---
abstract: 'We study a class of topological black hole solutions in RSII braneworld scenario in the presence of a localized Maxwell field on the brane. Such a black hole can carry two types of charge, one arising from the extra dimension, the tidal charge, and the other one from a localized gauge field confined to the brane. We find that the localized charge on the brane modifies the bulk geometry and in particular the bulk Weyl tensor. The bulk geometry does not depend on different topologies of the horizons. We present the temperature and entropy expressions associated with the event horizon of the braneworld black hole and by using the first law of black hole thermodynamics we calculate the mass of the black hole.'
address: |
    $^1$Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran\
    $^2$Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran\
    $^3$ Department of Physics, Fudan University, Shanghai 200433, China
author:
- 'Ahmad Sheykhi $^{1,2}$[^1] and Bin Wang $^{3}$[^2]'
title: On topological charged braneworld black holes
---

In the past years there has been a lot of interest in the braneworld scenario, based on the assumption that all gauge fields in standard model of particle physics are confined on a $3$-brane, playing the role of our $4$-dimensional universe, embedded in a higher dimensional spacetime, while the gravitational field, in contrast, is usually considered to live in the whole spacetime. The first picture appeared in braneworld scenarios was the second Randall-Sundrum model (RSII) in which, our universe observed as a positive tension $3$-brane embedded in a $5$-dimensional anti de-Sitter bulk [@RS]. In this model the localization of gravity happens on the brane due to the negative bulk cosmological constant and the cross over between $4$-dimensional and $5$-dimensional gravity is set by the anti de-Sitter radius. Within the context of the RSII scenario, it is important that the induced metric on the brane is, in the low energy regime, the solution predicted by standard general relativity in four dimensions. Otherwise the usual astrophysical properties of black holes and stars would not be recovered. Therefore it is natural to assume the formation of black hole in the braneworld due to gravitational collapse of matter trapped on the brane. In fact, the construction and study of black hole solutions on the brane has been one of the most important and intriguing challenge in braneworld physics. There are several reasons why this problem is so challenging. First, the effective gravitational field equation on the brane is not the usual Einstein one but contains higher correction terms due to the nonlocal bulk effects on the brane and therefore is more complicated compared with the usual gravitational field equations. Second, even one finds the solution of the effective gravitational field equations on the brane, one can not regard it as a braneworld black hole solution. One can just consider this solution as an initial data for the evolution of the brane into the bulk. The first attack on this problem was done by Chamblin, Hawking and Reall who investigated the gravitational collapse of uncharged, non-rotating matter in RSII braneworld model [@Cham1]. They showed that a static uncharged black hole on the brane is described by a “black cigar" solution in five dimensions. If this cigar extends all the way down to the anti-de Sitter horizon, then we recover the metric for a black string in anti-de Sitter spacetime. However, such a black string is unstable near the anti-de Sitter horizon [@Gre1; @Gre2]. An exact braneworld black hole solution satisfies a closed system of effective gravitational field equations on the brane, describing an uncharged black hole in the RSII scenario was obtained in [@Dad]. By using the braneworld gravitational field equations derived in [@Shi], it was shown that a Reissner-Nordstrom geometry could arise on the brane provided that the bulk Weyl tensor takes a particular form. The solution in [@Dad] carries a “tidal charge", arising from the projection of the bulk free gravitational field effects onto the brane. However, it was argued in [@Cham2] that although the solution in [@Dad] was claimed to describe an uncharged black hole, one can not regard it as a braneworld black hole solution. One can just consider this solution as an initial data for the evolution of the brane into the bulk. Until this evolution is performed and boundary conditions in the bulk are imposed, it is not clear what this solution represents. For example, it might give rise to some pathology such as a naked curvature singularity. Therefore the main problem remains in the braneworld black hole physics is to study the effect of the braneworld black hole on the bulk geometry, and in particular the nature of the off-brane horizon structure. Indeed, the analytical solution for the bulk spacetime has not been found until now. The numerical calculations on the bulk geometry in the case of charged and uncharged braneworld black holes have been investigated in [@Cham2] and [@Shibata], respectively. Other attempts on the study of braneworld black holes and their physical properties have been carried out in [@Cas1; @Cas2; @Bro; @Gre3; @Ali; @kof; @Bwang; @Bwang2; @Yosh; @Yosh2].

The purpose of the present Letter is to tackle the first problem mentioned above in the braneworld black hole physics. We will consider the Maxwell gauge fields confined onto the brane. Employing a simple strategy, we solve gravitational field equations on the brane and obtain the charged topological braneworld black hole solutions. Our solution is the generalization of [@Cham2] to different horizon topologies. We also present the temperature and entropy expressions associated with the event horizon of the braneworld black hole and calculate the mass of the black hole by using the first law of black hole thermodynamics. Since the flux lines of gauge fields can pierce the horizon only when they intersect the brane, our bulk theory is the same as that of the uncharged case and one might expect that the “black cigar" solution still describes the bulk containing the charged braneworld black hole. Here we will not repeat the discussion on the bulk metric, since we see that the bulk geometry does not depend on different topologies of the horizons, thus our bulk metric is the same as that discussed in [@Cham2] for the spherically symmetric braneworld black hole.

We start with the effective field equations on a 3-brane embedded in the 5-dimensional anti de-Sitter spacetime with $\mathbb{Z}_2$ symmetry expressed as [@Shi] $$\begin{aligned}
\label{eqn}
G_{\mu\nu}=-\Lambda g_{\mu\nu} +8\pi G
T_{\mu\nu}+\kappa_5^4\pi_{\mu\nu}-E_{\mu\nu},\end{aligned}$$ where $$\begin{aligned}
G&=&\frac{\kappa^4_5}{48\pi} \lambda,  \hspace{0.7cm}
\Lambda=\frac{\kappa_5^2}{2}\Bigl(
\Lambda_5+\frac{\kappa_5^2}{6}\lambda^2
\Bigr).\\\end{aligned}$$ Here $\kappa_5$ and $\Lambda_5$ are, respectively, the five-dimensional gravity coupling constant and cosmological constant. The factor $\Lambda$ is the effective cosmological constant on the brane, $\lambda$ is the brane tension, and $T_{\mu\nu}$ is the stress energy tensor confined onto the brane, so $T_{AB}\,n^B=0$, where $n^A$ is the unit normal to the brane. The first correction term relative to Einstein’s gravity is the inclusion of a quadratic term $\pi_{\mu\nu}$ in the stress-energy tensor, arising from the extrinsic curvature term in the projected Einstein tensor, and is given by $$\pi_{\mu\nu}=\frac{1}{12}T T_{\mu\nu}-\frac{1}{4}
T_{\mu\alpha}T_{\nu}^{\ \alpha}{}+
\frac{1}{8}\,g_{\mu\nu}\left(T_{\alpha\beta}T^{\alpha\beta}-\frac{1}{3}T^2
\right) \,.
    \label{inducedEFE}$$ The second correction term, ${E}_{\mu\nu}$, is the projection of the five-dimensional bulk Weyl tensor onto the brane, which is defined as $ E_{\mu\nu} = {}^{(5)}C_{\mu\alpha\nu\beta} n^\alpha
n^\beta $ and encompasses the nonlocal bulk effect. The only general known property of this nonlocal term is that it is traceless, namely ${ E}^{\mu}{}_{\mu}=0$. Using the traceless property of the projected Weyl tensor, Eq. (\[eqn\]) can be simplified into $$\label{R}
R=4\Lambda-8\pi G \,T-
\frac{\kappa_5^4}{4}\left(T_{\alpha\beta}T^{\alpha\beta}-\frac{1}{3}
T^2 \right) \,.$$ We would like to find the topological black hole solutions of the field equations (\[eqn\]). We assume the induced metric on the brane in the form $$\label{metric}
ds^2=-f(r)dt^2 +{dr^2\over f(r)}+ r^2d\Omega_{k}^2 ,$$ where $d\Omega_{k}^2$ is the line element of a two-dimensional hypersurface $\Sigma$ with constant curvature, $$\label{met}
d\Omega_k^2=\left\{
  \begin{array}{ll}
    $$d\theta^2+\sin^2\theta d\phi^2$$,\quad \quad\!\!{\rm for}\quad $$k=1$$, &  \\
    $$d\theta^2+\theta^2 d\phi^2$$,\quad\quad\quad {\rm for}\quad $$k=0$$,&  \\
    $$d\theta^2+\sinh^2\theta d\phi^2$$, \quad {\rm for}\quad $$k=-1$$.&
  \end{array}
\right.$$ For $k = 1$, the topology of the event horizon is the two-sphere $S^2$, and the spacetime has the topology $R^2 \times S^2$. For $k
= 0$, the topology of the event horizon is a torus and the spacetime has the topology $R^2 \times T^2$. For $k = -1$, the surface $\Sigma$ is a two-dimensional hypersurface $H^2$ with constant negative curvature. In this case the topology of spacetime is $R^2 \times H^2$. It is not necessary to take the exact metric describing a topological braneworld black hole in the form (6). In general one may expect that $g_{rr}\neq -
{g_{tt}}^{-1}$. But, it is well known that the induced metric describing a charged black hole should be close to Reissner-Nordstrom metric, so our ansatz for the braneworld black hole metric is a good guess [@Cham2].

Assuming the localized gauge field on the brane is the Maxwell field with action $$S = -\frac{1}{16 \pi G} \int d^4 x \sqrt{-g} F_{\mu\nu}
F^{\mu\nu}.$$ The corresponding localized energy-momentum tensor on the brane can be written as $$\label{Tem}
T_{\mu\nu} = \frac{1}{4\pi G} \left(F_{\mu \rho} F_{\nu}\,^{\rho}
- \frac{1}{4} g_{\mu\nu} F_{\rho\sigma} F^{\rho\sigma} \right).$$ which is traceless, satisfying $T=T_{\mu}^{\ \mu}=0$. We also assume that there is a localized static point charge on the brane which produces an electric field $$\label{Ftr}
F_{tr}=\frac{q}{r^2},$$ where $q$ is the charge parameter. Using metric (\[metric\]), the electric field (\[Ftr\]) and Eq. (\[Tem\]) for the total energy-momentum tensor localized on the brane, one can show that Eq. (\[R\]) has a solution of the form $$\label{f}
f(r)=k-{\frac {2m}{r}}-\frac{\Lambda}{3}{r}^2+{\frac
{\beta+q^2}{{r}^{2}}}+{\frac {1}{240}}\,{\frac
{{\kappa_5}^{4}{q}^{4}}{{r}^{6}}},$$ where $m$ and $\beta$ are arbitrary integration constants and we have assumed $4\pi G =1$, for simplicity. Although in [@Dad], $\beta>0$ has been interpreted as a tidal charge associated with the bulk Weyl tensor, in the presence of localized charge on the brane, it is quite possible to take $\beta<0$ as pointed out in [@Cham2]. Indeed, the projected Weyl tensor, transmits the tidal charge stresses from the bulk to the brane. One may also interpret $\beta$ as a five-dimensional mass parameter [@Cham2]. The horizons can be found by solving Eq. $f(r)=0$. This equation cannot be solved analytically except for $q=0$. The event horizon of the charged braneworld black hole locates at $r_{+}$ where $r=r_{+}$ is the largest root of equation $f(r)=0$. Inserting solution (\[f\]) into field equations (\[eqn\]), we obtain the components of the five-dimensional bulk Weyl tensor. The result is $$\label{weyl}
E^{t}_{\ t}=E^{r}_{\ r}=-E^{i}_{\
i}=\frac{\beta}{r^4}+\frac{1}{24}\frac{\kappa_5^4 q^4}{r^8},$$ where $i=1,2$. Clearly the traceless nature of the Weyl tensor is obeyed. Eqs. (\[eqn\]) with solutions (\[f\]) and (\[weyl\]) form a closed system of equations on the brane.

Some discussions on our solution are needed. In the special case $k=1$ and $\Lambda=0$, $q=0$, our solution (11) reduces to the uncharged braneworld black hole solution found in [@Dad]. In the case $k=1$ and $\Lambda=0$, our solution (11) reduces to the charged black hole solution presented in [@Cham2]. With the presence of the charge on the brane, the bulk geometry has to change, since now $T_{\mu\nu}\neq0$. In other words, the localized charge on the brane will induce changes in the bulk geometry and therefore modifies the bulk Weyl tensor. This property keeps for different topologies of the horizon. Further from Eq. (\[weyl\]) we see that the horizon topology of the braneworld black hole does not affect the bulk geometry and therefore the bulk Weyl tensor is independent of the constant curvature $k$.

In the following we are going to calculate the conserved and thermodynamic quantities of the braneworld black hole. We will adopt a simple strategy based on the profound connection between gravity and thermodynamics which has recently been revealed in various gravity theories [@Jac]-[@Sheywang], showing the deep correspondence between the gravitational equation describing the gravity in the bulk and the first law of thermodynamics on the apparent horizon. This connection sheds the light on holography since the gravitation equations persist the information in the bulk while the first law of thermodynamics on the apparent horizon contains the information on the boundary. Besides, this connection was shown as a useful tool to extract the entropy of the braneworld. In the general case, gravity on the brane does not obey the Einstein theory and the usual area formula for the black hole entropy does not hold on the brane. The relation between the braneworld black hole horizon entropy and its geometry is not known. It was argued in [@Shey1; @Shey2] that the entropy associated with the apparent horizon on the brane can be extracted from the obtained gravity and thermodynamics correspondence. The entropy and temperature associated with the apparent horizon of the FRW universe on the brane, in the RSII braneworld model, are found in [@Cai4; @Shey1] with form $$\begin{aligned}
\label{S}
 S&=&\frac{2 \pi \ell
}{G_{5}}{\displaystyle\int^{\tilde r_A}_0\frac{\tilde{r}_A^{2}
}{\sqrt{\tilde{r}_A^2+\ell^2}}d\tilde{r}_A}
=\frac{2\pi{\tilde{r}_A}^{3}}{3G_{5}}
 \times
{}_2F_1\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},
-\frac{{\tilde{r}_A}^2}{\ell^2}\right),\\
T&=&\frac{1}{2\pi {\tilde{r}_A}},\label{T}\end{aligned}$$ where $\tilde{r}_A$ is the apparent horizon radius and $\ell$ is the AdS radius of the bulk spacetime which is related to the bulk cosmological constant. Here ${}_2F_1(a,b,c,z)$ is a hypergeometric function and $G_{5}=\kappa_{5}^2/8\pi$ is the gravitational constant in five dimensions. Recently, we have shown that the extracted apparent horizon entropy, given in Eq. (\[S\]), satisfies the generalized second law of thermodynamics [@Sheywang]. The satisfaction of the generalized second law of thermodynamics further supports that the entropy (13) is a reasonable thermodynamical entropy describing the brane.

Now we suppose that the temperature and entropy formula (\[S\]) and (\[T\]) also hold on the event horizon of the black hole on the brane. Replacing the apparent horizon radius $\tilde r_A$ by the black hole horizon radius $r_{+}$, we have the temperature and entropy on the event horizon of the braneworld black hole $$\begin{aligned}
\label{S2}
S&=&\frac{2 \pi \ell
}{G_{5}}{\displaystyle\int^{r_{+}}_0\frac{r_{+}^{2}
}{\sqrt{r_{+}^2+\ell^2}}dr_{+}}=\frac{2\pi{r_{+}}^{3}}{3G_{5}}
 \times
{}_2F_1\left(\frac{3}{2},\frac{1}{2},\frac{5}{2},
-\frac{r_{+}^2}{\ell^2}\right),\\
T&=&\frac{1}{2\pi r_{+}}.\label{T2}\end{aligned}$$ Eq. (16) is exactly the Hawking temperature on the event horizon. The validity of (15) to describe the event horizon entropy of the braneworld black hole can be justified by considering its limiting case with $\tilde{r}_{+} \ll\ell$. Physically this limit means that the size of extra dimension is very large if compared with the black hole event horizon radius. In this limit Eq. (15) reduces to the five-dimensional area formula for the black hole entropy $S =2\Omega_{3}{\tilde{r}_+}^{3}/4G_{5}$, where $\Omega_{3}=4\pi/3$ is the volume of a unit sphere. The factor $2$ comes from the $\mathbb{Z}_2$ symmetry in the bulk. This is an expected result since in this regime the anti de-Sitter bulk reduces to the Minkowski spacetime. And due to the absence of the negative cosmological constant in the Minkowski bulk, no localization of gravity happens on the brane. Thus the gravity on the brane is still five-dimensional and the entropy formula on the black hole event horizon obeys the five-dimensional area formula [@Shey1].

Adopting the first law of black hole thermodynamics on the event horizon $r_{+}$ and considering that the electric charge of black hole does not affect its mass, we just need to discuss the uncharged case with the first law $$\label{frst}
dM=TdS.$$ Integrating (\[frst\]) and inserting (\[S2\]) and (\[T2\]), we obtain the mass of the braneworld black hole $$\label{mass2}
M=\frac{\ell}{G_{5}}\int^{r_{+}}_0 \frac{ r_{+}d
r_{+}}{\sqrt{r_{+}^2+\ell^2}}=\frac{\ell}{G_{5}}\left(\sqrt{r_{+}^2+\ell^2}-\ell
\right).$$ It is interesting to see that in the limiting case $\tilde{r}_{+}
\ll\ell$, the mass formula (\[mass2\]) reduces to $$\label{mass3}
M=\frac{r_{+}^2}{2 G_{5}},$$ which is exactly the mass of the five-dimensional black hole in Einstein gravity.

In conclusion, we have obtained a class of topological black hole solutions in RSII braneworld scenario in the presence of a localized Maxwell field on the brane. We have shown that the localized charge on the brane modifies the bulk geometry and in particular the bulk Weyl tensor. The horizon topology of the braneworld black holes does not affect the geometry of extra dimension. We presented the temperature and entropy expressions associated with the event horizon of the braneworld black hole. We also obtained the mass of the braneworld black holes through the use of the first law of black hole thermodynamics.

We would like to mention here that in this Letter we have not studied fully the effect of the braneworld black hole on the bulk geometry, and in particular the nature of the off-brane horizon structure. This has been done for solutions which reduce to the Schwarzschild black hole on the brane [@Cham1]. We have adopted a different approach: instead of starting from an induced metric on the brane, we have solved the closed system of the effective field equations for the induced metric on the brane in RSII model, and found a class of topological braneworld black holes. Therefore the main problem remains to find the exact bulk metric that describes a topological braneworld black hole. This was solved for uncharged black holes in three dimensions [@Emp]. Unfortunately, the higher dimensional generalization of this metric is still not known. In general the bulk spacetime may be given, by solving the full five-dimensional equations, and the geometry of the embedded brane is then deduced. Due to the complexity of the five-dimensional equations, one may follow the strategy outlined in this Letter, by considering the intrinsic geometry on the brane, which encompasses the imprint from the bulk, and consequently evolve the metric off the brane. However, in this Letter we did not study the effects of the braneworld black hole on the bulk geometry, and in particular the nature of the topological horizon structure in the bulk. Indeed, determining the bulk geometry is an extremely difficult task which needs numerical calculations, so it was not explored here.

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[^1]: sheykhi@mail.uk.ac.ir

[^2]: wangb@fudan.edu.cn
