# Jacobi equations of geodetic brane gravity

Riccardo Capovilla‡, Giovany Cruz†, and Efraín Rojas
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Departamento de Física, Cinvestav-IPN, Av. Instituto Politécnico Nacional
2508, Col. San Pedro Zacatenco, 07360, Gustavo A. Madero, Ciudad de México,
México.‡†
Facultad de Física, Universidad Veracruzana, Campus Sur, Paseo No. 112,
Desarrollo Habitacional, Nuevo Xalapa, 91097, Xalapa-Enríquez, Veracruz,
México.∗

(April 4, 2022)

###### Abstract

We consider brane gravity as described by the Regge-Teitelboim geometric
model, in any co-dimension. In brane gravity our spacetime is modeled as the
time-like world volume spanned by a space-like brane in its evolution, seen as
a manifold embedded in an ambient background Minkowski spacetime of higher
dimension. Although the equations of motion of the model are well known,
apparently their linearization has not been considered before. Using a direct
approach, we linearize the equations of motion about a solution, obtaining the
Jacobi equations of the Regge-Teitelboim model. They take a formidable aspect.
Some of their features are commented upon. By identifying the Jacobi
equations, we derive an explicit expression for the Morse index of the model.
To be concrete, we apply the Jacobi equations to the study of the stability of
a four-dimensional Schwarzschild spacetime embedded in a six-dimensional
Minkowski spacetime. We find that it is unstable under small linear
deformations.

##  I Introduction

In the 70’s, T. Regge and C. Teitelboim (RT) considered a geometric model for
our spacetime as the world volume of a three-dimensional brane evolving
geodesically in a fixed higher-dimensional background Minkowski spacetime. For
their motivation, the title of their proceedings contribution, Gravity à la
string: a progress report, says it all [1]. The action they considered in
their geometric model is identical to the Einstein-Hilbert action of general
relativity. The crucial difference are the field variables. Rather than the
spacetime metric as in general relativity, in the RT model the field variables
are the embedding functions of the world volume, so that the world volume
metric becomes a composite field variable.

The equations of motion of the RT model are of second order in derivatives,
and weaker than the Einstein equations. The feature of geometric models with
equations of motion of second order is shared by a larger class of geometric
models, to which the RT model belongs, called Lovelock branes [2, 3]. All
solutions of the Einstein equations are also solutions of the RT model, but
the solution space of the latter is larger [4, 5, 6]. The extra part can be
interpreted as “geometrical dark matter” [7], in competition with present
efforts to describe dark matter/energy that add exotic terms to the energy-
momentum tensor [8, 9], or modifications of the geometric part, like $f(R)$
theories, see e.g. [10, 11, 12]. In this context, we note that the addition of
a world volume cosmological constant is equivalent, from a brane point of
view, to the addition of a Dirac-Nambu-Goto (DNG) term to the action.

The pioneering work of Regge and Teitelboim has received relatively recently
renewed interest in the context of brane-world scenarios [13, 14, 15], and in
particular through the studies of Davidson and collaborators, that introduced
the suggestive term “Geodetic Brane Gravity” , see e.g. [16, 17]. In addition,
one expects that the RT action will emerge as an effective action in any
geometrical treatment of branes that takes into account their physical
thickness. At the same order of magnitude, one finds possible geometric models
quadratic in the extrinsic curvature tensor, known as “rigidity terms” [18,
19].

It is important to note that, at a basic geometric level, in order to ensure
the local existence of an embedding framework, at most $N=n(n+1)/2$ dimensions
are needed for the ambient spacetime background. For $n=4$, at most 10
dimensions are needed. In addition, it is known that if the world volume
metric admits Killing vectors, that number can be reduced [20, 21, 22, 23]. In
particular, we remark that not every spacetime solution of the Einstein
equations can be embedded as a hypersurface. For example the embedding of the
Schwarzschild solution requires at least co-dimension two [24]. This provides
one important motivation to consider arbitrary co-dimension, even though it
does complicate things. Needless to say, this is relevant about the stability
of this type of geometric configurations, since it is also necessary to
analyse the conditions to ensure or not its stability. In particular, higher
co-dimension implies the necessity to use geometrical structures that take
into account the rotational freedom in the normal fields to the world volume.
In our opinion, this issue was overlooked in many contributions on the RT
model, and does not appear to have been addressed before.

The aim of this note is to derive the Jacobi equations for the RT geometric
model. For a relativistic particle that satisfies the geodesic equation, the
Jacobi equations are simply the geodesic deviation equation, that quantifies
the behaviour of two nearby particle in a curved background spacetime [25]. In
the higher dimensional brane generalization, one can adopt the same point of
view. More in general, the Jacobi equations provide an essential tool for the
study of the stability of a solution of the equations of motion. The
derivation of the Jacobi equations can be accomplished with a direct approach,
that consists in the linearization of the equations of motion, and this the
one adopted in this paper. We consider also the possible presence of matter
fields that live on the brane. An equivalent and alternative avenue would be a
variational approach, that considers the second variation of the action when
the equations of motion are satisfied, i.e. on-shell. In both cases, the
Jacobi equations can be derived as Euler-Lagrange equations of a suitable
action principle that is a cousin of the Jacobi’s accessory variational
principle for systems with a finite number of degrees of freedom, see e.g.
[26]. For the geometric model under consideration, this is accomplished by the
identification of interesting conserved geometrical structures. The accessory
action is proportional to the index of the geometrical model under
consideration. The index is a crucial tool in the study of the geometry in the
large, using Morse theory, and it has proven to be a valuable tool in
geometric stability problems, such as minimal surfaces, see e.g. [27, 28]. In
addition, the index ought to provide an interesting avenue towards a path
integral quantization of the model, since the accessory action is quadratic in
the fields.

Once established the general form of the Jacobi equations, we focus on a
specific solution of the equations of motion, namely a four dimensional
Schwarzschild geometry embedded in a six dimensional Minkowski spacetime.
Exploiting the symmetries of the solution, we derive a set of equations that
determine the quasi-normal modes of the system. Following the guideline
developed in [29], a numerical calculation allows us to draw some conclusions
about the linear stability of the geometry. Indeed, we find signals of
instability for this configuration in absence of matter. Although this is a
widely accepted feature in black hole theories in higher dimensions, it seems
likely that stable solutions can be found in higher-dimensional embedded black
hole geometries [30].

This paper is organized as follows. In Sec. II , a brief description of the
geometry and notation used is given. In Sec. III, we introduce the RT
geometrical model. In Sec. IV, we derive an explicit covariant form for the
Jacobi equations of the model, via a direct linearization of the equations of
motion. We also derive the index of the model. In Sec. V, we study the
stability of a four-dimensional Schwarzschild spacetime embedded in a six-
dimensional Minkowski spacetime background, using the results previously
derived for the general case. Sec. VI provides a brief discussion. Some
technical details are presented in two Appendices.

##  II Brane geometry in higher co-dimension

Consider a $p+1$ dimensional manifold $m$ that represents the world volume of
a spacelike brane $\Sigma$. $m$ is embedded in a $n+1$ dimensional flat
Minkowski spacetime background $\\{\mathcal{M}$,$\eta\\}$ with metric
$\eta_{\mu\nu}=\text{diag}(-1,1,...,1)$ ($\mu,\nu=0,1,2,\ldots,n$). The world
volume is described by the embedding functions $X^{\mu}(u^{a})$ where $u^{a}$
are local coordinates for $m$ ($a,b=0,1,2,\ldots p$). The tangent vectors to
$m$ are given by $X^{\mu}_{a}:=\partial X^{\mu}/\partial u^{a}$. The inner
product of the tangent vectors gives the induced metric on $m$,
$g_{ab}:=\eta_{\mu\nu}X^{\mu}_{a}X^{\nu}_{b}=X_{a}\cdot X_{b}$. Here and
henceforth a dot denotes inner product using the background Minkowski metric.
By $g^{ab}$ we denote the inverse of $g_{ab}$, and by $g$ its determinant. The
world volume $m$ is assumed to be time-like so $g<0$. The normal vectors to
$m$ are represented by $n^{\mu\,i}$, ($i,j=1,2,...,n-p$), and $n-p$ is the co-
dimension of $m$. The normal vectors are orthogonal to the tangent vectors,
$n^{i}\cdot X_{a}=0$, and orthonormal between themselves, $n_{i}\cdot
n_{j}=\delta_{ij}$. These expressions define the normal vectors up to a sign
and a local $O(n-p)$ rotation. This gauge freedom requires the introduction of
a suitable gauge field, known as twist potential, given by
$\omega_{a}{}^{ij}=n^{i}\cdot\partial_{a}n^{j}$, see e.g. [31]. We also
introduce the extrinsic curvature of $m$ with
$K_{ab}{}^{i}=-n^{i}\cdot\nabla_{a}X_{b}$, and the mean extrinsic curvature as
its trace, $K^{i}=g^{ab}K_{ab}{}^{i}$. Here $\nabla_{a}$ denotes the torsion-
less world volume covariant derivative, compatible with the induced metric,
$\nabla_{a}g_{bc}=0$. In addition, we will use the gauge covariant derivative
given by $\widetilde{\nabla}_{a}=\nabla_{a}-\omega_{a}$. The Riemann tensor of
$\nabla_{a}$ is denoted by $R_{abc}{}^{d}$, with Ricci tensor
$R_{ab}=R_{acb}{}^{c}$, scalar curvature $R=g^{ab}R_{ab}$, and Einstein tensor
$G_{ab}=R_{ab}-(1/2)g_{ab}R$. We follow the conventions of [25].

The intrinsic and extrinsic geometries of the world volume $m$ are related by
the Gauss-Codazzi-Mainardi equations

$R_{abcd}=K_{ac}{}^{i}K_{bd\,i}-K_{ab}{}^{i}K_{cd\,i},$ (1a)
$0=\widetilde{\nabla}_{a}K_{bc}{}^{i}-\widetilde{\nabla}_{b}K_{ac}{}^{i}.$
(1b)

For completeness, it should be mentioned that, for co-dimension higher that
one, there are additional equations that involve the curvature of the twist
potential, see e.g. [31], but they will not be used in this paper.

We note also the contractions with the covariant metric,

$R_{ab}=K^{i}K_{ab\,i}-K_{ac\,i}K_{b}{}^{c\,i},$ (2a)
$R=K^{i}K_{i}-K^{ab\,i}K_{ab\,i},$ (2b)
$0=\widetilde{\nabla}_{a}K^{i}-\widetilde{\nabla}^{b}K_{ab}{}^{i}\,.$ (2c)

Additionally, for any tensor $S_{ab}$, $S_{[ab]}$ indicates anti-
symmetrization under the convention $S_{[ab]}=(S_{ab}-S_{ba})/2$. Similarly,
$S_{(ab)}$ indicates symmetrization according to $S_{(ab)}=(S_{ab}+S_{ba})/2$.

##  III Regge-Teitelboim model

The RT model is the integral over the trajectory of a $p$-dimensional brane
$\Sigma$, that depends on the scalar curvature $R$ of the world volume $m$
obtained from the world volume metric $g_{ab}$ [1],

$S_{\text{\tiny
RT}}[X,\varphi]=\int_{m}\sqrt{-g}\left[\frac{1}{2\kappa}R+L_{\text{\tiny
m}}(\varphi,X)\right],$ (3)

where $\kappa$ is a constant, and we have absorbed the infinitesimal
$d^{p+1}u$ in the integral sign over $m$, henceforth. To make contact with
general relativity, $\kappa=8\pi G_{N}$. The field variables are the embedding
functions $X^{\mu}(u^{a})$, and the possible matter fields living on the
brane, named collectively with $\varphi(u^{a})$, with matter Lagrangian
$L_{\text{\tiny m}}(\varphi,X)$. This action is identical in form to the
Einstein-Hilbert action of general relativity, but the field variables are
different. In particular, the induced metric $g_{ab}=X_{a}\cdot X_{b}$, that
determines the scalar curvature $R$, is a composite geometrical structure. We
assume that the world volume is without boundary, for simplicity. The
symmetries of the action are world volume reparametrizations, the Poincaré
symmetry of the background Minkowski spacetime and, since $p+1<n+1$,
invariance under rotations of the normal vectors adapted to the world volume.

The infinitesimal changes of the field variables, $X^{\mu}(u^{a})\to
X^{\mu}(u^{a})+\delta X^{\mu}(u^{a})$, can be decomposed into tangential and
normal deformations as follows $\delta
X^{\mu}=\phi^{a}(u^{b})X^{\mu}{}_{a}+\phi^{i}(u^{b})n^{\mu}{}_{i}$, where
$\phi^{a}$ and $\phi^{i}$ denote tangential and normal deformations fields,
respectively [31]. The tangential deformations can always be associated to an
infinitesimal reparametrization, and can be ignored safely if there is no
boundary. We are left with the physical transverse deformations, that we
denote with $\delta_{\perp}X^{\mu}=\phi^{i}n^{\mu}{}_{i}$. In particular, we
note that

$\delta_{\perp}g_{ab}=2X_{(a}\cdot\delta_{\perp}X_{b)}=2K_{ab}{}^{i}\phi_{i}\,,$
(4)

together with $\delta_{\perp}g^{ab}=-2K^{ab}{}_{i}\phi^{i}$.

The first variation of the action (3) reads [25]

$\displaystyle\delta S_{\text{\tiny RT}}[X,\varphi]$ $\displaystyle=$
$\displaystyle\int_{m}\left[\frac{\sqrt{-g}}{2\kappa}G_{ab}+\frac{\partial(\sqrt{-g}L_{\text{\tiny
m}})}{\partial g^{ab}}\right]\delta g^{ab},$ $\displaystyle=$
$\displaystyle\int_{m}\left[\frac{\sqrt{-g}}{2\kappa}G_{ab}+\frac{\partial(\sqrt{-g}L_{\text{\tiny
m}})}{\partial g^{ab}}\right]\delta_{\perp}g^{ab},$

where we have neglected a boundary term in the first line, whereas in the
second line we have considered that the complete variation of the induced
metric, namely, $\delta g_{ab}=2K_{ab}{}^{i}\phi_{i}+2\nabla_{(a}\phi_{b)}$,
implies that the tangential deformations of the world volume give a tangential
variation of the action $\delta S_{\parallel}$ that is only a boundary term.
It follows from (4) that

$\delta S_{\text{\tiny
RT}}[X,\varphi]=-\frac{1}{2\kappa}\int_{m}\sqrt{-g}\left(G_{ab}-\kappa
T_{ab}\right)K^{ab}{}_{i}\phi^{i}\,,$ (5)

Here, $G_{ab}$ denotes the world volume Einstein tensor, and
$T_{ab}:=(-2/\sqrt{-g})\partial(\sqrt{-g}L_{\text{\tiny m}})/\partial g^{ab}$
is the world volume stress-energy tensor. The extremization of the action,
$\delta S_{\text{\tiny RT}}=0$, gives the $n-p$ equations of motion

$\mathcal{E}_{i}=\left(G_{ab}-\kappa T_{ab}\right)K^{ab}{}_{i}=0.$ (6)

These equations are of second-order in derivatives of the fields $X^{\mu}$,
with the extrinsic curvature tensor playing the role of an acceleration. This
type of gravity has a built-in Einstein limit since every solution of Einstein
equations, $G_{ab}-\kappa T_{ab}=0$, is necessarily a solution of geodetic
brane gravity. On the other hand, equations (6) are weaker in the sense that a
more general solution of the form $G_{ab}-\kappa T_{ab}=\mathcal{T}_{ab}$ may
exist as long as

$\mathcal{T}_{ab}K^{ab}_{i}=0\quad\text{and}\quad\mathcal{T}_{ab}\neq 0.$ (7)

Certainly, it has been speculated in [17] that the geometrical structure
$\mathcal{T}_{ab}$ can interpreted as a non-ordinary matter contribution,
labelled as dark matter, since it is not included in the standard matter
contribution $T_{ab}$.

Notice further that, in the absence of matter, in the spirit of classical
string theory, we may think of the equations of motion (6) as the
generalization of the condition for extremal surfaces in the sense that we
have the vanishing of a trace of the extrinsic curvature, where the Einstein
tensor $G_{ab}$ plays the role of the induced metric.

##  IV Jacobi equations

Not all solutions of the equations of motion (6) lead to stable configurations
of the extended object. In this sense, the Jacobi equations provide conditions
to explore this issue. Certainly, their solutions, also named Jacobi fields,
help us to understand more deeply how the geometry behaves under deformations
of the embedding functions in the background spacetime. Since we are
interested in obtaining a covariant expression for such equations, a
convenient strategy is to directly linearize the equations of motion (6). As
formally discussed in [31], for co-dimension higher than one, the application
of the normal deformation operator $\tilde{\delta}_{\perp}$ to the equations
of motion together with the assumption that the equations of motion
$\mathcal{E}_{i}=0$ are fulfilled, afford the linearized equations. Hence, we
focus on the expressions

$\tilde{\delta}_{\perp}\left(G_{ab}K^{ab\,i}-\kappa
T^{ab}K_{ab}{}^{i}\right)=0.$ (8)

Here, $\tilde{\delta}_{\perp}$ denotes a deformation operator covariant under
normal frame rotations, associated with the physical transverse motions, and
constructed in analogy to the covariant derivative $\tilde{\nabla}_{a}$, [31].
In reference to this subject, it is worth mentioning the existence of another
connection, $\gamma_{ij}=\gamma_{[ij]}$, analogous to $\omega_{a}{}^{ij}$,
that is necessary to satisfy the requirement of the covariance of the
deformations under normal frame rotations. Fortunately, its incorporation has
no physical consequences once the on-shell condition is met since
$\tilde{\delta}_{\perp}\mathcal{E}_{i}=\delta_{\perp}\mathcal{E}_{i}-\gamma_{i}{}^{j}\mathcal{E}_{j}$.

The variation $\tilde{\delta}_{\perp}(G_{ab}K^{ab}{}_{i})$ is rather involved.
To perform it we will proceed in steps. By considering the explicit form of
the world volume Einstein tensor we have that

$\displaystyle\tilde{\delta}_{\perp}(G_{ab}K^{ab}_{i})$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2}RK_{abi}-\frac{1}{2}R_{ab}K_{i}+2G_{a}{}^{c}K_{cbi}\right)\delta_{\perp}g^{ab}$
(9) $\displaystyle+$ $\displaystyle
G^{ab}\tilde{\delta}_{\perp}K_{ab\,i}+K^{ab}{}_{i}\delta_{\perp}R_{ab}$
$\displaystyle-$ $\displaystyle\frac{1}{2}K_{i}g^{ab}\delta_{\perp}R_{ab}.$

Before we continue, we need to compute the variations involving the Ricci
tensor. From (2a), after a straightforward computation we get

$\displaystyle g^{ab}\delta_{\perp}R_{ab}$ $\displaystyle=$ $\displaystyle
2(g^{ab}K_{j}-K^{ab}{}_{j})\tilde{\delta}_{\perp}K_{ab}{}^{j}+R_{ab}\delta_{\perp}g^{ab},$
(10) $\displaystyle K^{ab}_{i}\delta_{\perp}R_{ab}$ $\displaystyle=$
$\displaystyle\left(g^{ab}K^{cd}{}_{i}K_{cd\,j}-2K^{ac}{}_{i}K^{b}{}_{c\,j}\right.$
(11) $\displaystyle+$
$\displaystyle\left.K^{ab}{}_{i}K_{j}\right)\tilde{\delta}_{\perp}K_{ab}{}^{j}+R_{acbd}K^{cd}{}_{i}\delta_{\perp}g^{ab}.$

Inserting these expressions into (9) yields

$\displaystyle\tilde{\delta}_{\perp}(G_{ab}K^{ab}{}_{i})$ $\displaystyle=$
$\displaystyle\left(R_{acbd}K^{cd}{}_{i}-R_{ab}K_{i}+\frac{1}{2}RK_{ab\,i}+2G_{a}{}^{c}K_{bc\,i}\right)\delta_{\perp}g^{ab}$
(12) $\displaystyle+$
$\displaystyle\left[K_{i}K^{ab}{}_{j}+K_{j}K^{ab}{}_{i}-K^{a}{}_{c\,i}K^{bc}{}_{j}-K^{a}{}_{c\,j}K^{bc}{}_{i}-g^{ab}(K_{i}K_{j}-K^{cd}{}_{i}K_{cd\,j})+G^{ab}\delta_{ij}\right]\tilde{\delta}_{\perp}K_{ab}{}^{j}.$

In turn, this expression suggests to introduce the geometric tensor

$\displaystyle\mathbb{G}^{ab}{}_{ij}$ $\displaystyle:=$
$\displaystyle\left[K_{i}K^{ab}{}_{j}+K_{j}K^{ab}{}_{i}-K^{a}{}_{c\,i}K^{bc}{}_{j}-K^{a}{}_{c\,j}K^{bc}{}_{i}\right.$
(13) $\displaystyle-$
$\displaystyle\left.g^{ab}\left(K_{i}K_{j}-K^{cd}{}_{i}K_{cd\,j}\right)\right].$

Note that it is symmetric in both pairs of indices
$\mathbb{G}^{ab}{}_{ij}=\mathbb{G}^{ba}{}_{ij}=\mathbb{G}^{ab}{}_{ji}.$

A fact that will be used below is that this tensor quadratic in the extrinsic
curvature is divergence-free with respect to the tangential indices

$\widetilde{\nabla}_{a}\mathbb{G}^{ab}{}_{ij}=0.$ (14)

This is not self-evident, and it requires the use of the contracted Codazzi-
Mainardi equations (2c), as shown explicitly in Appendix A.

Note that for a hypersurface, using the contracted Gauss-Codazzi equations
(2c), $\mathbb{G}^{abi}{}{ij}$ takes the simple form
$\mathbb{G}^{ab}=2G^{ab}$, a fact that emphasizes its geometrical nature, for
co=dimension higher than one.

By inserting (12) into (8), and taking into account the definition (13), we
get

$\left[\mathbb{G}^{ab}{}_{ij}+(G^{ab}-\kappa
T^{ab})\delta_{ij}\right]\tilde{\delta}_{\perp}K_{ab}{}^{j}+\left(R_{acbd}K^{cd}{}_{i}-R_{ab}K_{i}+R_{a}{}^{c}K_{bc\,i}+G_{a}{}^{c}K_{bc\,i}\right)\delta_{\perp}g^{ab}-\kappa
K_{ab\,i}\delta_{\perp}T^{ab}=0.$ (15)

At this stage, we are ready to insert the needed normal deformations of the
first and second fundamental forms

$\displaystyle\delta_{\perp}g^{ab}$ $\displaystyle=$
$\displaystyle-2K^{ab}{}_{i}\,\phi^{i},$ (16)
$\displaystyle\tilde{\delta}_{\perp}K_{ab}{}^{i}$ $\displaystyle=$
$\displaystyle-\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\phi^{i}+K_{ac}{}^{i}K^{c}{}_{b\,j}\,\phi^{j},$
(17)

into (15) to obtain

$\displaystyle\left[\mathbb{G}^{ab}{}_{ij}+(G^{ab}-\kappa
T^{ab})\delta_{ij}\right]\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\phi^{j}$
$\displaystyle+$
$\displaystyle\left\\{2(R_{acbd}K^{cd}{}_{i}-R_{ab}K_{i}+R_{a}{}^{c}K_{bc\,i}+G_{a}{}^{c}K_{bc\,i})K^{ab}{}_{l}\right.$
(18) $\displaystyle\left.-\left[\mathbb{G}^{ab}{}_{ij}+(G^{ab}-\kappa
T^{ab})\delta_{ij}\right]K_{a}{}^{c\,j}K_{bc\,l}\right\\}\phi^{l}+\kappa\,K_{ab\,i}\delta_{\perp}T^{ab}=0.$

To illustrate the overall structure of these equations, quite formidable in
their appearance and content, it is convenient at this point to define the
tensor

$\displaystyle\mathbb{M}^{ab}{}_{ij}:=\mathbb{G}^{ab}{}_{ij}+(G^{ab}-\kappa
T^{ab})\delta_{ij}\,.$ (19)

By virtue of this definition, we have the useful identity

$\displaystyle\mathbb{M}^{ab}{}_{ij}K_{a}{}^{c\,j}K_{bc\,l}$ $\displaystyle=$
$\displaystyle(R_{acbd}K^{cd}{}_{i}-R_{ab}K_{i}+R_{a}{}^{c}K_{bc\,i}$
$\displaystyle+$ $\displaystyle
G_{a}{}^{c}K_{bc\,i})K^{ab}{}_{l}-\kappa\,T_{a}{}^{c}K_{bc\,i}K^{ab}{}_{l},$

that coincides with the terms appearing on the r.h.s. of (18). This identity
allows us to rewrite (18) in a compact form as

$\displaystyle\mathbb{M}^{ab}{}_{ij}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\phi^{j}$
$\displaystyle+$
$\displaystyle\left(\mathbb{M}^{ab}{}_{il}K_{a}{}^{c\,l}K_{bc\,j}+2\kappa
T_{a}{}^{c}K_{bc\,i}K^{ab}{}_{j}\right)\phi^{j}$ (20) $\displaystyle+$
$\displaystyle\kappa\,K_{ab\,i}\delta_{\perp}T^{ab}=0.$

These are the Jacobi equations for geodetic brane gravity describing small
normal deformations of the world volume. As a matter of course, the inclusion
of arbitrary matter fields confined to the world volume, affects the dynamics
of the Jacobi fields by including the derivatives of the matter fields. Note
that (20) are second-order partial differential equations in the unknown
functions $\phi^{i}$, which is a feature that characterizes brane theories
with second-order derivative equations of motion [32]. As mentioned earlier,
the solutions to the Jacobi equations address the issue of stability through
the nature of the normal modes $\phi^{i}$, so appropriate boundary conditions
must enter the game.

On the other hand, if we focus our attention on the case where there are no
brane matter fields, $T_{ab}=0$, and assume the fulfillment of the equations
of motion, we obtain a more compact and elegant expression for the Jacobi
equations for a pure RT geometrical model. Indeed, by defining now a new
tensor

$\displaystyle\mathcal{M}^{ab}{}_{ij}:=\mathbb{G}^{ab}{}_{ij}+G^{ab}\delta_{ij},$
(21)

we can write (20) in the form

$\mathcal{M}^{ab}{}_{ij}\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\phi^{j}+\mathcal{V}_{ij}\,\phi^{j}=0,$
(22)

where we identify a geometrical “potential”

$\mathcal{V}_{ij}:=\mathcal{M}^{ab}{}_{il}K_{a}{}^{c\,l}K_{bc\,j}.$ (23)

The formal similarity of (22) with a set of Klein-Gordon equations is indeed
striking. In passing, note that the matrix structure (23) is symmetric in the
normal indices.

This set up paves the way to construct an accessory variational problem. Under
these conditions, observe that the kinetic “mass matrix”
$\mathcal{M}^{ab}{}_{ij}$ is divergenceless, as follows from the geometric
identity (14), and the divergenceless property of the Einstein tensor,

$\displaystyle\widetilde{\nabla}_{a}\mathcal{M}^{ab}{}_{ij}=0.$ (24)

The accessory action can be written then as

$S_{\text{\tiny
RT}}[\phi]=-\frac{1}{2}\int_{m}\sqrt{-g}\left[\mathcal{M}^{ab}{}_{ij}\widetilde{\nabla}_{a}\phi^{i}\widetilde{\nabla}_{b}\phi^{j}-\mathcal{V}_{ij}\phi^{i}\phi^{j}\right].$
(25)

Up to a boundary term, variation with respect to the normal deformations gives
the Jacobi equations in the form (22) as its Euler-Lagrange equations. It is
interesting to note that the accessory principle, up to factor of one half,
gives the index of the RT geometric model,

$I_{\text{\tiny RT}}[\phi]=\int_{m}{\mathcal{I}}(\phi,\phi)\,.$ (26)

As all accessory variational principles, (25) is quadratic in the field
variables. This makes it suitable for a quantization using a path integral
approach, and a determination of the effect of quantum fluctuations [33]. We
plan to address this issue in future work.

Regarding the hypersurface case, by considering the reductions
$\mathbb{G}^{ab}=2G^{ab}$ and $\mathcal{M}^{ab}=3G^{ab}$, the Jacobi equations
(22) specialize to

$G^{ab}\left(\nabla_{a}\nabla_{b}\phi+K_{a}{}^{c}K_{bc}\phi\right)=0.$ (27)

This result is in agreement with the one found in [32], where the RT model is
considered as a special case of Lovelock branes.

Clearly, the Jacobi equation (27) can be obtained from the extremization of
the action functional

$S[\phi]=-\frac{1}{2}\int_{m}\sqrt{-g}\left[G^{ab}\nabla_{a}\phi\nabla_{b}\phi-G^{ab}K_{a}{}^{c}K_{bc}\,\phi^{2}\right],$
(28)

when varied with respect to the $\phi$ field. If we consider a world volume
such that $G_{ab}\sim g_{ab}$, i.e. an Einstein manifold, this action reduces
to the action of a massive scalar field where the variable mass term is
proportional to $K_{ab}K^{ab}$. This result is equivalent to what is obtained
when one linearizes the equation of the DNG model in a flat background, see
[34].

Another interesting case is provided by the inclusion of the DNG action,
playing the role of a cosmological constant $\Lambda$, in our development. In
such a case, $L_{\text{\tiny{m}}}=\Lambda$ so that $T_{ab}=\Lambda\,g_{ab}$.
The form of the Jacobi equations, (22), remains unchanged except that the
matrix $\mathbb{M}^{ab}{}_{ij}$ now becomes

$\mathbb{M}^{ab}{}_{ij}=\mathbb{G}^{ab}{}_{ij}+(G^{ab}-\kappa\Lambda\,g^{ab})\delta_{ij}.$
(29)

Notice that we still have at hand the divergenceless property
$\widetilde{\nabla}_{a}\mathbb{M}^{ab}{}_{ij}=0$.

##  V Linear stability of Schwarzschild geometry in $\mathcal{M}^{6}$

To appreciate the formalism previously developed, we consider the special case
of a Schwarzschild geometry for the world volume, with no brane matter fields,
embedded in a 6-dim Minkowski spacetime, $\mathcal{M}^{6}$. A Schwarzschild
solution of general relativity is also automatically a particular solution of
the equations of motion (6). We use the Jacobi equations to study its linear
local stability. An illustrative case, with still a high degree of complexity
in the search for its analytical solution, is provided by the particular setup
$G_{ab}=R_{ab}=T_{ab}=0$. Under these assumptions (22) become

$\mathbb{G}^{abij}\left(\widetilde{\nabla}_{a}\widetilde{\nabla}_{b}\phi_{j}+K_{ac\,j}K^{c}{}_{b\,l}\,\phi^{l}\right)=0.$
(30)

Among the different embeddings for a 4-dim Schwarzschild geometry, see e.g.
[24], we choose to consider the Fronsdal embedding [35, 36] given by

$\begin{split}&X^{1}=2R\sqrt{1-\frac{R}{r}}\sinh\left(\frac{t}{2R}\right),\\\
&X^{2}=2R\sqrt{1-\frac{R}{r}}\cosh\left(\frac{t}{2R}\right),\\\
&X^{3}=\int\sqrt{\frac{R}{r}+\left(\frac{R}{r}\right)^{2}+\left(\frac{R}{r}\right)^{3}}dr,\\\
&X^{4}=r\sin\theta\sin\phi,\\\ &X^{5}=r\sin\theta\cos\phi,\\\
&X^{6}=r\cos\theta\end{split}$ (31)

where $\\{t,r,\theta,\phi\\}$ are the local brane coordinates, and $R$ denotes
the event horizon. For this parametrization, we have two non-vanishing
extrinsic curvature components given by

$\begin{split}&K_{ab}{}^{1}=\text{diag}\left(0,ba^{\prime}-ab^{\prime},-rb,-rb\sin^{2}\theta\right),\\\
&K_{ab}{}^{2}=\text{diag}\left(-\frac{\gamma^{2}}{2Ra},\frac{2aR^{2}}{r^{3}-Rr^{2}},-\frac{aR^{2}}{r},-\frac{aR^{2}}{r\sin^{2}\theta}\right),\end{split}$
(32)

where we have introduced

$\begin{split}&a=\sqrt{\frac{r^{3}}{r^{3}+r^{2}R+rR^{2}+R^{3}}},\\\
&b=\sqrt{\frac{R(r^{2}+rR+R^{2})}{r^{3}+r^{2}R+rR^{2}+R^{3}}},\\\
&\gamma=\sqrt{1-\frac{R}{r}}.\end{split}$ (33)

By considering the ansatz $\Phi=e^{-i\omega t}Y_{lm}(\theta,\phi)\rho(r)$,
where the normal deformation field is $\Phi=(\phi^{1},\phi^{2})$, and $\rho$
is also be considered as a vector field, $\rho=(\rho^{1},\rho^{2})$, the
Jacobi equations (22) are separable, and can be written as a matrix
arrangement of radial equations in the form

${\bf A}\rho^{\prime\prime}+{\bf B}\rho^{\prime}+({\bf C}-\omega^{2}{\bf
D})\rho=0,$ (34)

where $\rho^{\prime}=d\rho/dr$. Here, ${\bf A}$, ${\bf B}$, ${\bf C}$, and
${\bf D}$ are $2\times 2$ matrices that depend on the radial coordinate $r$.
Their explicit components are given in Appendix B. It is worth mentioning that
the matrix ${\bf C}$ is the only one that contains the angular momentum
information through $l$. Multiplying equation (34) by ${\bf D}^{-1}$, and
introducing the tortoise-like radial coordinate $r_{*}=\int(dr/f_{g})$ with
$f_{g}=br^{2}\gamma^{2}/\sqrt{3}aR^{2}$, and considering
$\rho=\mathbf{M}\chi$, where $\mathbf{M}$ is a matrix defined in such a way
that the term proportional to $d\chi/dr_{*}$ vanishes, then the system of
equations (34) acquires a form familiar in black hole theory stability
analysis

$\frac{d^{2}\chi}{dr_{*}^{2}}+\omega^{2}\chi-\mathbf{V}\chi=0,$ (35)

where, as in (34), $\chi$ must be understood as a vector. The matrix potential
$\mathbf{V}$ is explicitly written in terms of matrices ${\bf B,\ C,}$ and,
${\bf D}$ in Appendix B. The system of equations (35) provides a system of
coupled harmonic oscillators with quasi-normal frequencies
$\omega=\omega_{R}+i\omega_{I}$. One can test the stability of this
configuration by studying these frequencies $\omega$.

Regarding the asymptotic behavior of the fields, at spatial infinity for non
zero angular momentum, $l\neq 0$, the values of the diagonal components of the
potential matrix $\mathbf{V}$ diverge. It follows that one can assume that the
field $\chi$ must be zero at $r\to\infty$. On the other hand, the matrix
potential $\mathbf{V}$ vanishes at the event horizon $R$, so the solution to
the resulting equation is $\chi\sim e^{-i\omega r_{*}}+e^{i\omega r_{*}}$,
where the exponential with sign - (+) represents an incoming (outgoing) wave
to the black hole configuration. Additionally, since nothing can escape from
the black hole, the part of the solution corresponding to an outgoing wave is
not allowed. Thus, the field $\chi$ must be of the form $\chi=e^{-i\omega
r_{*}}\psi$. Substituting this in (35), the following set of equations is
obtained

$f_{g}\frac{d}{dr}\left(f_{g}\frac{d\psi}{dr}\right)-2i\omega
f_{g}\frac{d\psi}{dr}-\mathbf{V}\psi=0.$ (36)

We divide this equation by $f_{g}$, and then multiply it by $\psi^{{\dagger}}$
so that we get

$\psi^{{\dagger}}\frac{d}{dr}\left(f_{g}\frac{d\psi}{dr}\right)-2i\omega\psi^{{\dagger}}\frac{d\psi}{dr}-\psi^{{\dagger}}\mathbf{V}_{g}\psi=0,$
(37)

where $\mathbf{V}_{g}=\mathbf{V}/f_{g}$. Now, integrating by parts (37), and
taking into account that $\psi(\infty)=0$ and $f_{g}(R)=0$, we obtain

$\int_{R}^{\infty}dr\left[f_{g}\left|\frac{d\psi}{dr}\right|^{2}+2i\omega\psi^{{\dagger}}\frac{d\psi}{dr}+\psi^{{\dagger}}\mathbf{V}_{g}\psi\right]=0.$
(38)

The transpose and conjugate operations applied to the last equation result in

$\int_{R}^{\infty}dr\left[f_{g}\left|\frac{d\psi}{dr}\right|^{2}-2i\omega^{*}\frac{d\psi^{{\dagger}}}{dr}\psi+\psi^{{\dagger}}\mathbf{V}_{g}^{{\dagger}}\psi\right]=0.$
(39)

We proceed to integrate by parts the second term of the previous equation, and
then we take the difference of the result with (38). We get

$\displaystyle\int_{R}^{\infty}dr\left[(\omega-\omega^{*})\psi^{{\dagger}}\psi^{\prime}-\frac{i}{2}\psi^{{\dagger}}\left(\mathbf{V}_{g}-\mathbf{V}_{g}^{{\dagger}}\right)\psi\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad=\omega^{*}\left|\psi(R)\right|^{2},$
(40)

where $\psi^{\prime}=d\psi/dr$. From (40) we can solve for
$\psi^{{\dagger}}\psi^{\prime}$ and substituting this in the (38), we obtain

$\int^{\infty}_{R}dr\left[f_{g}\left|\psi^{\prime}\right|^{2}+\frac{i\omega_{R}}{2\omega_{I}}\psi^{{\dagger}}\left(\mathbf{V}_{g}-\mathbf{V}_{g}^{{\dagger}}\right)\psi+\frac{1}{2}\psi^{{\dagger}}\left(\mathbf{V}_{g}+\mathbf{V}_{g}^{{\dagger}}\right)\psi\right]=-\frac{\left|\omega\right|^{2}\left|\psi(R)\right|^{2}}{\omega_{I}},$
(41)

where we assume that $\omega_{I}\neq 0$. If the integrand in (41) is positive
definite, then the imaginary part of frequency must be negative, that is an
indication of having stable deformations of this black hole geometry. Indeed,
since the first term in (41) is positive, it only remains to analyse the
nature of the second term

$\begin{split}\nu:=&\frac{i\omega_{R}}{2\omega_{I}}\psi^{{\dagger}}\left(\mathbf{V}_{g}-\mathbf{V}_{g}^{{\dagger}}\right)\psi+\frac{1}{2}\psi^{{\dagger}}\left(\mathbf{V}_{g}+\mathbf{V}_{g}^{{\dagger}}\right)\psi\\\
=&V_{g11}\left|\psi_{1}\right|^{2}+V_{g22}\left|\psi_{2}\right|^{2}-\left|\left(V_{g12}+V_{g21}\right)+\frac{\omega_{R}}{\omega_{I}}\left(V_{g12}-V_{g21}\right)\right||\psi_{1}\psi_{2}|\\\
&+\left(\left|\psi_{1}\right|^{2}+\left|\frac{1}{2}\left(V_{g12}+V_{g21}\right)\psi_{2}\right|^{2}+\left|V_{g12}+V_{g21}\right||\psi_{1}\psi_{2}|-\left|\psi_{1}-\frac{1}{2}\left(V_{g12}+V_{g21}\right)\psi_{2}\right|^{2}\right)\\\
&+\left(\left|\psi_{1}\right|^{2}+\left|\frac{\omega_{R}}{2\omega_{I}}\left(V_{g12}-V_{g21}\right)\psi_{2}\right|^{2}+\left|\frac{\omega_{R}}{\omega_{I}}\right||V_{g12}-V_{g21}||\psi_{1}\psi_{2}|-\left|\psi_{1}-\frac{i\omega_{R}}{2\omega_{I}}\left(V_{g12}-V_{g21}\right)\psi_{2}\right|^{2}\right),\end{split}$
(42)

where a matrix multiplication has been performed in the second line, and we
have used the triangle inequality. Note that the two terms that appear in the
parentheses are positive definite. Therefore, the sign of $\nu$ depends
strongly on the values of $V_{g11}$ and $V_{g22}$. In this sense, $\nu$ could
be negative if $V_{g11}$ and $V_{g22}$ are both negative enough. Indeed, to
illustrate this fact, in Figure 1 we show the functions $V_{g11}$ and
$V_{g22}$ for $l=1$ and $R=2$.

Figure 1: $V_{g11}(r)$ and $V_{g22}(r)$ for $l=1$ and $R=2$.

This fact is an indication that, for $r>2$, there are negative values of
$V_{g11}$ and $V_{g22}$, so there could exist frequencies with positive
imaginary part that are associated with an unstable oscillation mode. However,
this analysis is not conclusive since the value of the integral in (41) could
still be positive.

A suitable strategy to further the analysis of the frequencies is to apply
numerical methods. Fortunately, a variety of numerical methods have been
considered in order to solve this type of problems, such as continued
fraction, series method, [37, 38, 29, 39], and so on, and one has the option
to choose the more suitable one according to the asymptotic behavior of the
potential both at spatial infinity and at the event horizon. The matrix
potential ${\bf V}$, at the boundaries, behaves as the potential of
Schwarzschild black hole in anti-de Sitter spacetime evaluated in the same
regions. In this sense, we rely on the numerical method carried out in [29]
for guidance in our case, where a similar analysis is performed. By
considering a Taylor expansion of the components of field $\chi$ as follows

$\chi_{i}=e^{-i\omega r_{*}}\sum_{n}a^{(i)}_{n}(\omega)\left(r-R\right)^{n},$
(43)

we can substitute this into (35), and expand those differential equations in a
Taylor series around the event horizon. Then, by solving the resulting
algebraic equations, order by order, the coefficients $a^{(i)}_{n}(\omega)$
can be found. We observe that they only depend on the 2-dimensional vector
$\mathbf{a_{0}}=\\{a_{0}^{(1)},a_{0}^{(2)}\\}$. In this sense, we can choose a
suitable orthogonal basis for the 2-dimensional space of the initial
coefficients $a_{0}^{(1)}$ and $a_{0}^{(2)}$. For each element of the basis,
one constructs the $2\times 2$ matrix

$\mathbf{S}(\omega)=\lim_{r\rightarrow\infty}\left(\begin{matrix}\chi_{1}^{(1)}&\chi_{1}^{(2)}\\\
\chi_{2}^{(1)}&\chi_{2}^{(2)}\end{matrix}\right),$ (44)

where the superscripts denote a particular vector of the basis. Certainly,
$\chi_{i}^{(1)}$ corresponds to $\mathbf{a_{0}}=\\{1,0\\}$ and
$\chi_{i}^{(2)}$ corresponds to $\mathbf{a_{0}}=\\{0,1\\}$ (see [39] for more
specific details in the procedure). Following the line of reasoning in [39],
to obtain the values of the frequencies $\omega$ it is necessary to find the
zeros of (43) by imposing the condition

$\det\mathbf{S}(\omega)=0.$ (45)

As mentioned previously, the process indicated in (45) can be coded in
Mathematica, and one can find the lowest eigenfrequencies for different values
of $l$ and $R$. Indeed, taking the following variable change $r\rightarrow Rr$
in (35), we are able to find the values shown in Table 1.

$l$ | $R\,\omega_{R}$ | $R\,\omega_{I}$
---|---|---
1 |  $\sim$ 0 | 0.83
2 | 1.01 | 0.36
3 | 1.62 | 0.56
4 | 2.45 | 0.75
Table 1: Frequencies for different values of $l$.

The value $\omega_{R}$ is associated with the oscillation of the deformation
while $\omega_{I}$ is related to its attenuation or increasing. Note that the
imaginary part of $\omega$ is always positive for any value of $R$. These
frequencies are associated with unstable deformations since the field is
$\Phi\sim e^{-i\omega t}$, and this diverges when $t\rightarrow\infty$.
Although a Schwarzschild four-dimensional black hole is stable in general
relativity, our results show that linear instabilities are present in the
embedded Schwarzschild black hole that satisfies the geodetic brane gravity
equation. This kind of instabilities is also found in the study of higher-
dimensional black holes, see e.g. [40].

##  VI Discussion

In this paper we have developed a covariant approach for the analysis of
perturbations of branes governed by geodetic brane gravity through the Regge-
Teitelboim geometric model, in any co-dimension.

We focus on normal deformations since they are the only ones that represent
the physical perturbations. We have obtained the covariant Jacobi equations
for the model, that determine the behaviour of such breathing perturbations
modes $\phi^{i}$. In particular, the equations are explicitly covariant under
local normal frame rotations. Within this geometric framework, some conserved
geometric structures enter the game, and help to express the Jacobi equations
as a set of wave-like forms from which geometric “mass” terms can be read off.
There is consistency in the theory since by making the appropriate reduction
for co-dimension one, results previously found are met [32].

In addition, we were able to exploit the conserved geometrical structures to
construct an action, or Jacobi accessory variational principle, whose
extremalization produces the Jacobi equations as its equations of motion. This
action is proportional to the geometric index for the RT model. It is worth
emphasizing that it is quadratic in the field modes, opening an interesting
avenue in the path integral quantization of the model, that we plan to
investigate in future work.

As an illustration of the general framework, using a Fronsdal scheme for a
black-hole geometry, our general result has been specialized to a
Schwarzschild geometry embedded in $\mathcal{M}^{6}$. A study of the Jacobi
equations for this case using analytical methods is instructive, but not
conclusive. On the other hand, supported by a numerical analysis, unstable
small deformations are found. In this sense, it should be emphasized that our
discussion of the properties of the eigenmodes $\omega$, related to the
stability of this type of black holes embedded in a higher dimensional space,
is neither exhaustive nor complete, but we believe that our approach opens the
door to exploring the stability of interesting relativistic systems from
covariant expressions obtained in the framework of extended objects.
Additionally, the results presented indicate that they can be generalized to
the whole class of Lovelock branes, and we consider it as a next step in the
understanding of this type of geometrical models for branes.

One crucial assumption, both in physical and geometrical terms, is the
embedding in a flat background spacetime of the world volume. A generalization
to arbitrary background spacetimes is difficult, but for maximally symmetric
ambient spacetimes like a de Sitter or anti-de Sitter background it appears to
be doable. We plan to address this issue in a future communication.

## Acknowledgements

GC acknowledges support from a CONACYT-México doctoral fellowship. ER
acknowledges encouragement from ProDeP-México, CA-UV-320: Álgebra, Geometría y
Gravitación. Also, RC and ER thank partial support from Sistema Nacional de
Investigadores, México.

##  Appendix A Proof for the conservation law (14).

In this appendix, we show explicitly that the tensor $\mathbb{G}_{ab}{}^{ij}$
is divergenceless. We choose to define the tensorial matrices

$\mathbb{G}_{ab}{}^{ij}:=2\mathbb{R}_{ab}{}^{ij}-g_{ab}\mathbb{R}^{ij},$ (46)

where

$\displaystyle\mathbb{R}_{ab}{}^{ij}$ $\displaystyle:=$ $\displaystyle
K^{(i}K_{ab}{}^{j)}-K_{a}{}^{c(i}K_{bc}{}^{j)}$ (47)
$\displaystyle\mathbb{R}^{ij}$ $\displaystyle:=$ $\displaystyle
K^{i}K^{j}-K^{cd\,i}K_{cd}{}^{j}.$ (48)

The divergence of (46) is

$\widetilde{\nabla}^{a}\mathbb{G}_{ab}^{ij}=2\widetilde{\nabla}^{a}\mathbb{R}_{ab}^{ij}-g_{ab}\widetilde{\nabla}^{a}\mathbb{R}^{ij}.$
(49)

On the one hand, we have for the first term

$\displaystyle\widetilde{\nabla}^{a}R_{ab}^{ij}$ $\displaystyle=$
$\displaystyle(\widetilde{\nabla}^{a}K^{(i})\,K_{ab}{}^{j)}+(\widetilde{\nabla}^{a}K_{ab}{}^{(i})\,K^{j)}$
(50) $\displaystyle-$
$\displaystyle(\widetilde{\nabla}^{a}K_{a}{}^{c\,(i})\,K_{bc}{}^{j)}-K_{a}{}^{c\,(i}\,\widetilde{\nabla}^{a}K_{bc}^{j)}$
$\displaystyle=$
$\displaystyle(\widetilde{\nabla}^{a}K^{(i})\,K_{ab}{}^{j)}+(\widetilde{\nabla}_{b}K^{(i})\,K^{j)}$
$\displaystyle-$
$\displaystyle(\widetilde{\nabla}^{c}K^{(\,i})\,K_{bc}^{j)}-K_{a}{}^{c\,(i}\,\widetilde{\nabla}^{a}K_{bc}^{j)}$
$\displaystyle=$
$\displaystyle(\widetilde{\nabla}_{b}K^{(i})\,K^{j)}-K_{a}{}^{c\,(i}\,\widetilde{\nabla}^{a}K_{bc}^{j)}$

where we have used the contracted integrability condition
$\widetilde{\nabla}^{a}K_{ab}^{i}=\widetilde{\nabla}_{b}K^{i}$ (Codazzi-
Mainardi) to obtain the second line.

On the other hand, observe that for the second term

$\displaystyle\widetilde{\nabla}_{b}\mathbb{R}^{ij}=2\left[(\widetilde{\nabla}_{b}K^{(i})\,K^{j)}-K_{a}{}^{c\,(i}\,\widetilde{\nabla}^{a}K_{bc}^{j)}\right]\,.$
(51)

The difference between twice (50) and (51) reads

$\displaystyle
2\widetilde{\nabla}^{a}R_{ab}^{ij}-g_{ab}\widetilde{\nabla}^{a}R^{ij}=\widetilde{\nabla}^{a}G_{ab}^{ij}=0\,.$
(52)

##  Appendix B Explicit form of transformation matrix ${\bf M}$ and effective
potential ${\bf V}$.

We provide here the explicit form of the matrices appearing in (34), and write
the matrices ${\bf M}$ and ${\bf V}$ in terms of them. We have

$\begin{split}&\mathbf{A}=\left(\begin{matrix}a_{11}&a_{12}\\\
a_{12}&-a_{11}\end{matrix}\right),\ \
\mathbf{B}=\left(\begin{matrix}b_{11}&b_{12}\\\
b_{12}&-b_{11}\end{matrix}\right),\\\
&\mathbf{C}=\left(\begin{matrix}c_{11}&c_{12}\\\
c_{12}&-c_{11}-\frac{6R^{2}}{r^{6}}\end{matrix}\right),\ \
\mathbf{D}=\left(\begin{matrix}d_{11}&d_{12}\\\
d_{12}&-d_{11}\end{matrix}\right).\end{split}$ (53)

Notice that ${\bf A}$, ${\bf B}$ and ${\bf D}$ are traceless symmetric
matrices while ${\bf C}$ is not. In fact, this is responsible for not being
able to decouple the system of equations (34). The explicit components of
these matrices are

$\begin{split}&a_{11}=\frac{2R\left(r^{3}-R^{3}\right)}{r^{3}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)},\\\
&a_{12}=-\frac{(r-R)(r^{3}+r^{2}R+rR^{2}-R^{3})\sqrt{r^{3}R(r^{2}+rR+R^{2})}}{r^{5}R(r^{3}+r^{2}R+rR^{2}+R^{3})},\\\
&b_{11}=\frac{2R\left(-r^{6}+r^{4}R^{2}+6r^{3}R^{3}+3r^{2}R^{4}+2rR^{5}+R^{6}\right)}{r^{4}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)^{2}},\\\
&b_{12}=\frac{-r^{9}-3r^{8}R-6r^{7}R^{2}-22r^{6}R^{3}-20r^{5}R^{4}-16r^{4}R^{5}+6r^{3}R^{6}+6r^{2}R^{7}+5rR^{8}+3R^{9}}{2r^{3}\sqrt{r^{3}R\left(r^{2}+rR+R^{2}\right)}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)^{2}},\\\
&c_{11}=l(l+1)\frac{R(r-R)\left(r^{2}+2rR+3R^{2}\right)}{r^{5}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)}-\frac{R^{2}\left(9r^{9}+27r^{8}R+54r^{7}R^{2}+42r^{6}R^{3}+17r^{5}R^{4}-21r^{4}R^{5}+4rR^{8}+12R^{9}\right)}{2r^{6}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)^{3}},\\\
&c_{12}=l(l+1)\frac{\sqrt{\frac{r^{3}R}{r^{2}+rR+R^{2}}}\left(r^{6}+2r^{5}R+3r^{4}R^{2}+16r^{3}R^{3}+15r^{2}R^{4}+14rR^{5}-3R^{6}\right)}{4r^{7}R\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)}\\\
&-\frac{R\left(3r^{11}+12r^{10}R+30r^{9}R^{2}+90r^{8}R^{3}+182r^{7}R^{4}+296r^{6}R^{5}+254r^{5}R^{6}+150r^{4}R^{7}-5r^{3}R^{8}-56r^{2}R^{9}-56rR^{10}-36R^{11}\right)}{4r^{11/2}\sqrt{R\left(r^{2}+rR+R^{2}\right)}\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)^{3}},\\\
&d_{11}=-\frac{6R^{4}}{r^{6}-r^{2}R^{4}},\\\
&d_{12}=\frac{3R^{2}\left(r^{3}+r^{2}R+rR^{2}-R^{3}\right)\sqrt{r^{3}R\left(r^{2}+rR+R^{2}\right)}}{(r^{7}-r^{4}R^{3})\left(r^{3}+r^{2}R+rR^{2}+R^{3}\right)}.\end{split}$
(54)

On account of the definition

${\bf h}:=-\left(\frac{{\bf D}^{-1}{\bf B}}{f_{g}}+{\bf
I}\frac{df_{g}}{dr}\right),$ (55)

where ${\bf I}$ denotes the $2\times 2$ unit matrix, we find that the matrix
$\mathbf{M}$ can be written as

$\mathbf{M}=\exp\left(-\frac{1}{2}\int\frac{{\bf h}}{f_{g}}dr\right).$ (56)

In the same way, the potential matrix ${\bf V}$ in terms of these matrices,
becomes

${\bf V}=\frac{f_{g}}{2}\frac{d{\bf h}}{dr}+\frac{1}{4}{\bf h}^{2}+{\bf
M}^{-1}{\bf D}^{-1}{\bf C}{\bf M}.$ (57)

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