}+r^{2})\omega$, $K_{\theta}=m-a\omega\sin^{2}\theta$, and
$\sigma=a\beta^{2}(m-\omega a)-\omega$. Then, with the solution found, one
obtains the Proca field as

$\displaystyle A^{a}(\beta)=B^{ab}\partial_{b}Z(\beta)\,$ (61)

for each value $\beta$. Note that one may interpret the different values of
$\beta$ found from the equations as corresponding to different polarizations.
This can be seen by putting Eq. (61) in the form
$A^{a}=\sum_{\beta}c_{\beta}B^{ab}\partial_{b}Z(\beta)$, where $c_{\beta}$ are
constants and each $Z(\beta)$ is a different independent solution since they
obey different equations. It is unclear whether or not all solutions can be
described using this Ansatz. It will be seen now that at least in the
Schwarzschild limit there are solutions that are not described by the Ansatz.

####  IV.2.2 Schwarzschild-AdS limit

The special case of Schwarzschild-AdS can be obtained by taking the limit
$a=0$. The Kerr-AdS line element given in Eq. (52) reduces to the
Schwarzschild-AdS line element given in Eq. (11). In addition, from Eqs. (57)
and (58), the tensor $B^{ab}$ in the nonrotating $a=0$ limit becomes

$\displaystyle\bm{B}_{\text{S}}=-\frac{1}{q_{r}f}\partial_{t}^{2}+\frac{f}{q_{r}}\partial_{r}^{2}+\frac{1}{r^{2}}\partial_{\theta}^{2}+\frac{1}{r^{2}\sin^{2}\theta}\partial_{\phi}^{2}\,,$
(62) $\displaystyle\bm{B}_{\text{A}}=\frac{\beta
r}{q_{r}}(\partial_{r}\partial_{t}-\partial_{t}\partial_{r})\,.$ (63)

Equation (59) for $R(r)$ is now given by

$\displaystyle\partial_{r}\Big{[}\frac{r^{2}f}{q_{r}}\partial_{r}R\Big{]}+\Big{[}\frac{\omega^{2}r^{2}}{fq_{r}}-i\omega\frac{2-q_{r}}{q_{r}^{2}\beta}+\frac{\mu^{2}}{\beta^{2}}\Big{]}R=0\,.$
(64)

The angular equation (60) turns into

$\displaystyle\frac{1}{\sin\theta}\partial_{\theta}\Big{[}\sin\theta\partial_{\theta}S\Big{]}-\frac{m^{2}}{\sin^{2}\theta}S+\Big{[}i\frac{\omega}{\beta}-\frac{\mu^{2}}{\beta^{2}}\Big{]}S=0\,.$
(65)

The solutions for this equation are the spherical harmonics $Y^{\ell m}$.
Thus, it is possible to obtain the expression for the covariant components of
the massive vector field as a function of the scalar $R(r)$ and the spherical
harmonics $Y^{\ell m}$ for each $\beta$. In the Schwarzschild-AdS background,
i.e., $a=0$, Eq. (61), with the help of Eqs. (62) and (63), give

$\displaystyle A_{a}=\Big{(}-\frac{i\omega}{q_{r}}+\frac{\beta
rf}{q_{r}}\partial_{r},\frac{1}{q_{r}}\partial_{r}-i\frac{\omega\beta
r}{q_{r}f},\partial_{\theta},\partial_{\phi}\Big{)}R(r)Y^{\ell m}\,,$ (66)

where we have dropped the explicit dependence on $\beta$ to not overcrowd the
notation. Moreover, from Eq. (65) the values for the parameter $\beta$ can
then be found by setting
$i\frac{\omega}{\beta}-\frac{\mu^{2}}{\beta^{2}}=\ell(\ell+1)$. Thus, there
are two different values for $\beta$ for each $\ell>0$. Calling these values
$\beta_{+}$ and $\beta_{-}$, one has

$\displaystyle\beta_{\pm}=i\omega\frac{1\pm\sqrt{1+\frac{4\mu^{2}\ell(\ell+1)}{\omega^{2}}}}{2\ell(\ell+1)}\,.$
(67)

The two different $\beta$ values each correspond to a different polarization.
A further analysis on the expression suggests that $\beta_{-}$ describes the
scalar-type polarization, since setting the Proca mass $\mu=0$ makes it vanish
as one expects for the massless Maxwell field. Moreover, setting the Proca
$\ell$ to zero yields a definite value for $\beta_{-}$, namely
$\beta_{\text{monopole}}=-i\frac{\mu^{2}}{\omega}$, so that $\beta_{-}$ is the
polarization that contains the monopole case. The case $\beta_{+}$ describes
then the vector-type polarization, since setting the Proca mass $\mu=0$ yields
$\beta_{+}=i\omega\frac{1}{\ell(\ell+1)}$, as one expects for the massless
Maxwell field, whereas setting the Proca $\ell$ to zero, $\ell=0$, gives an
infinite $\beta$ which seems to have no meaning.

Two important features can be drawn from the FKKS Ansatz of Eq. (66). The
first is that there is a natural decoupling of the two polarizations related
to the electric modes, in contrast to the VSH method of Sec. III. The second
feature is that when comparing Eq. (66) with Eq. (14), it can be seen that the
Ansatz in the Schwarzschild-AdS limit, $a=0$, does not describe the vector-
type polarization related to the magnetic modes, i.e., it does not describe
the function $u_{(3)}$ in (14), again in contrast to the VSH method of Sec.
III. By inspecting the principal tensor in Eq. (53), two of the eigenvalues of
$h^{a}_{\,\,b}$ are given by $\pm ix_{2}=\pm ia\cos{\theta}$. By taking the
Schwarzschild-AdS limit, both eigenvalues go to zero and so the principal
tensor becomes degenerate. This violates the initial requirement that the
principal tensor needs to be nondegenerate in order to characterize all the
symmetries of the spacetime. This fact surely has implications on the absence
of the magnetic modes of the Proca field in the FKKS approach.

###  IV.3 Quasinormal modes in Schwarzschild-AdS in the FKKS method

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (FKKS) |  $\omega l$ (FKKS)
$0.01$ | $3.330-2.489\,\,i$ | $184.968-266.395\,\,i$
$0.10$ | $3.339-2.501\,\,i$ | $185.578-267.524\,\,i$
$0.20$ | $3.362-2.534\,\,i$ | $187.355-270.817\,\,i$
$0.40$ | $3.444-2.652\,\,i$ | $193.650-282.498\,\,i$
$0.50$ | $3.498-2.729\,\,i$ | $197.761-290.138\,\,i$
(a) Scalar-type polarization ($\beta_{-}$).

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (FKKS) |  $\omega l$ (FKKS)
$0.01$ | $1.554-0.542\,\,i$ | $0-149.984\,\,i$
$0.10$ | $1.557-0.552\,\,i$ | $0-152.099\,\,i$
$0.20$ | $1.568-0.584\,\,i$ | $0-158.432\,\,i$
$0.30$ | $1.584-0.633\,\,i$ | $0-168.817\,\,i$
$0.40$ | $1.606-0.699\,\,i$ | $0-183.291\,\,i$
$0.50$ | $1.632-0.777\,\,i$ | $0-202.684\,\,i$
(b) Vector-type polarization ($\beta_{+}$).

Table 3:  Quasinormal mode frequencies $\omega l$ of the Proca field electric
modes with $\ell=1$, using the FKKS method in Schwarzschild-AdS for $r_{+}=l$
and $r_{+}=100l$ and for several values of the Proca field mass $\mu l$. (a)
The frequencies of the scalar-type polarization $\beta_{-}$ of the electric
modes are displayed. (b) The frequencies of the vector-type polarization
$\beta_{+}$ of the electric modes are displayed.

The aim now is to solve Eq. (64) for $R$. Note beforehand that, as proved
analytically in thesis , the Ansatz of Eq. (66) continues to obey the Proca
equations given in Eqs. (20) and (21), and the Bianchi identity given in Eq.
(24), by considering the correspondence $u_{(0)}=-\frac{i\omega
r}{q_{r}}R(r)+\frac{\beta r^{2}f}{q_{r}}\partial_{r}R(r)$,
$u_{(1)}=\frac{rf}{q_{r}}\partial_{r}R(r)-i\frac{\omega\beta
r^{2}}{q_{r}}R(r)$, and $u_{(2)}=\ell(\ell+1)R(r)$, for each $\beta$. This
remarkable correspondence between the VSH method and the FKKS method given by
these transformations, decouples the two polarizations in the Proca equations
with the relevant scalar function still obeying a second-order partial
differential equation, a result that has also been shown in Percival:2020skc .

We can now solve Eq. (64) for $R$. Indeed, in Eq. (64), we must swap the
derivatives in $r$ in terms of derivatives in $r_{*}$. Afterwards by
multiplying the obtained equation by a factor $\frac{q_{r}f}{r^{2}}$, we
obtain Eq. (21) with $u_{(1)}$ and $u_{(2)}$ given by the correspondence
above, i.e., $u_{(1)}=\frac{rf}{q_{r}}\partial_{r}R(r)-i\frac{\omega\beta
r^{2}}{q_{r}}R(r)$, and $u_{(2)}=\ell(\ell+1)R(r)$. We then write
$R=\mathcal{R}e^{-i\omega(t+r_{*})}$ and substitute it into Eq. (21) with the
given correspondence to yield

$\displaystyle\Bigg{[}(x-x_{+})v(x)\partial_{x}^{2}+w(x)\partial_{x}+\frac{y(x)}{x-x_{+}}\Bigg{]}\mathcal{R}=0\,,$
(68)

where again $x=\frac{1}{r}$ and $x_{+}=\frac{1}{r_{+}}$, and the polynomials
are

$\displaystyle v(x)=x^{2}\frac{f}{r^{2}}\frac{(x^{2}-\beta^{2})}{x-x_{+}}\,,$
(69) $\displaystyle w(x)=(x^{2}-\beta^{2})(2i\omega
x^{2}+2x^{3}-6Mx^{4})-2x^{3}\frac{f}{r^{2}}\,,$ (70) $\displaystyle
y(x)=(x-x_{+})\Big{[}(x^{2}-\beta^{2})(\mu^{2}+\ell(\ell+1)x^{2})$
$\displaystyle\quad-2i\omega(x^{3}+\beta x^{2})\Big{]}\,.$ (71)

The clear differences between the above polynomials and the ones in Eqs.
(30)-(32) are, first, the dependence on the parameter $\beta$, which is
characteristic for each polarization and, second, the higher order in $x$. The
explicit dependence of the polynomials on $\omega$, without considering the
$\beta$ dependence, at first glance will be at most linear for the same
reasons stated for the polynomials in Eqs. (30)-(32). Another difference,
perhaps more concealed, is that when introducing $\beta$, using Eq. (67), into
Eq. (68) together with Eqs. (69)-(71), one finds that the polynomials will
also have a different dependence on $\omega$, $\ell$, and $\mu$ compared with
the ones in Eqs. (30)-(32). The polynomials defined in Eqs. (69)-(71) can be
expanded around $x_{+}$ so that it is possible to write

$\displaystyle v(x)=\sum_{j=0}^{\infty}v_{j}(x-x_{+})^{j}\,,$ (72)
$\displaystyle w(x)=\sum_{j=0}^{\infty}w_{j}(x-x_{+})^{j}\,,$ (73)
$\displaystyle y(x)=\sum_{j=0}^{\infty}y_{j}(x-x_{+})^{j}\,,$ (74)

respectively, where $v_{j}$, $w_{j}$, and $y_{j}$ are expansion coefficients
that vanish for $j$ higher than some value. Then the scalar $\mathcal{R}$ can
be expanded as

$\mathcal{R}=\sum_{n=0}^{\infty}a_{(R)n}(x-x_{+})^{n}\,,$ (75)

where the coefficients $a_{(R)}$ are calculated using the recurrence relation

$\displaystyle
a_{(R)n}=-\frac{1}{P_{n}}\sum_{j=0}^{n-1}\Big{(}j(j-1)v_{n-j}+jw_{n-j}+y_{n-j}\Big{)}a_{(R)j}\,,$
(76)

where $P_{n}$ is given by

$\displaystyle P_{n}=n(n-1)v_{0}+nw_{0}\,,$ (77)

and the $v_{j}$, $w_{j}$, and $y_{j}$ are the coefficients of the expansion in
Eqs. (72)-(74), considering the polynomials in Eqs. (69)-(71). In putting the
problem in this way we have just shown that the quasinormal modes derived from
the FKKS Ansatz in the nonrotating limit of Kerr-AdS, i.e., for Schwarzschild-
AdS, can now be computed by applying the Horowitz-Hubeny numerical procedure.
This means that the quasinormal modes can be computed by requiring that $R$
vanishes at $x\rightarrow 0$, thus

$\displaystyle\sum_{j=0}^{N}a_{(R)j}(-x_{+})^{j}=0\,,$ (78)

where $N$ is formally infinite but for numerical purposes is a large integer.
We do not present the mode frequencies for $\ell=0$ which would be taken from
$\beta_{-}$, but they were calculated by us using this method and they agree
with frequencies calculated using the VSH method of Sec. III and with the
results in Konoplya:2006 . The numerical calculations of the quasinormal modes
for $\ell=1$ for Schwarzschild-AdS in the FKKS method are presented in Table
3. These quasinormal modes were computed for each of the two values of $\beta$
taken from Eq. (67). Since $\beta_{-}$ can be identified as corresponding to
the scalar-type polarization, and $\beta_{+}$ can be identified as
corresponding to the vector-type polarization, the modes for each polarization
are easily distinguished. The mode frequencies for the scalar-type
polarization $\beta_{-}$ are shown in Table 3(a) and the mode frequencies for
vector-type polarization $\beta_{+}$ are shown in Table 3(b). It was taken
$N=40$, and the method seems to converge even though the polynomials given in
Eqs. (69)-(71) have higher order dependence in $x$. The values of the
quasinormal mode frequencies for other higher monotones, higher values of
$\mu$ and $r_{+}$, are found in thesis . We must emphasize that the
quasinormal modes for the other vector-type polarization, encoded in
$u_{(3)}$, cannot be found since the FKKS Ansatz in the Schwarzschild-AdS
limit does not describe the magnetic modes.

##  V Comparison of results between the VSH method and the FKKS method

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (VSH) |  $\omega l$ (FKKS) |  $\omega l$ (VSH) |  $\omega l$ (FKKS)
$0.01$ | $3.331-2.489\,\,i$ | $3.330-2.489\,\,i$ | $184.968-266.394\,\,i$ | $184.968-266.395\,\,i$
$0.10$ | $3.339-2.500\,\,i$ | $3.339-2.501\,\,i$ | $185.604-267.461\,\,i$ | $185.578-267.524\,\,i$
$0.20$ | $3.362-2.531\,\,i$ | $3.362-2.534\,\,i$ | $187.452-270.612\,\,i$ | $187.355-270.817\,\,i$
$0.40$ | $3.446-2.645\,\,i$ | $3.444-2.652\,\,i$ | $193.925-282.119\,\,i$ | $193.650-282.498\,\,i$
$0.50$ | $3.501-2.722\,\,i$ | $3.498-2.729\,\,i$ | $198.077-289.799\,\,i$ | $197.761-290.138\,\,i$
(a) Scalar-type polarization.

| $r_{+}=l$ | $r_{+}=100l$
---|---|---
$\mu l$ |  $\omega l$ (VSH) |  $\omega l$ (FKKS) |  $\omega l$ (VSH) |  $\omega l$ (FKKS)
$0.01$ | $1.554-0.542\,\,i$ | $1.554-0.542\,\,i$ | $0-149.984\,\,i$ | $0-149.984\,\,i$
$0.10$ | $1.557-0.552\,\,i$ | $1.557-0.552\,\,i$ | $0-152.099\,\,i$ | $0-152.099\,\,i$
$0.20$ | $1.568-0.583\,\,i$ | $1.568-0.584\,\,i$ | $0-158.432\,\,i$ | $0-158.432\,\,i$
$0.30$ | $1.585-0.633\,\,i$ | $1.584-0.633\,\,i$ | $0-168.817\,\,i$ | $0-168.817\,\,i$
$0.40$ | $1.607-0.699\,\,i$ | $1.606-0.699\,\,i$ | $0-183.291\,\,i$ | $0-183.291\,\,i$
$0.50$ | $1.634-0.777\,\,i$ | $1.632-0.777\,\,i$ | $0-202.684\,\,i$ | $0-202.684\,\,i$
(b) Vector-type polarization.

Table 4:  Quasinormal mode frequencies $\omega l$ of the Proca field electric
modes $u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ with $\ell=1$, comparing the VSH and
FKKS methods in Schwarzschild-AdS for $r_{+}=l$ and $r_{+}=100l$ and for
several values of the Proca field mass $\mu l$. (a) The frequencies of the
scalar-type polarization of the electric modes in both methods are displayed.
(b) The frequencies of the vector-type polarization of the electric modes in
both methods are displayed. The numbers are taken from Tables 1 and 3.

The computations of the quasinormal modes were performed numerically using
Mathematica. In the computations we have put $N=40$, and higher $N$ would not
change the results as presented.

The quasinormal modes for the monopole, $\ell=0$, using both the VSH method
and the FKKS method were calculated by us and are consistent with
Konoplya:2006 , so we do not need to present them here.

The quasinormal modes for $\ell=1$, in the electric mode sector using both the
VSH and the FKKS Ansätze are put together and displayed in Table 4 for a
comparison between both methods. The values shown in Table 4 are taken
directly from Tables 1 and 3. In Table 4(a) the quasinormal frequencies for
the scalar-type polarization are shown, and in Table 4(b) the quasinormal
frequencies for the vector-type polarization are shown. The magnetic sector
does not appear in the FKKS method so there is no possibility of comparison in
this sector.

In the VSH method it is hard to distinguish the polarizations in the electric
modes for low $r_{+}$, an example being $r_{+}=l$. On the other hand, for
large $r_{+}$, $r_{+}\geq 100l$, one can distinguish them by comparison with
the frequencies of the modes in the magnetic sector, since here both vector-
type polarizations have modes with negligible real frequency. In the FKKS
method polarization is well characterized by the different values of $\beta$,
allowing a direct distinction. By convenience, our strategy to distinguish the
electric modes in the VSH method was to use the frequencies given by the FKKS
Ansatz and check if they were present.

The massless limit $\mu=0$ of the quasinormal modes shall now be analyzed in
detail. As referred to in Sec. III, when analyzing the quasinormal modes in
the VSH method, the $\mu=0$ scalar-type polarization becomes nonphysical but
its quasinormal modes, which correspond to the ones of massless Klein-Gordon
scalar field as it can be checked both analytically and numerically, will not
disappear automatically. We now proceed to explain how this polarization can
be removed. We notice that for $\mu=0$, i.e., for a Maxwell field, there are
only two physical degrees of freedom, and the corresponding equations are
governed by the $\mu=0$ Proca equations with the Bianchi identity of Eq. (24)
now being a gauge condition, specifically, the Lorenz condition. But even with
the Lorenz condition being imposed, one is still left with a spurious degree
of freedom which corresponds to the contribution of a gradient of a scalar
field which obeys the massless Klein-Gordon equation. By counteradding the
gradient of this scalar field with the same modes, one is able to remove the
spurious degree of freedom and reset the results for the quasinormal modes of
a Maxwell field in Schwarzschild-AdS. As referred to in Sec. IV, when
analyzing the quasinormal modes in the FKKS method, the $\mu=0$ scalar-type
polarization becomes nonphysical as it should. Here, we note that Eq. (35) is
equivalent to Eq. (21) with the correspondence
$u_{(1)}=\frac{rf}{q_{r}}\partial_{r}R(r)-i\frac{\omega\beta
r^{2}}{q_{r}}R(r)$, $u_{(2)}=\ell(\ell+1)R(r)$. Setting $\mu=0$ one has from
Eq. (67) that $\beta_{-}=0$, and then Eq. (67) turns into an equation for a
massless Klein-Gordon scalar field. The same reasoning that we have performed
for the VSH method applies now, and again one is able to remove the spurious
degree of freedom and reset the results for the quasinormal modes of a Maxwell
field in Schwarzschild-AdS.

The maximum relative deviation between the quasinormal frequencies of both
treatments was found to be $0.2\%$ which is the case of the last row of Tab.
4(a) for $r_{+}=l$. The relative deviation $\sigma$ was calculated through the
formula $\sigma(\%)=\sqrt{\frac{\absolutevalue{{{\omega}}_{\rm
VSH}-{{\omega}}_{\rm FKKS}}}{\absolutevalue{{{\omega}}_{\rm VSH}}}}\times
100$. This confirms that the electric modes of the VSH method are described in
the FKKS Ansatz. We should also add that the Horowitz-Hubeny numerical
procedure applied to the VSH and the FKKS methods for the electric modes in
obtaining the quasinormal frequencies work fine with the FKKS method
converging faster see the Appendix A. Furthermore, the VSH method does not
capture part of the quasinormal modes for the same $N$ of the FKKS method.
These modes certainly should appear for the ideal $N\rightarrow\infty$ limit.
Surely, this adds value to the FKKS decomposition of the vector $A_{a}$, Eq.
(66), in the electric mode sector to decouple the two polarizations.

In thesis , the quasinormal modes for each method were computed for higher
monotones for different values of $\mu l$ and for different values of
$\frac{r_{+}}{l}$. There were higher monotones of the quasinormal modes of
$u_{(0)}$, $u_{(1)}$, and $u_{(2)}$ in the VSH method that could not be found,
but which were present within the FKKS Ansatz. The same occurred when
computing the frequencies for higher values of $\mu$, for example $\mu l=8$.
The convergence of the VSH method, generalized here for a system of equations,
was not demonstrated, thus it may be possible that the fact of having two
polarizations in the system requires a much higher value of $N$ so that these
monotones can be found in the VSH method. To show rigorously that both the VSH
and FKKS methods describe the same quasinormal modes, we would need to compare
every frequency, for all the space of variables $\mu l$ and $\frac{r_{+}}{l}$,
and also compare all the higher monotones. This is an impossible task but the
fact that it was shown that $R(r)$ obeys Eqs. (20), (21), and (24) with the
correspondence $u_{(0)}=-\frac{i\omega r}{q_{r}}R(r)+\frac{\beta
r^{2}f}{q_{r}}\partial_{r}R(r)$,
$u_{(1)}=\frac{rf}{q_{r}}\partial_{r}R(r)-i\frac{\omega\beta
r^{2}}{q_{r}}R(r)$, and $u_{(2)}=\ell(\ell+1)R(r)$ substantiates that such is
the case thesis .

##  VI Conclusions

We have separated the Proca equations in a Schwarzschild-AdS spacetime by
using the VSH method, which employs vector spherical harmonics in a
spherically symmetric spacetime, in our case in the Schwarzschild-AdS
spacetime. Specifically, the Proca field was taken to satisfy an Ansatz in
terms of the vector spherical harmonics and of time and radial dependent
functions $u_{(i)}$, with $i=0,1,2,3$. These functions can be classified into
electric modes ($u_{(0)}$, $u_{(1)}$, and $u_{(2)}$) and magnetic modes
($u_{(3)}$). The Proca equations thus separate and give a system of partial
differential equations which are coupled for the electric modes and decoupled
for the magnetic modes. The dynamical solutions of the system will have three
degrees of freedom which we call polarizations, and each polarization will
have its sets of eigenvectors. The polarization is of scalar-type if the set
of eigenvectors corresponding to it behave similarly to a scalar, otherwise
the polarization is vector-type. The electric modes have a scalar and a
vector-type polarization, whereas the magnetic modes have the remaining
vector-type polarization. Since the equations for the electric modes are
coupled, it is hard to distinguish their polarizations. In the massless limit,
the equations can be decoupled without increasing their order and the two
vector-type polarizations degenerate, while the scalar-type polarization is
described by a gradient of a massless scalar field which obeys the Klein-
Gordon equation. The quasinormal modes for the Proca field can then be found
and were displayed. In the limit of massless field with $r_{+}=100l$, the
quasinormal modes of the vector-type polarizations approach the values given
in Cardoso:2001 . For the monopole mode $\ell=0$ we have not shown the values
but our calculations agree with those in Konoplya:2006 . The quasinormal modes
were found using an extension of the Horowitz-Hubeny numerical procedure
Horowitz:2000  for the electric modes and the original method to the magnetic
modes.

We have also studied the Proca equations in a Schwarzschild-AdS spacetime
using the FKKS method. The complete set of the spacetime symmetries is
generated by the principal tensor and allows a separation of the Proca
equations for generic spinning geometries, in particular in the Kerr-AdS
spacetime Frolov:2018 . The Ansatz describes the Proca field as a contraction
of the polarization tensor with the gradient of a complex scalar and enables
the Proca equations to reduce to an angular and radial equation for that
complex scalar. The polarization tensor depends on the principal tensor and a
complex constant $\beta$, whose discrete values are determined by the
equations and each value corresponds to a different polarization. It remains
unclear whether the FKKS Ansatz captures all the degrees of freedom of the
Proca vector field. In order to study this issue, an analysis of the Proca
system in the nonrotating limit of the Kerr-AdS spacetime, i.e., in the
Schwarzschild-AdS spacetime, was made. We analyzed this Ansatz in
Schwarzschild-AdS to check if it is able to describe all the polarizations. We
showed that the FKKS Ansatz in the Schwarzschild-AdS limit describes two
polarizations of the massive vector field, namely the scalar-type and the
vector-type polarizations corresponding to the electric modes of the field and
it was verified that the method allows for an easier identification of each
polarization. Moreover, for these electric modes an analytical correspondence
between the VSH method and the FKKS method was obtained, revealing a
remarkable transformation that decouples the two polarizations in the Proca
equations with the relevant scalar function still obeying a second-order
partial differential equation. In the massless limit, we observed that the
complex scalar field of the FKKS Ansatz associated to the scalar-type
polarization obeys the Klein-Gordon equation and the expression for the vector
field is equivalent to the gradient of this complex scalar. Thus, the FKKS
method in the massless limit yields the same result as the VSH method. On the
other hand, the FKKS Ansatz does not capture the magnetic modes. Since it is
known that magnetic modes are present in the Kerr geometry in the FKKS Ansatz
Dolan:2018dqv ; Baumann:2019eav ; Percival:2020skc , the reason for the
absence of the magnetic modes may be due to the degeneracy of the principal
tensor in the nonrotating limit. The quasinormal modes of the electric sector
of the Proca field with the FKKS Ansatz were computed using the Horowitz-
Hubeny numerical procedure.

We performed a numerical comparison of the quasinormal modes of the electric
sector that were obtained by the VSH method and the FKKS method. The
quasinormal modes of the electric sector in both methods coincide well, having
a maximum relative deviation of $0.2\%$. Even though only the fundamental
quasinormal modes are displayed here for $\ell=1$, this corroborates that the
FKKS method is not only able to describe the electric modes but also is able
to decouple both polarizations naturally in the electric mode sector. In the
massless limit, the quasinormal modes for the scalar-type polarization do not
vanish in both methods, coinciding with the quasinormal modes of a Klein-
Gordon scalar field. Nevertheless, these modes are nonphysical since they can
be removed by the gauge freedom. Since the FKKS Ansatz does not describe the
magnetic modes, the quasinormal modes associated to this sector cannot be
compared.

Further study of the polarizations described by the FKKS Ansatz in spinning
geometries should be undertaken. For the Kerr metric an analytical comparison
between the Teukolsky method and the FKKS method in the massless limit was
presented in Dolan:2018dqv , and so it would be interesting to see an
extension of such a comparison for Kerr-AdS, and even for Kerr-dS. This is
achievable since the Newman-Penrose formalism used by Teukolsky in Kerr can in
principle be extended to Kerr-AdS.

###### Acknowledgements.

This work was supported through the European Research Council Consolidator
Grant No. 647839, the Portuguese Science Foundation FCT Project No.
IF/00577/2015, the FCT Project No. PTDC/MAT-APL/30043/2017, the FCT Project
No. UIDB/00099/2020, and the FCT Project No. UIDP/00099/2020. This work has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No.
101007855. The authors would like to acknowledge networking support by the
GWverse COST Action CA16104, “Black holes, gravitational waves and fundamental
physics.”

##  Appendix A Comparison of numerical convergence between the VSH and FKKS
methods

In Sec.V, when comparing the VSH and FKKS methods we made notice that the FKKS
method converges faster. To see this convergence explicitly we show two
figures, Figs. 1 and 2, where the quasinormal modes corresponding to the last
row of Tab. 4(a) are computed with varying $N$.

(a) The frequency $\omega_{R}$, the real part of $\omega$, as a function of
$N$.

(b) The frequency $\omega_{I}$, the imaginary part of $\omega$, as a function
of $N$.

Figure 1: Quasinormal mode of the scalar-type polarization of the Proca field
for $\ell=1$, $\mu l=0.50$, $r_{+}=l$ computed with varying computational
number $N$, for the VSH method (blue) and the FKKS method (orange). The values
in blue seem to converge to the values of orange.

(a) The frequency $\omega_{R}$, the real part of $\omega$, as a function of
$N$.

(b) The frequency $\omega_{I}$, the imaginary part of $\omega$, as a function
of $N$.

Figure 2: Quasinormal mode of the scalar-type polarization of the Proca field
for $\ell=1$, $\mu l=0.50$, $r_{+}=100l$ computed with varying computational
number $N$, for the VSH method (blue) and the FKKS method (orange). The values
in blue seem to converge to the values of orange.

The value obtained by the VSH method seems to converge to the value obtained
by the FKKS method. Even though the value corresponding to the FKKS method
also changes, this change only occurs in the fifth significant digit. It was
not possible to compute for higher $N$ since it requires higher machine
precision.

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