aainstitutetext: Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, PA 19104, USAbbinstitutetext: Center for Applied
Mathematics and Theoretical Physics, University of Maribor, SI2000 Maribor,
Sloveniaccinstitutetext: Departamento de Física, Universidade Federal da
Paraíba, Caixa Postal 5008, 58051-970, João Pessoa, Paraíba,
Brazilddinstitutetext: Sarah Lawrence College, Bronxville, NY 10708, USA

# Gaussian null coordinates, near-horizon geometry and conserved charges on
the horizon of extremal non-dilatonic black $p$-branes111Based in part on a
lecture given by M.C. at Fifth Amazonian Symposium on Physics, Belém, November
18-22, 2019.

Mirjam Cvetič a,c Paulo J. Porfírio d Alejandro Satz<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we examine the emergence of conserved charges on the horizon of
a particular class of extremal non-dilatonic black $p$-branes (which reduce to
extremal dilatonic black holes in $D=4$ dimensions upon toroidal
compactification) in the presence of a probe massless scalar field in the
bulk. This result is achieved by writing the black $p$-brane geometry in a
Gaussian null coordinate system which allows us to get a non-singular near-
horizon geometry description. We find that the near-horizon geometry is
$AdS_{p+2}\times S^{2}$ and that the $AdS_{p+2}$ section has an internal
structure which can be seen as a warped product of $AdS_{2}\times S^{p}$ in
Gaussian null coordinates. We show that the bulk scalar field satisfying the
field equations is expanded in terms of non-normalizable and normalizable
modes, which for certain suitable quantization conditions are well-behaved at
the boundary of $AdS_{p+2}$ space. Furthermore, we show that picking the
normalizable modes results in the existence of conserved quantities on the
horizon. We discuss the impact of these conserved quantities in the late time
regime.

##  1 Introduction

The search for a full theory of quantum gravity has intensified over the last
decades, with string theory as one of the most prominent candidates. In this
scenario, black holes and extended objects as $p/D$-branes play an important
role (see f.e. Stelle:1996tz ; Mohaupt:2000gc ; Horowitz:1991cd ;
Skenderis:1999bs ; Duff:1994an  for a review). These objects display
remarkable properties, for example, black holes from General Relativity (GR)
are thermodynamical systems characterized by a temperature and an entropy
which is proportional to the area of the horizon. It is expected that the
microscopic origin of the entropy of black holes should be explained by a
fully quantum gravity theory. In the context of extended objects, the first
law of black branes mechanics has been discussed in Townsend:2001rg .

A particular wave of interest has risen recently around extremal black holes
and the near-extremality condition. In particular, Aretakis has investigated
the stability of extremal black holes Aretakis:2011gz ; Aretakis:2011ha ;
Aretakis:2011hc ; Aretakis:2012ei  by perturbing the spacetime geometry with
probe fluctuating fields. An especially interesting result from this study is
the existence of a conserved charge on the horizon called Aretakis charge or
constant which itself sets up as a “hair". This effect is remarkable because
the radial derivative of a massless scalar field ($\partial_{r}\psi$) decays
outside the horizon while in the horizon it becomes a constant. Aretakis also
showed that any higher radial derivative ($\partial_{r}^{k}\psi$) blows up at
the horizon. The Aretakis charge was recently calculated Cvetic:2020zqb  for
extremal rotating STU black holes in four-dimensions Cvetic:1996xz  and five-
dimensions Cvetic:1996kv .

It has been shown in Godazgar:2017igz ; Cvetic:2018gss  that the Aretakis
charge for extremal Reissner-Nordstrom (RN) black holes perturbed by a
massless scalar field is intimately related to conserved charges at the future
null infinity, called Newman-Penrose charges Newman:1968uj , which are present
in any asymptotically flat spacetime. The duality between both charges is
related to a conformal symmetry linking the RN metric to its transformation
under inversion of the radial coordinate, which converts the horizon into
future null infinity. Aside from this, the massless wave equation displays an
additional conformal symmetry on this background, called Couch-Torrence
symmetry couch , which allows a direct mapping between the scalar field near
the horizon and the scalar field at the future null infinity say.
Consequently, it is possible to map Aretakis charges to NP charges.

Most papers have explored these subjects in the context of four-dimensional
black holes; however, in Figueras:2008qh  the authors have treated such issues
in higher-dimensional black holes. On the other hand, the approach used there
does not apply in the context of extended objects as black $p$-branes since it
requires the assumption of the horizon should be compact, which is not the
case for higher-dimensional branes. Hence, a question should be raised: Are
Aretakis charges conserved on non-compact horizons? This study aims at to
addressing this question, which has not been explored in the literature. Our
main goal is to examine the existence of conserved charges on the horizon of
extremal non-dilatonic black branes. At this moment, it is worth calling
attention for some aspects in this scenario. The first one is that there is no
general Gaussian null coordinates prescription to describe extended objects in
a distinguishing way from black holes. In fact, such a coordinate system
becomes crucial in order to gather information near the horizon. Secondly, it
is well known that a large set of $p/D$-branes interpolates between
$AdS_{p+2}\times S^{D-(p+2)}$ geometry in the near-horizon limit and
$D$-dimensional Minkowski space in the asymptotic limit Gibbons:1993sv .

It has been shown in Gibbons:1994vm  that extremal dilatonic black holes can
be uplifted to extremal non-dilatonic black $p$-branes. This relation between
both allows us to resolve in many cases of interest the problem of the metric
and dilaton blowing up at the Killing horizon by just uplifting the Einstein-
Maxwell-dilaton theory in four dimensions to the Einstein-Maxwell one in
higher dimensions. This was first explored in Gibbons:1993sv , where the
authors have considered the asymptotic behavior of the metric and dilaton near
the horizon. As aforementioned, the asymptotic metric of the extremal black
$p$-brane near the horizon is just a product of a lower-dimensional anti de-
Sitter space with a sphere, and upon double-dimensional reduction it reduces
to a conformally $AdS_{2}\times S^{2}$ metric of the Einstein-Maxwell-dilaton
theory, describing the near-horizon of extremal dilatonic black holes with the
dilaton coupling being $a=\sqrt{\dfrac{p}{p+2}}$. Furthermore, the holographic
description of such near-horizon geometries upon Kaluza-Klein reduction on the
compact space was laid out in Maldacena:1997re ; Witten:1998qj ;
Klebanov:1999tb .

Our goal in this work will be to provide a new avenue for testing black
$p$-branes instabilities. We start our analysis by perturbing an extremal non-
dilatonic black $p$-brane in the near-horizon limit with a massless scalar
field. Such a scalar field is dealt with as a probe that does not have enough
energy to warp the space-time geometry. Therefore, the problem reduces to
solving the massless Klein-Gordon equation in the near-horizon background. We
will proceed with a coordinate transformation of the background in terms of
Gaussian null coordinates, which allow a non-singular description of the probe
scalar field at the horizon. Upon Kaluza-Klein (KK) reduction on a respective
sphere, the problem reduces to examining the bulk massive wave equation on
$AdS_{p+2}$ space by imposing boundary conditions. However, as we will see, by
writing the background through Gaussian null coordinates, $AdS_{p+2}$ can be
viewed as a conformal compactification of bi-dimensional anti de-Sitter space
with a $p$-sphere, $AdS_{2}\times S^{p}$. Next, we shall discuss the bulk
solutions through expanding the scalar field in modes, namely, normalizable
and non-normalizable ones. From the AdS/CFT point of view, the non-
normalizable modes are sources for the CFT operator living on the boundary of
$AdS_{p+2}$ while normalizable modes vanish on the boundary. We will restrict
ourselves to examine the normalizable modes which means that the boundary
effects will drop out.

This paper is organized as follows. In section 2 we give an overview on
extremal black $p$-branes from effective string theory. In section 3, we focus
on non-dilatonic extremal black $p$-branes and we express their near-horizon
geometries in Gaussian null coordinates, which provide a regular coordinate
chart on the horizon. In subsection 3.1, we probe the near-horizon geometry of
the brane with a scalar field. Next, we expand the scalar field in modes
fulfilling the isometries of the near horizon geometry. In subsection 3.2 we
find explicit the Aretakis quantities on the horizon. In subsection 3.3, we
examine the late time behavior of the scalar field in the $AdS_{2}$ section.
In Appendix A we solve the bulk massless Klein-Gordon equation in Gaussian
null coordinates taking into account the isometries of the near-horizon
geometry. We find all regular solutions satisfying the boundary conditions,
namely, normalizable and non-normalizable modes. Finally, In Appendix B we
display the quantization conditions to get regular solutions on the boundary
of $AdS_{p+2}$.

##  2 Extremal black p-brane ansatz

Our starting point will be the relevant bosonic part of the $D$-dimensional
effective Lagrangian arising from string theory Stelle:1996tz :

$\mathcal{L}=e\left(R-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{2(d+1)!}e^{a\phi}F_{d+1}^{2}\right),$
(1)

where $\phi$ is the dilaton, $F_{d+1}$ is the $d+1$-form field strength
corresponding to the $(d)$-form gauge field $A_{d}$, thus $F_{d+1}=dA_{d}$.
The parameter $a$ controls the interaction between the dilaton and the field
strength.

The equations of motion obtained varying Eq. (1) with respect to the dynamical
fields are

$\begin{split}&R_{MN}=\frac{1}{2}\partial_{M}\phi\partial_{N}\phi+\frac{1}{2(d)!}e^{a\phi}\big{(}F_{M...}F^{...}_{N}-\frac{d}{(d+1)(D-2)}g_{MN}F^{2}\big{)},\\\
&\nabla_{K}(e^{a\phi}F^{KM_{1}...M_{N}})=0,\\\
&\square\phi=\frac{a}{2(d+1)!}e^{a\phi}F^{2}.\end{split}$ (2)

The equations of motion can be easily solved after requiring symmetries of the
metric with or without preserving some supersymmetries. In our case we require
the metric ansatz to possess a
$(\mbox{Poincar\'{e}})_{p+1}\times\mbox{SO}(D-(p+1)$) Duff:1996hp ; Lu:1995yn
symmetry. This leads to the spacetime which can be interpreted as a
$(p+1)$-dimensional hyperplane embedded in the whole $D$-dimensional manifold.
A suitable chart covering the whole spacetime may be chosen through splitting
the coordinates into two components, namely, $x^{M}=(x^{\mu},y^{m})$, where
the Greek letters label coordinates which are “parallel” to the worldvolume,
$(\mu=0,...,p=d-1)$, while the lowercase Latin letters label coordinates which
are “transverse” to the worldvolume $(m=p+1,...,D-1)$. In this notation, the
ansatz for the metric takes the form

$ds^{2}=e^{A(r)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{B(r)}dy^{m}dy^{m},$ (3)

where $\eta_{\mu\nu}$ is the $(p+1)$-dimensional Minkowski metric, and
$r=\sqrt{y^{m}y^{m}}$ is the radial coordinate in the transverse space. The
former ansatz only deals with extremal black $p$-branes, which will be
sufficient for our purposes in this work. The ansatz is
$(\mbox{Poincar\'{e}})_{p+1}\times\mbox{SO}(D-(p+1)$) invariant because the
metric functions depend only on $r$. The corresponding ansatz for the dilaton
and the gauge field requiring
$(\mbox{Poincar\'{e}})_{p+1}\times\mbox{SO}(D-(p+1)$) symmetry takes the form

$\begin{split}&\phi=\phi(r),\\\
&F_{mM_{1}...M_{p+1}}=\epsilon_{M_{1}...M_{p+1}}\partial_{m}e^{C(r)}\,\,\,\mbox{or}\,\,\,F_{N_{1}...N_{p+2}}=\epsilon_{N_{1}...N_{p+2}m}\frac{y^{m}}{r^{p+3}},\end{split}$
(4)

where $\epsilon_{M_{1}...M_{p+1}}$ is the volume $p$-form and the first and
second field strengths describe the elementary $p$-brane and solitonic
$p$-brane (dual to the elementary case), respectively.

Substituting the ansatz in the field equations, one finds

$\displaystyle
ds^{2}=H^{\frac{-4\tilde{d}}{\Delta(D-2)}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H^{\frac{4d}{\Delta(D-2)}}dy^{i}dy^{i},$
(5) $\displaystyle e^{\phi}=H^{\frac{2a}{\xi\Delta}},$ (6) $\displaystyle
H(r)=1+\frac{k}{r^{\tilde{d}}},$ (7)

where $\tilde{d}=D-d-2$ and $\xi=+1$ or $-1$ for the elementary and solitonic
cases, respectively. For the explicit definitions of the function $C(r)$ and
the parameter $k$, see Lu:1995yn . The parameter $\Delta$ is related to the
prefactor $a$ through the relation

$a^{2}=\Delta-\frac{2d\tilde{d}}{D-2}.$ (8)

In order to get supersymmetric black $p$-branes one requires that the number
$\Delta$ is related to the number of preserving supersymmetries $N$ by
$N=\dfrac{4}{\Delta}$. Physically speaking, $N$ is an integer number
corresponding to the number of strength fields were put in a compact form
displayed in Eq.(1) once a particular basis has been chosen Klebanov:1996un .
In spite of this, black $p$-branes can exist for arbitrary $N$, even for non-
integer values of $N$ which correspond to non-supersymmetric black $p$-branes.
In next section we shall focus on a particular kind of black $p$-brane; non-
dilatonic ($a=0$).

##  3 The near-horizon limit

The aim of this section is to find a near-horizon description for extremal
non-dilatonic black branes, not necessarily supersymmetric since we are
considering arbitrary $N$. We concentrate our discussion on a particular sort
of extremal non-dilatonic black $p$-branes discussed in Gibbons:1994vm  which
are obtained by setting the dilaton to be zero, $a=0$, and $D=p+4$. Such a
class of extremal black $p$-branes are interesting because they reduce upon
dimensional reduction to extremal dilatonic black holes in four dimensions
Gibbons:1994vm . In the $p$-branes of our interest the metric and the field
strength take respectively the form:

$ds^{2}=\bigg{(}1+\frac{k}{r}\bigg{)}^{-\frac{2}{p+1}}\bigg{[}(-dt^{2}+dx^{a}dx^{a})+\bigg{(}1+\frac{k}{r}\bigg{)}^{\frac{2(p+2)}{p+1}}(dr^{2}+r^{2}d\Omega^{2}_{2})\bigg{]},$
(9)
$F_{rM_{1}...M_{p+1}}=-\frac{2}{\sqrt{2}}\sqrt{\frac{p+2}{p+1}}\epsilon_{M_{1}...M_{p+1}}\frac{k}{r^{2}},$
(10)

where $a=1,...,p$. Note that the degenerate Killing horizon is located at
$r=0$. Furthermore, the metric can be rewritten in Schwarzschild-like form by
identifying $r=\tilde{r}-k$:

$ds^{2}=\bigg{(}1-\frac{k}{\tilde{r}}\bigg{)}^{\frac{2}{p+1}}(-dt^{2}+dx^{a}dx^{a})+\bigg{(}1-\frac{k}{\tilde{r}}\bigg{)}^{-2}d\tilde{r}^{2}+\tilde{r}^{2}d\Omega^{2}_{2}.$
(11)

The horizon is now located at $\tilde{r}=k$. In particular, the case $p=0$
recovers the extremal Reissner-Nordström solution in GR. Henceforth, we can
use the metric in the form (9).

As aforementioned the extremal non-dilatonic black $p$-brane considered above
can be viewed as an extremal dilatonic black hole in four dimensions. Upon
carrying out the double-dimensional reduction from $D=p+4$ to $D=4$, one finds
a Lagrangian similar to Eq.(1) in four dimensions whose solution describes an
extremal magnetically-charged black hole Garfinkle:1990qj  given by

$\begin{split}ds^{2}_{4}&=-\left(1+\frac{k}{r}\right)^{-\frac{p+2}{p+1}}dt^{2}+\left(1+\frac{k}{r}\right)^{\frac{p+2}{p+1}}\left(dr^{2}+r^{2}d\Omega^{2}_{2}\right);\\\
e^{a\phi}&=\left(1+\frac{k}{r}\right)^{\frac{p}{2(p+1)}};\\\
F_{2}&=Q\epsilon_{2},\end{split}$ (12)

where the dilaton coupling takes a particular value $a^{2}=\dfrac{p}{(p+2)}$
and $\epsilon_{2}$ is the volume two-form. This setup mimics a non-dilatonic
black $p$-brane as embedding in $(p+4)$-dimensional space-time. Moreover, the
solution (12) has a BPS bound which relates the ADM mass $(M)$ and the charge
$(Q)$ of the black hole to the constant $k$ by the equation

$\frac{p+2}{2(p+1)}k=M\geq\sqrt{\frac{p+2}{2(p+1)}}Q.$ (13)

Note that in the non-dilatonic black $p$-brane description, the ADM mass and
charge should be taken into account per unit volume of $p$-branes instead. For
our purposes we have found more convenient to work with the extremal non-
dilatonic black $p$-brane description since it has a simple near-horizon
geometry, which is not the case for the extremal dilatonic black hole
Gibbons:1994vm ; Gibbons:1993sv . On the face of it, both metrics are related
to each other by

$ds^{2}_{p+4}=\left(1+\frac{k}{r}\right)^{-\frac{2}{p+1}}dx^{a}dx^{a}+\left(1+\frac{k}{r}\right)^{\frac{p}{p+1}}ds^{2}_{4},$
(14)

where $ds^{2}_{p+4}$ is the extremal non-dilatonic black $p$-brane space-time
(9).

In order to examine the behavior of the metric (9) near the horizon, one needs
a non-singular near-horizon description of it. In the limit $r\rightarrow 0$
it is easily seen that the near-horizon geometry of (9) is given by
$AdS_{p+2}\times S^{2}$. More explicitly, let us perform the following
coordinate transformation:
$\bigg{(}\dfrac{k}{r}\bigg{)}^{-\frac{1}{p+1}}=\lambda$. As a result,
$\lambda\rightarrow 0$ near the horizon. Thus, the near-horizon metric takes
the form

$ds^{2}_{NH}=\lambda^{2}\big{[}(-dt^{2}+dx^{a}dx^{a})\big{]}+\frac{k^{2}(p+1)^{2}}{\lambda^{2}}d\lambda^{2}+k^{2}d\Omega^{2}_{2},$
(15)

which can be rewritten in a more familiar form after one more coordinate
transformation $\lambda^{\prime}=(p+1)k\lambda$ and defining
$k^{\prime}=k(p+1)$, so the metric takes the form

$ds^{2}_{NH}=\bigg{(}\frac{\lambda^{\prime}}{k^{\prime}}\bigg{)}^{2}\bigg{[}(-dt^{2}+dx^{a}dx^{a})\bigg{]}+\bigg{(}\frac{k^{\prime}}{\lambda^{\prime}}\bigg{)}^{2}d\lambda^{\prime
2}+k^{2}d\Omega^{2}_{2}.$ (16)

Additionally, by taking the following coordinate transformation
$\tau=\dfrac{k^{\prime 2}}{\lambda^{\prime}}$ with $\tau>0$, we get

$ds^{2}_{NH}=\frac{k^{\prime
2}}{\tau^{2}}\left[-dt^{2}+dx^{a}dx^{a}+d\tau^{2}\right]+k^{2}d\Omega^{2}_{2},$
(17)

which is a product of $AdS_{p+2}$ with $S^{2}$. In the former equation,
$AdS_{p+2}$ is represented in terms of the upper half plane representation.
The $AdS$ radius is $k^{\prime}$ which in turn depends only on $k$ and the
dimension of the space-time. The metric (9) is characterized by two distinct
asymptotic behaviors for large $r$ and near the horizon: it interpolates
between the $D$-dimensional flat space and $AdS_{p+2}\times S^{2}$, when $r$
goes to infinity and in the near-horizon limit, respectively. A further
problem comes up since the metric in any of the coordinate systems introduced
above is singular at the horizon $\tau=+\infty$ ($\lambda^{\prime}=0$) and
therefore none of these coordinates are appropriate to describe the near-
horizon geometry.

To get a complete non-singular metric on the horizon, we should describe the
near-horizon metric in Gaussian null coordinates Kunduri:2007vf , assuming
that the new coordinates in the neighborhood on the horizon are
$(v,r,y^{b},\Omega_{2})$, with ($b=1,...,p$) parameterizes a particular
$p$-dimensional spatial section of the full space. They are related to the
Poincaré coordinates, $(t,\lambda^{\prime},x^{a},\Omega_{2})$, explicitly:

$\lambda^{\prime}=r\cosh(\eta),\,\,\,t=\bigg{(}v+\frac{1}{r}\bigg{)}k^{\prime
2},\,\,\,x^{a}=\frac{\mu^{a}\tanh(\eta)}{r}k^{\prime 2},$ (18)

where $0\leq\eta=y^{(1)}<+\infty$, $v$ is an ingoing (retarded) null
coordinate and $\mu^{a}$ parametrize a sphere, $S^{p}$, $\mu^{a}\mu^{a}=1$. In
terms of the new coordinates the metric takes the form

$ds^{2}_{NH}=k^{\prime
2}\bigg{[}\cosh^{2}(\eta)\bigg{(}-r^{2}dv^{2}+2dvdr\bigg{)}+d\eta^{2}+\sinh^{2}(\eta)d\Omega^{2}_{p-1}\bigg{]}+k^{2}d\Omega^{2}_{2},$
(19)

which obviously describes the space $AdS_{p+2}\times S^{2}$. In fact, the
metric in Gaussian null coordinate is completely non-singular at the horizon
$r=0$, and is extended to the region behind the horizon through the reflection
invariance of the full metric

$\begin{split}r&\rightarrow-r;\\\ v&\rightarrow-v.\end{split}$ (20)

We now observe that the former metric can be seen as a warped product of
$AdS_{2}$ with a $p$-dimensional hyperbolic space $\mathcal{H}^{p}$ plus
$S^{2}$. It is worth pointing out that Eq. (19) is singular at $\eta=+\infty$.
However, let us take the coordinate transformation $\tan(\alpha)=\sinh(\eta)$
in Eq. (19) with $0\leq\theta<\pi/2$. We get

$ds^{2}_{NH}=\frac{k^{\prime
2}}{\cos^{2}(\alpha)}\bigg{[}\bigg{(}-r^{2}dv^{2}+2dvdr\bigg{)}+d\alpha^{2}+\sin^{2}(\alpha)d\Omega^{2}_{p-1}\bigg{]}+k^{2}d\Omega^{2}_{2},$
(21)

where the geometry is clearly $AdS_{p+2}\times S^{2}$ as expected. However,
now the first term in the former metric is can be viewed as a conformal
compactification of a lower-dimensional anti-de Sitter space, $AdS_{2}$, times
a $p$-sphere, $S^{p}$. The conformal boundary of $AdS_{p+2}$ is defined by the
divergence of the conformal factor $\frac{k^{\prime 2}}{\cos^{2}(\alpha)}$ at
$\alpha=\frac{\pi}{2}$. In Gaussian null coordinates the conformal boundary
takes the form $AdS_{2}\times S^{p-1}$, which is isometric to the
$p+1$-dimensional Minkowski space. The $AdS_{p+2}$ topology in Gaussian null
coordinates is schematically presented in Fig.(1).

It is interesting to note that the near-horizon geometry of the extremal non-
dilatonic black $p$-brane written in the form given by Eq.(21) shows
“external” and “internal” structures. Looking at Fig.(1), the external
structure displays the behavior of $AdS_{p+2}$ in terms of $p$-extra spatial
dimensions which has its own boundary attained at $\alpha=\pi/2$. It turns out
that the vertical axis in Fig.(1) represents the $AdS_{2}$, a section of the
thorough space. Thus, the internal structure of the $AdS_{2}$ section, which
has its own horizon located at $r=0$, is warped by the $\alpha$-coordinate as
we have said previously. We are dealing in this work with near-horizon
geometries, so it is enough to restrict our analysis to $r\rightarrow 0$ .
From a physical perspective, $AdS_{p+2}$ is nothing more than a warped
geometry of $AdS_{2}\times S^{p}$: the $AdS_{p+2}$ radius is no longer a
constant, but depends on the radial coordinate of the compact space $S^{p-1}$
instead. This is not a novel property since many examples of warped geometries
arises from dimensional reduction in string theory and supergravity
Maldacena:1998uz ; vanNieuwenhuizen:1984ri . In particular, the warped
$AdS_{3}$ is built up as a Hopf fibration over $AdS_{2}$ Anninos:2008qb ;
Castro:2014ima . Hence, the near-horizon geometry of extremal dilatonic black
holes Eq.(12) is reached upon the double-dimensional reduction of extremal
non-dilatonic black $p$-branes near the horizon by using Eq.(14), and their
forms are conformal deformations of $AdS_{2}\times S^{2}$. More explicitly, by
doing this we find

$ds^{2}_{NH}=\Omega(r)\left[k^{\prime
2}\left(-r^{2}dv^{2}+2dvdr\right)+k^{2}d\Omega_{2}^{2}\right],$ (22)

where the conformal factor is
$\Omega(r)=\left(\dfrac{r}{k^{\prime}}\right)^{p}$ and the metric inside the
brackets is $AdS_{2}\times S^{2}$. For example, if we take $p=0$ we recover
the near-horizon geometry of the extremal RN black hole which is just
$AdS_{2}\times S^{2}$, though for $p\neq 0$ the remaining geometry is a
deformation of $AdS_{2}\times S^{2}$ as we have seen by re-interpreting the
black hole as a black $p$-brane in higher dimensions.

Figure 1:  The figure displays the topology of $AdS_{p+2}$. The circles stand
for $S^{p-1}$-spheres whose radii are $\alpha$ and the boundary is located at
$\alpha=\pi/2$ axis. The vertical axis consists of copies of $AdS_{2}$.

It is convenient to rewrite the metric (21) in a more compact fashion:

$\begin{split}ds^{2}_{NH}&=\frac{k^{\prime
2}}{\cos^{2}(\alpha)}\bigg{[}\bigg{(}-r^{2}dv^{2}+2dvdr\bigg{)}+d\Omega^{2}_{p}\bigg{]}+\gamma_{IJ}dx^{I}dx^{J}\\\
&=\frac{k^{\prime
2}}{\cos^{2}(\alpha)}\bigg{[}\bigg{(}-r^{2}dv^{2}+2dvdr\bigg{)}+\omega_{ab}dy^{a}dy^{b}\bigg{]}+\gamma_{IJ}dx^{I}dx^{J},\end{split}$
(23)

where $d\Omega^{2}_{p}=d\alpha^{2}+\sin^{2}(\alpha)d\Omega^{2}_{p-1}$ and,
$\omega_{ab}$ and $\gamma_{IJ}$ represent the $p$-dimensional spherical metric
and $2$-dimensional metric, respectively. The horizon is a degenerate Killing
horizon which is a null hypersurface defined by the vanishing of the norm of
the Killing vector $\dfrac{\partial}{\partial v}$. At the horizon the induced
metric looks like

$ds^{2}_{H}=\frac{k^{\prime
2}}{\cos^{2}(\alpha)}\omega_{ab}dy^{a}dy^{b}+\gamma_{IJ}dx^{I}dx^{J},$ (24)

where the first term represents a $p$-dimensional hyperbolic manifold
$\mathcal{H}^{p}$ and the second one stands for a $S^{2}$. So, black $p$-brane
horizons are non-compact surfaces.

$AdS$ spaces with a conformal boundary are not globally hyperbolic
Ishibashi:2003jd ; Ishibashi:2004wx . Therefore in order to define a well-
posed Cauchy problem, besides of picking adequate initial data, it is
necessary to impose suitable boundary conditions at the conformal boundary to
guarantee smooth solutions of the Klein-Gordon and/or Dirac equations in the
bulk. In the next section we shall examine a scalar field on this background
prescribing further boundary conditions.

###  3.1 Scalar field

In this section, we consider a probe massless scalar field minimally coupled
to gravity for the metric given by Eq. (23). The field action is given by

$S[\psi]=\frac{1}{2}\int
d^{D}x\sqrt{-g}\left\\{\partial_{A}\psi\partial^{A}\psi\right\\}.$ (25)

Since the $AdS$ space is not compact ($AdS$ has an infinite volume), the
action in principle diverges. So, in order to find finite physical quantities
one needs a regularization of the volume to avoid divergent terms
Polchinski:2010hw , as we shall see below. Varying the scalar field action
with respect to $\psi$, we have

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int
d^{p+1}y\int_{0}^{\pi/2-\epsilon}d\alpha\int
d^{2}x\sqrt{-g}\bigg{[}g^{AB}(\partial_{A}\psi)\partial_{B}\psi\bigg{]}$
$\displaystyle=$ $\displaystyle\int d^{2}x\int
d^{p+1}y\int_{0}^{\pi/2-\epsilon}d\alpha\left\\{\partial_{A}\bigg{(}\sqrt{-g}g^{AB}\partial_{B}\psi\delta\psi\bigg{)}-\bigg{[}\partial_{A}\bigg{(}\sqrt{-g}g^{AB}\partial_{B}\psi\bigg{)}\bigg{]}\delta\psi\right\\},$

where we have used $\epsilon$ as a cutoff parameter to regularize the infinite
volume of $AdS_{p+2}$ and integrated by parts from the first to second line.
Apart from this, the bulk has been restricted temporarily to the region
$0\leq\alpha\leq\ \pi/2-\epsilon$. Expanding all terms in Eq. (3.1), we arrive
at

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int
d^{2}x\int_{AdS_{p+2}}d^{p+2}y\bigg{[}-\left\\{\frac{1}{\sqrt{-g}}\partial_{A}\bigg{(}\sqrt{-g}g^{AB}\partial_{B}\psi\bigg{)}\right\\}\sqrt{-g}\delta\psi\bigg{]}+$
(27) $\displaystyle+$ $\displaystyle\int d^{2}x\int
d^{p+1}y\sqrt{-h}\,n^{\alpha}\partial_{\alpha}\psi\,\delta\psi\Big{|}_{\alpha=0}^{\alpha=\pi/2-\epsilon},$

where $h$ is the determinant of the induced metric $h_{\mu\nu}$ defined at the
boundary, $n^{\alpha}$ is a unit normal vector to the boundary, and the first
integral is performed in the bulk whilst the second integral is performed
between the initial and end points of $\alpha$. The first term in Eq.(27) is
proportional to the Klein-Gordon equation, and the second one is a boundary
term. As usual, in order to get a well-defined variation principle, the
boundary term must vanish for variations obeying special boundary conditions.
Afterwards, the problem reduces to solving the Klein-Gordon equation in the
bulk for particular boundary conditions which enforce the boundary term to
vanish.

Now, let us solve the Klein-Gordon equation on the background given by Eq.
(23). To start, we write

$\square\psi=\frac{1}{\sqrt{-g}}\partial_{M}\left(\sqrt{-g}g^{MN}\partial_{N}\psi\right)=0,$
(28)

where
$\sqrt{-g}=\bigg{(}\dfrac{k^{\prime}}{\cos\alpha}\bigg{)}^{p+2}\sqrt{\omega}\sqrt{\gamma}$.
The explicit form of the metric is as follows:

$\displaystyle(g_{MN})=\left(\begin{array}[]{cccc}-\dfrac{r^{2}k^{\prime
2}}{\cos^{2}(\alpha)}&\dfrac{k^{\prime 2}}{\cos^{2}(\alpha)}&0&0\\\
\dfrac{k^{\prime 2}}{\cos^{2}(\alpha)}&0&0&0\\\ 0&0&\dfrac{k^{\prime
2}}{\cos^{2}(\alpha)}\omega_{ab}&0\\\ 0&0&0&\gamma_{IJ},\end{array}\right),$
(33)

$\displaystyle(g^{MN})=\left(\begin{array}[]{cccc}0&\dfrac{\cos^{2}(\alpha)}{k^{\prime
2}}&0&0\\\ \dfrac{\cos^{2}(\alpha)}{k^{\prime
2}}&\dfrac{r^{2}\cos^{2}(\alpha)}{k^{\prime 2}}&0&0\\\
0&0&\dfrac{\cos^{2}(\alpha)}{k^{\prime 2}}\omega^{ab}&0\\\
0&0&0&\gamma^{IJ},\end{array}\right),$ (38)

where $\omega^{ab}$ and $\gamma^{IJ}$ represent the inverse of $\omega_{ab}$
and $\gamma_{IJ}$, respectively. As we noted earlier, our aim is to find
possible conserved charges in the near-horizon limit. Therefore, Eq. (28)
reduces to

$\begin{split}0&=\frac{1}{\cos^{(p)}(\alpha)}\sqrt{\omega}\sqrt{\gamma}2\partial_{v}\left[\partial_{r}\psi\right]+\frac{1}{\cos^{(p)}(\alpha)}\sqrt{\omega}\sqrt{\gamma}\partial_{r}\left[r^{2}\partial_{r}\psi\right]\\\
&+\sqrt{\gamma}\partial_{a}\left[\frac{1}{\cos^{(p)}(\alpha)}\sqrt{\omega}\omega^{ab}\,\partial_{b}\psi\right]+\frac{k^{\prime
2}}{\cos^{(p+2)}(\alpha)}\sqrt{\omega}\,\partial_{I}\left[\sqrt{\gamma}\gamma^{IJ}\partial_{J}\psi\right],\end{split}$
(39)

or, more explicitly,

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}2\sqrt{\omega_{(p-1)}}\sqrt{\gamma}\partial_{v}\left[\partial_{r}\psi\right]+\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\sqrt{\omega_{(p-1)}}\sqrt{\gamma}\partial_{r}\left[r^{2}\partial_{r}\psi\right]$
(40) $\displaystyle+$
$\displaystyle\sqrt{\omega_{(p-1)}}\sqrt{\gamma}\partial_{\alpha}\left[\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\partial_{\alpha}\psi\right]+k^{\prime
2}\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p+2)}(\alpha)}\sqrt{\omega_{(p-1)}}\,\partial_{I}\left[\sqrt{\gamma}\gamma^{IJ}\partial_{J}\psi\right]$
$\displaystyle+$
$\displaystyle\frac{\sin^{(p-3)}(\alpha)}{\cos^{(p)}(\alpha)}\sqrt{\gamma}\,\partial_{i}\left[\sqrt{\omega_{(p-1)}}\omega^{ij}_{(p-1)}\partial_{j}\psi\right],$

where $\omega_{(p-1)}$ is the determinant of the $(p-1)$-dimensional spherical
metric $\omega^{ij}_{(p-1)}$, and $i,j=1...p$. We propose as an ansatz the
following expansion in modes:

$\psi=\sum_{l,l^{\prime},m}\Phi_{m}(v,r)Y_{l^{\prime}}(\Omega_{(p-1)})Y_{l}(\Omega_{2})\phi_{l^{\prime}m}(\alpha),$
(41)

where $Y_{l}(\Omega_{2})$ and $Y_{l^{\prime}}(\Omega_{(p-1)})\equiv
Y_{l_{1},...,l_{p-1}}(\Omega_{(p-1)})$ are the spherical harmonics on $S^{2}$
and $S^{(p-1)}$ , their quantum numbers being respectively $l$, $l^{\prime}$.
More generically, $l_{1},...,l_{p-1}$ are integer numbers satisfying the
following condition:

$l_{p-1}\geq l_{p-2}\geq...\geq l_{2}\geq|l_{1}|.$ (42)

The spherical harmonics obey the equations Ammon:2015wua :

$\displaystyle\square_{S^{2}}Y_{l}(\Omega_{2})$ $\displaystyle=$
$\displaystyle-l(l+1)Y_{l}(\Omega_{2}),$ (43)
$\displaystyle\square_{S^{(p-1)}}Y_{l^{\prime}}(\Omega_{(p-1)})$
$\displaystyle=$
$\displaystyle-l^{\prime}(l^{\prime}+p-2)Y_{l^{\prime}}(\Omega_{(p-1)}),$ (44)

where we have used the shorthand notation
$\sum_{l^{\prime}}\equiv\sum_{l_{p-1}=1}^{\infty}\sum_{l_{p-2}=1}^{l_{p-1}}...\sum_{l_{1}=-l_{2}}^{l_{2}}$.
The quantum number $m$ stems from the fulfillment of regularity conditions of
the radial solutions ($\phi(\alpha)$) at the origin and boundary (see Appendix
A); in other words, $m$ is the radial quantum number.

Substituting Eqs. (43-44-41) into Eq. (40), we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\frac{\sqrt{\omega_{(p-1)}}\sqrt{\gamma}}{\Phi(v,r)}\left[2\partial_{v}\partial_{r}\Phi(v,r)+\partial_{r}\left(r^{2}\partial_{r}\Phi(v,r)\right)\right]+$
(45) $\displaystyle+$
$\displaystyle\sqrt{\omega_{(p-1)}}\sqrt{\gamma}\frac{1}{\phi(\alpha)}\partial_{\alpha}\left[\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\partial_{\alpha}\phi(\alpha)\right]-m^{2}_{0}\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p+2)}(\alpha)}\sqrt{\omega_{(p-1)}}\sqrt{\gamma}+$
$\displaystyle+$
$\displaystyle\frac{\sin^{(p-3)}(\alpha)}{\cos^{(p)}(\alpha)}\sqrt{\omega_{(p-1)}}\sqrt{\gamma}\,l^{\prime}(l^{\prime}+p-2),$

where we defined a “mass" term222Note that this term could be negative.,
$m^{2}_{0}=(p+1)^{2}l(l+1)$, and omitted the index of the functions for
convenience. Notice that this term comes up after Kaluza-Klein (KK) reduction
on $S^{2}$. The former equation can be split into two decoupled differential
equations: the first one, a ordinary differential equation for the radial
function and the second, a partial differential equation for the
$AdS_{2}$-section function:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\alpha}\left[\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\partial_{\alpha}\phi(\alpha)\right]+\bigg{[}\xi^{2}\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}-m^{2}_{0}\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p+2)}(\alpha)}-$
(46) $\displaystyle-$ $\displaystyle
l^{\prime}(l^{\prime}+p-2)\frac{\sin^{(p-3)}(\alpha)}{\cos^{p}(\alpha)}\bigg{]}\phi(\alpha),$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\partial_{v}\left[\partial_{r}\Phi(v,r)\right]+\partial_{r}\left[r^{2}\partial_{r}\Phi(v,r)\right]-\xi^{2}\Phi(v,r),$
(47)

with $\xi^{2}$ a separation constant. The explicit solutions for the radial
equation are displayed in Appendix A. Here it should be stressed that
$\xi^{2}$ (see Appendix B) is a combination of the set of quantum numbers
$\left\\{l,l^{\prime},m\right\\}$. As discussed in Balasubramanian:1998sn  for
$AdS$ spaces, the general solution for the scalar field is suitably split into
two parts: the normalizable and non-normalizable modes, such modes are
explicitly obtained in Appendix A for $AdS$ in Gaussian null coordinates. The
$AdS_{2}$-section equation is discussed in more details in Section 3.3.

###  3.2 Conserved charge on the horizon

In this subsection we shall discuss the existence of conserved charges on the
horizon of black branes. Similarly to the context of black holes in GR where
conserved charges on the horizon have been already obtained in Bizon:2012we ;
Godazgar:2017igz , it is reasonable to expect such conserved charges to hold
for black branes as well. We start by considering the wave equation 40 along
with the ansatz Eq.(41). Upon integration over the compact manifolds $S^{2}$
and $S^{p-1}$, the last two terms in it vanish as a consequence of spherical
harmonics properties. Now, integrating the remaining equation over the “extra”
dimension $\alpha$ and evaluating at the horizon, we obtain

$0=\int dV_{\mathcal{H}^{p}}\int
d^{2}x\,\sqrt{\gamma}\bigg{[}2\partial_{v}\partial_{r}\psi+\mathcal{A}\psi\bigg{]},$
(48)

where $V_{\mathcal{H}^{p}}$ stands for the volume of the $p$-dimensional
hyperboloid and $\mathcal{A}$ is an operator defined by:

$\mathcal{A}=\frac{\cos^{(p)}(\alpha)}{\sin^{(p-1)}(\alpha)}\partial_{\alpha}\bigg{(}\frac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}\partial_{\alpha}\bigg{)}.$
(49)

Note that the Eq.(48) is integrated over the horizon as one can see looking at
the Eq.(24). Recalling that the horizon is a hyperbolic space plus a sphere,
we shall proceed with a regularization scheme since the horizon is non-
compact. The regularized volume of the hyperboloid may be written in terms of
the volume of the $(p-1)$-dimensional sphere Maldacena:2012xp , namely,
$V_{\mathcal{H}^{p}}^{\mbox{reg.}}=V_{S^{p-1}}\int_{0}^{\pi/2-\epsilon}d\alpha\,\dfrac{\sin^{(p-1)}(\alpha)}{\cos^{(p)}(\alpha)}$,
with $\epsilon$ is a cutoff introduced in order to get finite results.
Introducing for convenience the shorthand notation
$\psi=\sum\Psi(X)\phi(\alpha)$ (we are omitting the indices in the sum), with
$X$ standing for the other coordinates, it is possible to carry out the
integration over $\alpha$ by using the asymptotic behavior of bulk modes. An
in-depth assessment of bulk modes of the wave equation in Gaussian null
coordinates is provided in Appendices A-B. Firstly, we shall explicit the bulk
modes for a particular case where the parameter $\nu$ is a non-integer
number333See Appendix A for the definition of $\nu$.. In this case they are
expanded in terms of hypergeometric functions

$\displaystyle\phi(\alpha)$ $\displaystyle=$ $\displaystyle
C\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{+}}(\alpha)\,{{}_{2}}F_{1}\left(e_{+},f_{+};(1+\nu);\cos^{2}(\alpha)\right)+$
(50) $\displaystyle+$ $\displaystyle
D\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{-}}(\alpha)\,{{}_{2}}F_{1}\left(e_{-},f_{-};(1-\nu);\cos^{2}(\alpha)\right),$

where all the constants are defined in Appendix A. As well-known the
hypergeometric functions form representations of the conformal group
$SL(2,\mathbb{R})$. As a consequence, we can truncate the series to the
leading terms near the boundary, they fall off as
$\cos^{2\lambda_{\pm}}(\alpha)$. Therefore, the full solution satisfies the
following asymptotic behavior,

$\psi(X,\alpha)\sim\sum_{\left\\{l,l^{\prime},m\right\\}}\Psi(X)\cos^{2\lambda_{+}}(\alpha)+\gamma(X)\cos^{2\lambda_{-}}(\alpha),$
(51)

near the boundary. In order to have well-behaved solutions at the boundary one
needs to impose boundary conditions (as displayed in Eqs.(117, 118)) which
lead to $\Psi(X)$ and $\gamma(X)$ modes to be quantized. The standard
quantization condition corresponds to fixing $\gamma(X)=0$ or, equivalently,
to taking $D=0$ as discussed in Appendix A and B. As a result, $\Psi(X)$ are
normalizable quantized modes satisfying the quantization condition (117). On
the other hand, there is an alternative quantization condition which
corresponds to fixing $\Psi(X)=0$ or, equivalently, to picking $C=0$ as
discussed in Appendix A and B. In this case, $\gamma(X)$ are non-normalizable
modes satisfying the quantization condition (118). Of course, the above
analysis only makes sense for $\nu>1$; otherwise, both modes are normalizable
and a more careful analysis must be done to choose the fluctuation as pointed
out in Appendix B.

The solution (50) may be written in a more convenient way after applying the
quantization conditions $(\ref{QC1},\ref{QC2})$:

$\displaystyle\phi_{1}(\alpha)$ $\displaystyle=$ $\displaystyle
C\frac{m!}{\left(2\lambda_{+}-\frac{(p-1)}{2}\right)_{m}}\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{+}}(\alpha)P_{m}^{(2\lambda_{+}-\frac{(p+1)}{2},\frac{p}{2}+l^{\prime}-1)}(1-2\cos^{2}(\alpha))+$
(52) $\displaystyle+$ $\displaystyle
D\frac{m!}{\left(2\lambda_{-}-\frac{(p-1)}{2}\right)_{m}}\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{-}}(\alpha)P_{m}^{(2\lambda_{-}-\frac{(p+1)}{2},\frac{p}{2}+l^{\prime}-1)}(1-2\cos^{2}(\alpha))$

where $P_{m}^{(a,b)}(x)$ are Jacobi polynomials, the quantity $(a)_{m}$ is
defined in Eq.(103) and, $C$ and $D$ are constants given by Eqs.(109,110). In
addition, $\lambda_{\pm}$ are the roots given by Eq.(94) defined from the
hypergeometric equation (90). Hence, in this case, the full solution may be
conveniently written in a more appropriate form as follows:

$\phi_{1}(\alpha)=C^{(+)}\phi^{(+)}(\alpha)+C^{(-)}\phi^{(-)}(\alpha),$ (53)

where

$\begin{split}C^{(+)}&=C\frac{m!}{\left(2\lambda_{+}-\frac{(p-1)}{2}\right)_{m}},\\\
C^{(-)}&=D\frac{m!}{\left(2\lambda_{-}-\frac{(p-1)}{2}\right)_{m}},\\\
\phi_{\pm}(\alpha)&=\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{\pm}}(\alpha)P_{m}^{(2\lambda_{\pm}-\frac{(p+1)}{2},\frac{p}{2}+l^{\prime}-1)}(1-2\cos^{2}(\alpha)).\end{split}$
(54)

The function $\phi^{(+)}(\alpha)$ is used to build the pure normalizable modes
for all non-integer $\nu$ since the Jacobi polynomials are orthonormal the by
applying the inner product 88. The other function $\phi^{(-)}$ only can be
cast into pure normalizable modes for the non-integer $\nu$ within the range
$\nu<1$, in this way, the Jacobi polynomials can be made orthonormal within
this range.

Now let us consider to the other case which corresponds to taking $\nu$ to be
an integer number. In this situation the solution is given by:

$\displaystyle\phi_{1}(\alpha)$ $\displaystyle=$ $\displaystyle
D\sin(\alpha)^{l^{\prime}}\cos(\alpha)^{2\lambda_{-}}\sum_{k=0}^{\nu-1}\frac{\left(a-\nu\right)_{k}\left(b-\nu\right)_{k}}{k!\left(1-\nu\right)_{k}}\cos(\alpha)^{2k}-$
(55) $\displaystyle-$ $\displaystyle
C\sin(\alpha)^{l^{\prime}}\cos(\alpha)^{2\lambda_{+}}\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{k!(k+\nu)!}\cos(\alpha)^{2k}\bigg{[}\ln(\cos^{2}(\alpha))-$
$\displaystyle-$ $\displaystyle\psi(k+1)-\psi(k+\nu+1)+\psi\left(a+k\right)+$
$\displaystyle+$ $\displaystyle\psi\left(b+k\right)\bigg{]},$

where now $C$ and $D$ are constants given by Eqs.(113,114). As in the former
case we select the pure non-normalizable modes by taking $C=0$, which is the
quantization condition (118). These modes falls off as
$\cos^{2\lambda_{-}}(\alpha)$ near the boundary as expected to such modes. On
the other hand, the normalizable modes are picked by taking $D=0$ which is the
quantization condition (117). However, one must be careful to do so since for
these modes poles of the gamma function are involved. Upon imposing the
normalizable boundary conditions, we have:

$\displaystyle\phi_{1}(\alpha)$ $\displaystyle=$
$\displaystyle-\frac{(-1)^{\nu}\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(-m-\nu)\Gamma(\frac{p}{2}+l^{\prime}+m)}\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{+}}(\alpha)\sum_{k=0}^{\infty}\frac{\left(-m\right)_{k}\left(\frac{p}{2}+l^{\prime}+m+\nu\right)_{k}}{k!(k+\nu)!}\times$
$\displaystyle\times$
$\displaystyle\cos^{2k}(\alpha)\bigg{[}\ln\left(\cos^{2}(\alpha)\right)-\psi(k+1)-\psi(k+\nu+1)+\psi(-m+k)+\psi(b+k)\bigg{]},$

where we have used Eq.(113) for $C$. Note that the first gamma function in the
denominator has negative integer numbers in its argument; at these points the
reciprocal gamma function is zero, i.e., $\dfrac{1}{\Gamma(-m-\nu)}=0$. On the
other hand, the third $\psi$ function inside the sum has poles for negative
integer number argument; so, this term is the only surviving one inside the
sum while the others do not contribute. The remaining term is:

$\displaystyle\phi_{1}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\nu}\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(-m-\nu)\Gamma(\frac{p}{2}+l^{\prime}+m)}\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{+}}(\alpha)\sum_{k=0}^{\infty}\frac{\left(-m\right)_{k}\left(\frac{p}{2}+l^{\prime}+m+\nu\right)_{k}}{k!(k+\nu)!}\times$
(57) $\displaystyle\times$
$\displaystyle\cos^{2k}(\alpha)\bigg{[}\psi(-m+k)\bigg{]},$

We should extract a finite limit from the term
$\dfrac{\psi(-m+k)}{\Gamma(-m-\nu)}$. Let us expand both functions around
negative integer numbers using the regulator $\varepsilon$, namely:

$\begin{split}\psi(-m+k+\varepsilon)&=-\frac{1}{\varepsilon}-\gamma+H_{(m-k)}+\sum_{j\geq
1}\bigg{[}H_{(m-k)}^{j+1}+(-1)^{k+1}\zeta(j+1)\bigg{]}\varepsilon^{j};\\\
\Gamma(-m-\nu+\varepsilon)&=\frac{(-1)^{m+\nu}}{(m+\nu)!}\bigg{[}\frac{1}{\varepsilon}+\psi(m+\nu+1)+\frac{\varepsilon}{2}\bigg{(}\frac{\pi^{2}}{3}+\psi(m+\nu+1)^{2}-{\psi}^{\prime}(m+\nu)\bigg{)}+\mathcal{(}O)(\varepsilon^{2})\bigg{]},\end{split}$
(58)

where $H_{m}$ are harmonic numbers and $\zeta(s)$ is the Riemann zeta
function. Expanding the reciprocal gamma function and upon taking the limit
$\varepsilon\rightarrow 0$ we have the regularized expression:

$\lim_{\varepsilon\to
0}\frac{\psi(-m+k+\varepsilon)}{\Gamma(-m-\nu+\varepsilon)}=\frac{\psi(-m+k)}{\Gamma(-m-\nu)}=\frac{(m+\nu)\Gamma(m+\nu)}{(-1)^{m+\nu-1}}.$
(59)

Inserting this result into Eq.(57) one finds:

$\displaystyle\phi_{1}(\alpha)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{1-m}(m+\nu)\Gamma\left(\frac{p}{2}+l^{\prime}\right)\Gamma\left(m+\nu\right)}{\Gamma(\frac{p}{2}+l^{\prime}+m)}\sin^{l^{\prime}}(\alpha)\cos^{2\lambda_{+}}(\alpha)\sum_{k=0}^{\infty}\frac{\left(-m\right)_{k}\left(\frac{p}{2}+l^{\prime}+m+\nu\right)_{k}}{k!(k+\nu)!}\times$
(60) $\displaystyle\times$ $\displaystyle\cos^{2k}(\alpha).$

Now it is easy to check that the asymptotic behaviour near the boundary for
this case goes as $\cos^{2\lambda_{+}}(\alpha)$ as expected.

For our purposes we shall select the pure normalizable modes for all $\nu$
which in turn are arrived at imposing the quantization condition Eq.(117).
Such a choice enforces the vanishing of the second term in Eq.(48) evaluated
at the boundary. Physically speaking, invoking the AdS/CFT dictionary, only
the non-normalizable or non-fluctuating modes survive at the boundary (and are
identified as sources for dual CFT operators $\mathcal{O}$ living on the
boundary) while the normalizable or fluctuating modes vanish at the boundary.
The only surviving term in Eq.(48) implies a conserved charge on the horizon,
which we will call Aretakis charge due to its similarity to the black hole
context:

$H^{Aretakis}=\int dV_{\mathcal{H}^{p}}\int
d^{2}x\,\sqrt{\gamma}\,2\partial_{r}\psi.$ (61)

Note that this quantity is finite since $\psi$ should be restricted to the
normalizable modes. To proceed further we will define mode-by-mode Aretakis
quantities in terms of the quantum numbers. To do so we can split our analysis
into two parts: the first one, for $p$ odd and $\nu$ either integer or null;
the second one, for $p$ even and $\nu$ non-integer. The separation constant
$\xi$ given by the Eq.(117) for normalizable modes for all $\nu$ may be
rewritten in a simpler way, namely, defining the new quantum number

$\displaystyle L$ $\displaystyle=$ $\displaystyle
2m+l^{\prime}+2\lambda_{+}-1$ (62) $\displaystyle=$ $\displaystyle
2m+l^{\prime}+p+(p+1)l,\,\,\,\mbox{with}\,\,\,L=p,p+1,...,$ (63)

where we have used the definition $m^{2}_{0}=(p+1)^{2}l(l+1)$ in the second
line in the above equation. The quantum number $L$ is a combination of the
other ones so that we find $\xi^{2}=L(L+1)$. Plugging it in Eq.(47) we obtain:

$0=2\partial_{v}\left[\partial_{r}\Phi(v,r)\right]+\partial_{r}\left[r^{2}\partial_{r}\Phi(v,r)\right]-L(L+1)\Phi(v,r),$
(64)

where we have omitted the indices for convenience. Now by applying the
operator $\partial^{L}_{r}$ in the above equation and using mathematical
induction one can define conserved quantities on the horizon,

$H^{Aretakis}_{L}[\Phi]=\left(\partial_{r}^{L+1}\Phi\right)|_{r=0},$ (65)

which we call Aretakis quantities or constants. Like the full Aretakis charge
(61), they do not depend on the time $v$ on the horizon. This result is
generic for any extremal non-dilatonic black $p$-brane in (p+4) dimensions;
even though depending only on the $AdS_{2}$ section of the whole scalar field,
$\Phi(v,r)$, the influence of extra dimensions are encoded in the quantum
number $L$. Due to the fact that extremal non-dilatonic black $p$-branes can
be viewed as extremal dilatonic black holes in four dimensions as discussed in
section 2, the conserved quantities found above hold in the near-horizon
geometry of such black holes. So, the novel property in Eq.(65) is that it
provides a higher-dimensional generalization for the Aretakis quantities.

The near-horizon geometry of extreme RN black hole corresponds to $p=0$ and
the quantum numbers: $m=0$ and $l^{\prime}=0$; as a result $L$ reduces to $l$,
and we recover well-known results Lucietti:2012xr . Generically, as said
before, the quantum number $L$ is a combination of the set of quantum numbers
$\left\\{m,l^{\prime},l\right\\}$. As a consequence, for a particular choice
of $L$ there is a degenerate spectrum of possible choices of $m,l^{\prime}$
and $l$ leading to same Aretakis quantities. The lower-order mode is covered
by taking $L=p$; for this case the corresponding Aretakis constant is
expressed in terms of the space-time dimension. For example, for $p=1$
(corresponding to extremal non-dilatonic black strings) the lower-order
conserved quantity at the horizon is given by the $2$-nd radial derivative of
the scalar mode $\Phi(v,r)$. In general, for arbitrary extremal non-dilatonic
black $p$-brane, $\left(\partial_{r}^{p+1}\Phi\right)|_{r=0}$ are the lower-
order conserved quantities on the horizon.

###  3.3 Late time tails

In this section we will examine the physical consequences of the conserved
Aretakis quantities and what happens at the late time regime. The first step
is proceeding with a coordinate transformation on the $AdS_{2}$ section of the
full near-horizon metric $AdS_{p+2}\times S^{2}$. We introduce the coordinates
$(v,u)$, which are related to $(t,r)$ by:

$v=t-\frac{1}{r},\,\,\,u=t+\frac{1}{r}.$ (66)

The metric in this coordinates becomes:

$ds^{2}_{AdS_{2}}=-\frac{4}{(u-v)^{2}}dudv.$ (67)

Since $-\infty<u,v<\infty$ it is convenient to compactify the metric by
introducing new coordinates. By defining

$U=\tan^{-1}u,\,\,\,V=\tan^{-1}v,$ (68)

the former metric takes the form

$ds^{2}_{AdS_{2}}=-\frac{4}{\sin^{2}(U-V)}dUdV,$ (69)

where $U$ and $V$ are still outgoing and ingoing null coordinates,
respectively. However, now they are restricted to the region
$-\dfrac{\pi}{2}<V<U<\dfrac{\pi}{2}$. The future Poincaré horizon, previously
located at $r=0$, becomes located at $U=\dfrac{\pi}{2}$ in the new coordinate
frame.

In terms of the new coordinates Eq.(64) becomes

$\partial_{V}\partial_{U}\Phi(U,V)+\frac{L(L+1)}{\sin^{2}(U-V)}\Phi(U,V)=0.$
(70)

Similarly to the radial equation (see Appendix A), the former equation can be
solved in terms of hypergeometric functions. Nevertheless, for our purposes it
is enough to take into account the asymptotic behavior near the horizon. Thus,
we consider the asymptotic behavior near the horizon of the mode $\Phi(U,V)$
which corresponds to the limit $U\rightarrow\dfrac{\pi}{2}$ and
$V\rightarrow\dfrac{\pi}{2}$:

$\Phi(U,V)\rightarrow(U-V)^{a}\Big{(}f(V)+\mathcal{O}(U-V)\Big{)}.$ (71)

Substituting this into Eq.(70) we found two roots for $a$ which are

$a_{\pm}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1+4\xi^{2}},$ (72)

or, in terms of $L$,

$a_{+}=L+1,\,\,\,a_{-}=L.$ (73)

Once again we will make the choice for normalizable boundary conditions so
that only $a=a_{+}$ holds. Now we are ready to evaluate the Aretakis
quantities at late times. By substituting the asymptotic behavior Eq.(71) in
Eq.(70) and using the Leibniz rule we find:

$H_{L}=\frac{(-1)^{L+1}(L+1)!}{2^{2(L+1)}}f(\frac{\pi}{2}).$ (74)

Note that our result recovered those ones found in Lucietti:2012xr  for
$AdS_{2}$ space up to a normalization constant. For the lowest state, we have

$H_{p}=\frac{(-1)^{p+1}(p+1)!}{2^{2(p+1)}}f\left(\frac{\pi}{2}\right),$ (75)

which is completely expected since that has been describing a warped geometry
of $AdS_{2}$. In particular, for $p=0$ the warped factor is trivial and then
the lowest state for $AdS_{2}$ is recovered. Using Eq.(74) one can write the
asymptotic behaviour (71) near the horizon in terms of the constants:

$\Phi(U,V)=\frac{(-4)^{L+1}}{(L+1)!}H_{L}(U-V)^{L+1}+...$ (76)

or, in terms of $(v,r)$ coordinates, we have

$\Phi(v,r=0)=\frac{(-4)^{L+1}}{(L+1)!}\left(\frac{1}{v}\right)^{L+1}H_{L}+...$
(77)

where the ellipsis stand for subleading terms. Such result shows that $\Phi$
falls off as $v^{-L-1}$ at the late time while its $p$-th radial derivative
remains constant along the horizon. Applying the operator $\partial_{r}^{L+1}$
in Eq.(64) in order to examine highest derivatives than the $(L+1)$-order at
the horizon, we have

$\begin{split}0&=2\partial_{v}\partial_{r}^{L+2}\Phi(v,r)+\partial_{r}^{L+2}\left[r^{2}\partial_{r}\Phi(v,r)\right]-L(L+1)\partial_{r}^{L+1}\Phi(v,r)\\\
&=2\partial_{v}\partial_{r}^{L+2}\Phi(v,r)-L(L+1)\partial_{r}^{L+1}\Phi(v,r)+\sum_{n=0}^{L+2}{L+2\choose{\it
n}}\frac{2r^{2-n}}{(2-n)!}\partial_{r}^{(p+3-n)}\Phi(v,r),\end{split}$ (78)

and evaluating at the horizon,

$\begin{split}0&=2\partial_{v}\partial_{r}^{L+2}\Phi(v,r)|_{r=0}+(L+2)(L+1)\partial_{r}^{L+1}\Phi(v,r)|_{r=0}-L(L+1)\partial_{r}^{L+1}\Phi(v,r)|_{r=0}\\\
&=\partial_{v}\partial_{r}^{L+2}\Phi(v,r)|_{r=0}+(L+1)\partial_{r}^{L+1}\Phi(v,r)|_{r=0}.\\\
\end{split}$ (79)

Through Eq.(65) we find

$\partial_{r}^{L+2}\Phi(v,r)|_{r=0}=-(L+1)H_{L}v+g(r=0),$ (80)

where $g(r=0)$ is an arbitrary function evaluated at the horizon. The last
equation shows that $\partial_{r}^{L+2}\Phi(v,r)|_{r=0}$ grows linearly with
$v$ at late time. Thus it blows up as $v\rightarrow\infty$ suggesting the
presence of instabilities for $H_{L}\neq 0$. By contrast if $H_{L}=0$, then
$\partial_{r}^{L+2}\Phi(v,r)|_{r=0}$ is a constant and as long as this
constant is different from zero $\partial_{r}^{L+3}\Phi(v,r)|_{r=0}$ blows up
at late times.

##  4 Conclusions

In this work we have considered the near-horizon geometry of extremal non-
dilatonic black $p$-branes in $D=p+4$ in the presence of a probe massless
scalar field in order to explore the arising of conserved quantities on the
horizon. First, we constructed the near-horizon metric in Gaussian null
coordinates in order to define and compute the Aretakis quantities on the
horizon. We have argued that its near-horizon geometry becomes
$AdS_{p+2}\times S^{2}$, and that $AdS_{p+2}$ is written as a warped product
between $AdS_{2}$ and $\mathcal{H}^{p}$ or, equivalently, a conformal
compactification of $AdS_{2}\times S^{p}$. In addition, upon dimensional
reduction by $p$ times, we have shown that the near-horizon geometry reduces
to a warped geometry of $AdS_{2}\times S^{2}$. This result is a consequence of
the fact that extremal dilatonic black holes can be viewed as extremal non-
dilatonic black $p$-branes.

Finding the Aretakis quantities required knowledge of normalizable and non-
normalizable modes, which were found by solving directly the wave equation.
These modes play an important role as the choice for normalizable modes by
imposing boundary conditions leads to the vanishing of boundary contributions.
In this respect, the Aretakis quantities depends only on the $AdS_{2}$ section
of the scalar mode. Furthermore, we have obtained the explicit form for the
Aretakis quantities and their relation to the $p$ spatial dimensions. For
example, for the zero modes in the near-horizon geometry of RN black hole the
first transverse derivative of $\Phi$ is conserved, whilst for extremal
dilatonic black holes viewed as an extremal non-dilatonic black $p$-brane the
$p$-order transverse derivative of $\Phi$ is conserved. Therefore the
instability depends on the extra $p$ spatial dimensions.

The results obtained in this paper in some sense generalize those in the
literature focused on extremal black holes Lucietti:2012xr , but in contrast
to the cases explored previously, we have explored extremal black $p$-branes
whose horizon is not compact such that near-horizon theorems for black holes
do not hold. It is noteworthy that in spite of this difference conserved
Aretakis quantities can be defined. Further work will hopefully clarify the
minimum conditions for a spacetime to exhibit these properties.

##  Appendix A Explicit scalar modes in Gaussian null coordinates

In order to find regular solutions at the singular points, after imposing
boundary conditions it is convenient to use the well-known Sturm-Liouville
theory, see zettl2005sturm ; boyce2017elementary  for a detailed discussion.
First of all, Eq. (46) might be cast into the Sturm-Liouville form444We are
following the same conventions of boyce2017elementary . by taking the
transformation $z=\sin^{2}\alpha$ in Eq. (46). Thus we have

$L[\phi(z)]=\beta r(z)\phi(z),$ (81)

where

$\displaystyle L[\phi(z)]$ $\displaystyle=$
$\displaystyle-\frac{d}{dz}\left[p(z)\frac{d\phi(z)}{dz}\right]+q(z)\phi(z),$
(82) $\displaystyle p(z)$ $\displaystyle=$ $\displaystyle
z^{p/2}(1-z)^{(1-p)/2},$ (83) $\displaystyle q(z)$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{0}}{4}\frac{z^{(p-2)/2}}{(1-z)^{(p+3)/2}}+\frac{l^{\prime}(l^{\prime}+p-2)}{4}\frac{z^{(p-4)/2}}{(1-z)^{(p+1)/2}},$
(84) $\displaystyle r(z)$ $\displaystyle=$
$\displaystyle\frac{z^{(p-2)/2}}{(1-z)^{(p+1)/2}},$ (85) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle\frac{\xi^{2}}{4},$ (86)

valid in the range $0<z<1$, and satisfying boundary conditions of the general
form

$\alpha_{1}\phi(0)+\alpha_{2}\phi^{\prime}(0)=0,\quad\alpha_{3}\phi(1)+\alpha_{4}\phi^{\prime}(1)=0,$
(87)

where $\alpha$’s are constants. Such a setup constitutes a well-posed Cauchy
problem. As we shall see later, the equation above can be expressed in terms
of hypergeometric equations, and as a consequence the solutions are obtained
as power series around the singular points. Regularity conditions are required
in order to get well-defined solutions at the singular points. It is
convenient to first examine the behavior of the solutions of Eq. (81) near the
origin $(z=0)$, and then to consider these solutions near the boundary, using
the linear transformations of hypergeometric functions which convert the
argument of the function from $z$ to $(1-z)$. After that, quantization
conditions must be fulfilled in order to have well-behaved solutions at the
origin and the boundary. The solutions will be classified as normalizable
(square integrable) or non-normalizable (failing to be square integrable). Let
$\psi_{1}$ and $\psi_{2}$ be solutions of the Klein-Gordon equation, and their
inner product be:

$(\psi_{1},\psi_{2})=-i\int_{\Sigma_{v}}d\alpha
d^{p+2}x\sqrt{-h}\,n^{\mu}\,(\psi_{1}^{\ast}\partial_{\mu}\psi_{2}-\psi_{2}\partial_{\mu}\psi_{1}^{\ast}),$
(88)

where $\Sigma_{v}$ is a smooth space-like section, $h_{\mu\nu}$ is the induced
metric on $\Sigma_{v}$ and $n_{\alpha}$ is a unit normal vector to
$\Sigma_{v}$. The case $\psi_{1}=\psi_{2}$ defines the norm of the solution;
if this integral converges, the solution is called normalizable (square
integrable), while otherwise it is called non-normalizable.

In order to find the solutions we start by substituting the function

$\phi(z)=(1-z)^{\lambda}z^{\beta}\varphi(z),$ (89)

into Eq.(81), and we have

$\displaystyle
z(1-z)\frac{d^{2}\varphi}{dz^{2}}+\left(\left(-2\lambda-2\beta-\frac{1}{2}\right)z+2\lambda+\frac{1}{2}p\right)\frac{d\varphi}{dz}+$
(90) $\displaystyle+$
$\displaystyle\left(-\left(\beta+\lambda\right)^{2}+\frac{1}{2}\left(\lambda+\beta\right)+\frac{1}{4}\xi^{2}\right)\varphi=0,$

which is a hypergeometric differential equation. The coefficients satisfy the
following conditions:

$\displaystyle 2\beta(2\beta+p-2)$ $\displaystyle=$ $\displaystyle
l^{\prime}(l^{\prime}+p-2),$ (91) $\displaystyle 2\lambda(2\lambda-p-1)$
$\displaystyle=$ $\displaystyle m_{0}^{2},$ (92)

the solutions of which are given by:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1}{2}l^{\prime},\,\frac{1}{2}\left(2-p-l^{\prime}\right),$
(93) $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\lambda_{\pm}=\frac{1}{4}(p+1)\pm\frac{1}{4}\sqrt{(p+1)^{2}+4m_{0}^{2}}.$
(94)

As described in Grad ; abramowitz1965handbook , the hypergeometric equation
has two linear independent solutions which in turn depend on $\beta$ and
$\lambda$. For convenience and without loss of generality, we make the choice
$\lambda=\lambda_{+}$. Furthermore, we must distinguish the cases according to
the parity of the dimensional parameter $p$. Hence the solutions near the
origin, $z=0$, for $p$ odd, are:

$\displaystyle\phi_{1}(z)$ $\displaystyle=$ $\displaystyle
z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{+}}\,{{}_{2}}F_{1}(a,b;\frac{p}{2}+l^{\prime};z),$
(95) $\displaystyle\phi_{2}(z)$ $\displaystyle=$ $\displaystyle
z^{\frac{2-p-l^{\prime}}{2}}(1-z)^{\lambda_{+}}{{}_{2}}F_{1}\left(a-\frac{p}{2}-l^{\prime}+1,b-\frac{p}{2}-l^{\prime}+1;\frac{4-p-2l^{\prime}}{2};z\right),$
(96)

where we have defined the following quantities:

$\displaystyle a$ $\displaystyle=$
$\displaystyle\lambda_{+}-\frac{1}{4}(1-\sqrt{1+4\xi^{2}})+\frac{l^{\prime}}{2},$
(97) $\displaystyle b$ $\displaystyle=$
$\displaystyle\lambda_{+}-\frac{1}{4}(1+\sqrt{1+4\xi^{2}})+\frac{l^{\prime}}{2}.$
(98)

Since we want to obtain real solutions (95,96), the mass term should satisfy
the following inequality,

$m^{2}_{0}\geq-\frac{1}{4}(p+1)^{2},$ (99)

which is the well-known Breitenlohner-Freedman bound BF1 ; BF2 .

Let us consider first the case of even $p$, for which the first solution is
well-defined at $z=0$. However, the second one is no longer well-defined at
the origin. In such case, the second independent solution takes the form

$\begin{split}\tilde{\phi}_{2}(z)&=\phi_{1}(z)\ln(z)-\\\
&-\bigg{[}\sum_{k=1}^{\frac{p-2}{2}+l^{\prime}}\frac{\left(\frac{p-2}{2}+l^{\prime}\right)!(k-1)!}{\left(\frac{p-2}{2}+l^{\prime}-k\right)!\left(1-a\right)_{k}\left(1-b\right)_{k}}\,(-z)^{-k}+\\\
&+\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{\left(\frac{p}{2}+l^{\prime}\right)_{k}\,k!}\,g_{k}\,z^{k}\bigg{]},\end{split}$
(100)

where

$\displaystyle g_{k}$ $\displaystyle=$
$\displaystyle\psi(a+k)+\psi(b+k)-\psi(1+k)-\psi(\frac{p}{2}+l^{\prime}+k),$
(101) $\displaystyle\psi(z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma^{\prime}(z)}{\Gamma(z)},$ (102)
$\displaystyle(a)_{k}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(a+k)}{\Gamma(a)},$ (103)

are defined as in DLMF .

It is now necessary to impose boundary conditions near the origin. Using the
leading behavior of the solutions and the derivative of the hypergeometric
functions abramowitz1965handbook , we require that the boundary term
disappears at the origin. In other words, the second solution is unacceptable
since the first term in Eq. (27) vanishes for the classical solutions in the
bulk and then the second term (boundary term) evaluated at the origin only
vanishes for $\phi_{1}(z)$,

$\int d^{2}x\int d^{p+1}y\sqrt{-g}\,g^{zz}\partial_{z}\phi\Big{|}_{z=1}=0.$
(104)

This property is reminiscent of AdS/CFT where for the contributions coming
from the interior of the bulk the coupling with operators defined at the
conformal boundary is not allowed.

The next step is to examine the behavior near the boundary, $z=1$ or
$\alpha=\pi/2$. In order to do this, it is most convenient to rewrite the
argument of the hypergeometric function of the solution $\phi_{1}$ in terms of
$z$. This is reached by using the linear transformations for hypergeometric
functions prescribed in abramowitz1965handbook ; DLMF . In general, we can
write down:

$\phi_{1}(z)=C\phi_{3}(z)+D\phi_{4}(z),$ (105)

where $C$ and $D$ are constants while $\phi_{3}(z)$ and $\phi_{4}(z)$
represent the solutions near the boundary. Their explicit forms depend on
$\nu=\dfrac{1}{2}\sqrt{(p+1)^{2}+4m^{2}_{0}}$, which is only the difference
between $\lambda_{+}$ and $\lambda_{-}$.

i) The first case corresponds to $\nu$ being a non-integer number. Explicitly,
we have

$\displaystyle\phi_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(\nu)}{\Gamma(e_{+})\Gamma(f_{+})}z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{-}}{{}_{2}}F_{1}\left(e_{-},f_{-};(1-\nu);(1-z)\right)+$
(106) $\displaystyle+$
$\displaystyle\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(-\nu)}{\Gamma(e_{-})\Gamma(f_{-})}z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{+}}{{}_{2}}F_{1}\left(e_{+},f_{+};(1+\nu);(1-z)\right).$

where we have defined the following quantities:

$\displaystyle e_{\pm}$ $\displaystyle=$
$\displaystyle\lambda_{\pm}-\frac{1}{4}(1+\sqrt{4\xi^{2}+1})+\frac{l^{\prime}}{2},$
(107) $\displaystyle f_{\pm}$ $\displaystyle=$
$\displaystyle\lambda_{\pm}-\frac{1}{4}(1-\sqrt{4\xi^{2}+1})+\frac{l^{\prime}}{2},$
(108) $\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(-\nu)}{\Gamma(e_{-})\Gamma(f_{-})},$
(109) $\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(\nu)}{\Gamma(e_{+})\Gamma(f_{+})}.$
(110)

ii) The second one corresponds to $\nu$ being an integer number. We get

$\displaystyle\phi_{1}(z)$ $\displaystyle=$ $\displaystyle
z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{+}}\bigg{\\{}\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(\nu)}{\Gamma(a)\Gamma(b)}(1-z)^{-\nu}\sum_{k=0}^{\nu-1}\frac{\left(a-\nu\right)_{k}\left(b-\nu\right)_{k}}{k!\left(1-\nu\right)_{k}}(1-z)^{k}-$
(111) $\displaystyle-$
$\displaystyle\frac{(-1)^{\nu}\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(a-\nu)\Gamma(b-\nu)}\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{k!(k+\nu)!}(1-z)^{k}\big{[}\ln(1-z)-\psi(k+1)-$
$\displaystyle-$
$\displaystyle\psi(k+\nu+1)+\psi(a+k)+\psi(b+k)\big{]}\bigg{\\}},$

which is conveniently rewritten as follows

$\displaystyle\phi_{1}(z)$ $\displaystyle=$ $\displaystyle
z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{-}}\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(\nu)}{\Gamma(a)\Gamma(b)}\sum_{k=0}^{\nu-1}\frac{\left(a-\nu\right)_{k}\left(b-\nu\right)_{k}}{k!\left(1-\nu\right)_{k}}(1-z)^{k}-$
(112) $\displaystyle-$ $\displaystyle
z^{\frac{l^{\prime}}{2}}(1-z)^{\lambda_{+}}\frac{(-1)^{\nu}\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(a-\nu)\Gamma(b-\nu)}\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{k!(k+\nu)!}(1-z)^{k}\bigg{[}\ln(1-z)-$
$\displaystyle-$ $\displaystyle\psi(k+1)-\psi(k+\nu+1)+\psi\left(a+k\right)+$
$\displaystyle+$ $\displaystyle\psi\left(b+k\right)\bigg{]},$

where the constants in this case are given by:

$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\nu}\Gamma(\frac{p}{2}+l^{\prime})}{\Gamma(a-\nu)\Gamma(b-\nu)},$
(113) $\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{p}{2}+l^{\prime})\Gamma(\nu)}{\Gamma(a)\Gamma(b)}.$
(114)

iii)The last case is $\nu=0$. We have

$\displaystyle\phi_{1}(z)$ $\displaystyle=$ $\displaystyle
z^{\frac{l^{\prime}}{2}}(1-z)^{\frac{1}{4}(p+1)}\bigg{\\{}\frac{\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(a)\Gamma(b)}\sum_{k=0}^{\infty}\frac{\left(a\right)_{k}\left(b\right)_{k}}{(k!)^{2}}\bigg{[}2\psi(k+1)-\psi(a+k)-\psi(b+k)-$
(115) $\displaystyle-$ $\displaystyle\ln(1-z)\bigg{]}(1-z)^{k}\bigg{\\}},$

where the constants are

$\displaystyle
C=D=\frac{\Gamma\left(\frac{p}{2}+l^{\prime}\right)}{\Gamma(a)\Gamma(b)}.$
(116)

##  Appendix B Quantization conditions and (non)-normalizable modes

Now, let us examine the regularity condition, first, for both cases i) and
ii). For these cases, it is clear that the leading terms of Eq.(106,112) fall
off as $(1-z)^{\lambda_{\pm}}$ at the boundary, respectively. As a result, it
follows directly from Eq. (88) that for $\nu>1$ the solution $\phi_{4}(z)$
fails to be square integrable (normalizable) at $z=1$; on the other hand,
$\phi_{3}(z)$ is normalizable. So, as pointed out in Balasubramanian:1998sn ,
the combination of both modes sets up a non-normalizable mode. However, one
can select the normalizable modes by eliminating the first term on the right
hand side in Eqs.(106,112) by taking into consideration that $\Gamma$
functions in the denominator have poles when the following quantization
condition is satisfied, namely,

$\xi=\pm\sqrt{[2m+l^{\prime}+2\lambda_{+}][-1+2m+l^{\prime}+2\lambda_{+}]}.\quad
m=0,1,2,...\,$ (117)

Such a quantization condition leads to the vanishing of the coefficient $D$
and then the fluctuating modes are purely given by $\phi_{3}(z)$. Therefore,
we conclude that the normalizable modes are selected by the above condition.
On the other hand, the pure non-normalizable modes are selected by the set of
values

$\xi=\pm\sqrt{[2m+l^{\prime}+2\lambda_{-}][-1+2m+l^{\prime}+2\lambda_{-}]},\quad
m=0,1,2,...\,$ (118)

for the range of $\nu$ considered above; in this case, the non-fluctuating
modes are purely given by $\phi_{4}(z)$.

A different scenario comes up for $0<\nu<1$. In this situation, both solutions
are square integrable at the boundary for $\nu$ integer and non-integer, thus
$\phi_{1}(z)$ is normalizable. The quantization conditions Eqs. (117, 118)
hold. However, in this case Eq.(117) selects the $\phi_{3}(z)$ normalizable
mode whilst Eq. (118) picks the $\phi_{4}(z)$ normalizable mode. It is worth
to point out that the leading term of $\phi_{1}(z)$ falls off as
$(1-z)^{\lambda_{-}}$ at the boundary since $(1-z)^{\lambda_{-}}$ dominates
$(1-z)^{\lambda_{+}}$ near the boundary.

Lastly, in case iii) (i.e. when $\nu=0$), both $\phi_{3}(z)$ and $\phi_{4}(z)$
are square integrable.

###### Acknowledgements.

We are grateful to Chris Pope for helpful discussions and collaboration on
related topics. We would also like to thank Gonzalo Olmo and Albert Petrov for
discussions. The work of M.C. is supported in part by the DOE (HEP) Award de-
sc0013528, the Fay R. and Eugene L. Langberg Endowed Chair (M.C.) and the
Slovenian Research Agency (ARRS No. P1-0306). P.J.P. would like to thank the
Brazilian agency CAPES for the financial support (PDE/CAPES grant, process
88881.17175/2018-01) and Department of Physics and Astronomy, University of
Pennsylvania, for the hospitality.

## References

  * (1) K. S. Stelle, In *Trieste 1996, High energy physics and cosmology* 287-339 [hep-th/9701088].
  * (2) T. Mohaupt, Class. Quant. Grav. 17, 3429 (2000), [hep-th/0004098].
  * (3) G. T. Horowitz and A. Strominger, Nucl. Phys. B 360, 197 (1991).
  * (4) K. Skenderis, Lect. Notes Phys. 541, 325 (2000) [Subnucl. Ser. 36, 345 (2000)] [hep-th/9901050].
  * (5) M. J. Duff, R. R. Khuri and J. X. Lu, Phys. Rept. 259, 213 (1995) [hep-th/9412184].
  * (6) P. K. Townsend and M. Zamaklar, Class. Quant. Grav. 18, 5269 (2001) [hep-th/0107228].
  * (7) S. Aretakis, J. Funct. Anal. 263, 2770 (2012) [arXiv:1110.2006 [gr-qc]].
  * (8) S. Aretakis, Commun. Math. Phys. 307, 17 (2011) [arXiv:1110.2007 [gr-qc]].
  * (9) S. Aretakis, Annales Henri Poincare 12, 1491 (2011) [arXiv:1110.2009 [gr-qc]].
  * (10) S. Aretakis, Adv. Theor. Math. Phys. 19, 507 (2015) [arXiv:1206.6598 [gr-qc]].
  * (11) M. Cvetič, C. N. Pope, A. Saha and A. Satz, arXiv:2001.04495 [hep-th].
  * (12) M. Cvetič and D. Youm, Nucl. Phys. B 476, 118 (1996) [hep-th/9603100].
  * (13) M. Cvetič and D. Youm, Phys. Rev. D 54, 2612 (1996) [hep-th/9603147].
  * (14) H. Godazgar, M. Godazgar and C. N. Pope, Phys. Rev. D 96, 084055 (2017) [arXiv:1707.09804 [hep-th]].
  * (15) M. Cvetič and A. Satz, Phys. Rev. D 98, 124035 (2018), [arXiv:1811.05627 [hep-th]].
  * (16) E. T. Newman and R. Penrose, Proc. Roy. Soc. Lond. A 305, 175 (1968).
  * (17) W. E. Couch and R. J. Torrence, Gen. Rel. and Grav. 16, 789 (1984).
  * (18) P. Figueras, H. K. Kunduri, J. Lucietti and M. Rangamani, Phys. Rev. D 78, 044042 (2008) [arXiv:0803.2998 [hep-th]].
  * (19) G. W. Gibbons and P. K. Townsend, Phys. Rev. Lett. 71, 3754 (1993).
  * (20) G. W. Gibbons, G. T. Horowitz and P. K. Townsend, Class. Quant. Grav. 12, 297 (1995) [hep-th/9410073].
  * (21) J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. Math. Phys. 2, 231 (1998)] [hep-th/9711200].
  * (22) E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998)
  * (23) I. R. Klebanov and E. Witten, Nucl. Phys. B 556, 89 (1999) [hep-th/9905104].
  * (24) M. J. Duff, H. Lu and C. N. Pope, Phys. Lett. B 382, 73 (1996) [hep-th/9604052].
  * (25) H. Lu and C. N. Pope, Nucl. Phys. B 465, 127 (1996) [hep-th/9512012].
  * (26) I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B 475, 164 (1996) [hep-th/9604089].
  * (27) D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D 43, 3140 (1991); Phys. Rev. D 45, 3888 (1992).
  * (28) H. K. Kunduri, J. Lucietti and H. S. Reall, Class. Quant. Grav. 24, 4169 (2007) [arXiv:0705.4214 [hep-th]].
  * (29) J. M. Maldacena, J. Michelson and A. Strominger, JHEP 9902, 011 (1999) [hep-th/9812073].
  * (30) P. van Nieuwenhuizen and N. P. Warner, Commun. Math. Phys. 99, 141 (1985).
  * (31) D. Anninos, JHEP 0909, 075 (2009) [arXiv:0809.2433 [hep-th]].
  * (32) A. Castro and W. Song, arXiv:1411.1948 [hep-th].
  * (33) A. Ishibashi and R. M. Wald, Class. Quant. Grav. 20, 3815 (2003) [gr-qc/0305012].
  * (34) A. Ishibashi and R. M. Wald, Class. Quant. Grav. 21, 2981 (2004) [hep-th/0402184].
  * (35) J. Polchinski, arXiv:1010.6134 [hep-th].
  * (36) M. Ammon and J. Erdmenger, Gauge/gravity duality: Foundations and applications, (Cambridge University Press, Cambridge, England, 2015).
  * (37) V. Balasubramanian, P. Kraus and A. E. Lawrence, Phys. Rev. D 59, 046003 (1999) [hep-th/9805171].
  * (38) P. Bizon and H. Friedrich, Class. Quant. Grav. 30, 065001 (2013) [arXiv:1212.0729 [gr-qc]].
  * (39) J. Maldacena and G. L. Pimentel, JHEP 1302, 038 (2013), [arXiv:1210.7244 [hep-th]].
  * (40) J. Lucietti, K. Murata, H. S. Reall and N. Tanahashi, JHEP 1303, 035 (2013) [arXiv:1212.2557 [gr-qc]].
  * (41) A. Zettl Sturm-Liouville theory, mathematical surveys and monographs (American Mathematical Society, Providence, 2005)
  * (42) W. E. Boyce and R. C. DiPrima Elementary Differential Equations and Boundary Value Problems, 7th edition, John Wiley & Sons, 2011.
  * (43) I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (Academic Press, London, 1980)
  * (44) M. Abramowitz and I. A. Stegun Handbook of mathematical functions: with formulas, graphs, and mathematical tables, Vol.55, Courier Corporation, 1965.
  * (45) P. Breitenlohner and D. Z. Freedman, Phys. Lett. B 115, 197 (1982).
  * (46) P. Breitenlohner and D. Z. Freedman, Ann. Phys. 144, 249 (1982).
  * (47) Digital Library of Mathematical Functions, §15.10(i)(c).