# A Topologically Charged Black Hole in thefℛ Braneworld

Authors: Alexis Larrañaga; Andres Rengifo; Luis Cabarique
Journal: Advances in High Energy Physics
 (2013)

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## Abstract

We find a new general black hole solution in the braneworld scenario, considering a modified 5-dimensionalfℛ action in the bulk. We study the horizon structure and find the possibility of two, one, or no horizon depending on the value of the topological parameter β. On the thermodynamics side, we show that the value of the topological parameter determines the black hole temperature to have a divergent behaviour at small scales or to present a maximum value before cooling down towards a zero temperature remnant.


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## Body

## 1. Introduction

The braneworld scenario describes our 4-dimensional world as a brane that is embedded in a higher dimensional bulk and that supports all gauge fields excluding the gravitational field that lives in the whole spacetime. There are many braneworld models in the cosmological context as well as descriptions of local self-gravitating objects. One of the most successful models is that proposed by Randall and Sundrum [1, 2], where the 5-dimensional bulk has the geometry of an AdS space. The confinement of ordinary matter and gauge fields to the brane is achieved by the application of the Israel junction conditions. Even more, Shiromizu et al. [3] showed how to systematically project the Einstein field equations, written in the high dimensional bulk, onto the brane using the Gauss-Codazzi equations and the Israel junction conditions and assuming ℤ2 symmetry.On the other hand, theories of gravity, in which the Einstein-Hilbert action is replaced with a generic function of the Ricci scalar,f(R), the Einstein tensor, f(G), or both, f(R,G), have been studied recently as natural scenarios that unify and explain both, the inflationary paradigm and the dark energy problem [4]. However, all these models were considered in four dimensions. Therefore, it is interesting to consider both, the braneworld model and the modified gravity, as a unified scenario that we hope could be able to describe some of the cosmological puzzles and also give some new lights on the behaviour of black holes.The study of a Randall-Sundrum type model withf(ℛ) as the action in bulk (ℛ is the Ricci scalar in the bulk) has been presented by Borzou et al. [5]. They derive the field equations on the brane by using the Shiromizu-Maeda-Sasaki methods and the result is a set of field equations that differ from the standard Einstein field equations in 4D by additional terms like πμν, which depend on the energy-momentum tensor on the brane, the projection of the Weyl tensor, Eμν, and a quantity Qμν which originates in the geometry of the bulk space, specifically in the function f(ℛ). It is interesting that, for a conformally flat bulk, the quantity Qμν is conserved and could therefore be identified with a new kind of matter.Although black hole solutions on the brane are interesting because they have considerably richer physical aspects than black holes in general relativity and have been studied extensively in recent years [6–14], the obtention of such types of solutions in f(ℛ) braneworlds has been lacking. Therefore, the main purpose of this paper is to solve the gravitational field equations on the brane presented in [5] to obtain a topological braneworld black hole, which will be the generalisation of the one presented in [15] to include the effects of the modified action f(ℛ). In the last section of this work, we also present the temperature associated with the event horizon of the braneworld black hole and, from its characteristics, we conclude that our solution presents the interesting case of an extremal configuration for which the surface gravity vanishes and so does its temperature. As is well known, this kind of configurations present in general relativity [16], and alternative gravitational theories (see e.g., [17, 18]) and its properties have been studied extensively [19–21]. However, the possible interpretation of extremal configurations as elementary string excitations gives them a special interest in string theory [22].


## 2. The Field Equations on the Brane

The starting point is the action for the bulk,(1)S=∫‍d5x-𝔤f(ℛ)+SM,
where 𝔤 is the determinant of the bulk metric, ℛ is the bulk Ricci scalar, and SM is the matter action. This gives the 5-dimensional field equations:
(2)𝒢AB=ℛAB-12𝔤ABℛ=𝒯AB,
where
(3)𝒯AB=1f′(ℛ)[κ52TAB-(12ℛf′(ℛ)-12f(ℛ)+f′(ℛ))𝔤AB+∇A∇Bf′(ℛ)(12ℛf′(ℛ)-12f(ℛ)+f′(ℛ))],
and f′(ℛ)=(df/dℛ). Assuming that the brane is located at y=0, we write the energy-momentum tensor as
(4)TAB=-Λ5𝔤AB+SABδ(y),
where Λ5 is the cosmological constant in the bulk and
(5)Sμν=-λgμν+τμν
with λ being the brane tension and τμν the energy-momentum tensor on the brane. Contracting the 5-dimensional field equation and taking into account the decomposition of the energy-momentum tensor in (4) and (5), the 5-dimensional Ricci scalar can be written as
(6)ℛ=-5κ52Λ5f′(ℛ)+52f(ℛ)f′(ℛ)-4□f′(ℛ)f′(ℛ).In order to obtain the gravitational field on the brane, the Shiromizu-Maeda-Sasaki method considers the Gauss-Codazzi equations of 5-dimensional gravity and the Israel junction conditions to obtain the effective field equations [3]:
(7)Gμν=-Λgμν+8πGτμν+κ52πμν+Qμν-Eμν,
where Gμν=Rμν-(1/2)gμνR, is the 4-dimensional Einstein tensor, and κ5 is the 5-dimensional gravity coupling constant,
(8)κ54=(8πG5)2
with G5 being the gravitational constant in five dimensions and Λ the 4-dimensional cosmological constant that is given in terms of the 5-dimensional cosmological constant Λ5 and the brane tension λ by
(9)Λ=κ522(Λ5+κ526λ2).π
μ
ν is a quadratic tensor in the energy-momentum tensor given by
(10)πμν=112ττμν-14τμστνσ+18gμν(ταβταβ-13τ2),
with τ=τσσ, while Eμν is the projection of the 5-dimensional bulk Weyl tensor CABCD on the brane and can be written as Eμν=δμAδνBCABCDnAnB with nA the unit normal to the brane. Eμν represents the nonlocal bulk effect and its only known property is that it is traceless, Eσσ=0.Finally, the termQμν encompasses the f(ℛ) effects because it completely depends on this function and its derivatives. It is given by
(11)Qμν=[ℱ(ℛ)gμν+23∇A∇Bf′(ℛ)f′(ℛ)(δμAδνB+nAnBgμν)]y=0,
where
(12)ℱ(ℛ)=-415□f′(ℛ)f′(ℛ)-110ℛ(32+f′(ℛ))+14f(ℛ)-25□f′(ℛ).For simplicity, in order to obtain the black hole solution, we will consider the following traced field equation:(13)R=4Λ-8πGτ-κ544(ταβταβ-13τ2)-Q
with Q=Qσσ.


## 3. Black Hole on the Brane

We will propose a solution on the brane with the following form:(14)ds2=-A(r)dt2+dr2A(r)+r2dΩk2,
where dΩk2 is the line element of a two-dimensional hypersurface with constant curvature. For k=1, the topology of the event horizon is a two-sphere, for k=0, the event horizon corresponds to a torus, and for k=-1, it corresponds to a hypersurface with constant negative curvature. The corresponding line elements can be written as
(15)dΩk2={dθ2+sin2θdφ2ifk=1dθ2+θ2dφ2ifk=0dθ2+sinh2θdφ2ifk=-1.We are interested primarily on thek=1 solution, and we impose the condition grr=-gtt-1 because we want the induced metric to be close to Schwarzschild's solution. By considering a constant Ricci curvature scalar ℛ and solving the field equations (13) in vacuum (τμν=0), we obtain the line element coefficient as follows:
(16)A(r)=k-2Mr+βr2+Λeff3r2,
where M and β are integration constants, and
(17)Λeff=Λ-Q4
plays the role of an effective cosmological constant on the brane depending on the tension of the brane and the function f(ℛ). The constant β can be interpreted as a five-dimensional mass parameter [7, 15], but it is more useful to think β as a tidal charge associated with the bulk Weyl tensor, and therefore, it can take positive as well as negative values. Indeed, the projected Weyl tensor transmits the tidal charge stresses from the bulk to the brane. This fact can be seen by inserting solution (16) into field equations (7) to obtain the following components:
(18)Ett=Err=-Eθθ=-Eφφ=-βr4.It is clear that the traceless nature of the Weyl tensor is obeyed, and it is interesting to note that the horizon topology of the braneworld black hole does not affect the bulk geometry, that is, the bulk Weyl tensor is independent of the constant curvaturek.In the special casek=1 and Q=4Λ (i.e., Λeff=0), our solution reduces to the uncharged braneworld black hole solution found in [23]. For Q=0, our solution reduces to the topologically charged black hole solution in the braneworld presented in [15]. Now, let us consider two particular cases for the function f(ℛ).


### 3.1.f(ℛ)=ℛn Solution

Considering constant curvature and the particular functionf(ℛ)=ℛn, (6) and (11) give the components of Qμν as
(19)Qμν=[κ52Λ52-320(10κ52Λ55-2n)1/n]δμν,
and therefore, its trace is simply
(20)Q=2κ52Λ5-35(10κ52Λ55-2n)1/n.The effective cosmological constant on the brane induced by the tension of the brane and the functionf(ℛ) is
(21)Λeff=κ5412λ2+320(10κ52Λ55-2n)1/n.Note that the particular casen=1 gives the usual result Λeff=Λ, and the black hole reduces to Schwarzschild's metric with cosmological constant and tidal charge.


### 3.2.f(ℛ)=ℛ+(μ4/ℛ) Solution

The functionf(ℛ)=ℛ+(μ4/ℛ) is a good model to explain the positive acceleration of the expanding universe. In this model, a large value of ℛ gives f(ℛ)≈ℛ, so we may expect for those values of the 5-dimensional Ricci scalar a negligible modification of the usual solution. However, for small values of ℛ, gravity is modified.From (6) and (11), we obtain two possible values for components of Qμν as follows:
(22)Qμν=[-21μ420(5Λ5κ52±21μ4+25Λ52κ54)]δμν,
and therefore, its trace is
(23)Q=-21μ45(5Λ5κ52±21μ4+25Λ52κ54).The effective cosmological constant on the brane takes the values as follows:(24)Λeff=Λ+21μ420(5Λ5κ52±21μ4+25Λ52κ54).The resultΛeff=Λ is obtained when μ=0 (i.e., when the modification in f(ℛ) is turned off).


## 3.1.f(ℛ)=ℛn Solution

Considering constant curvature and the particular functionf(ℛ)=ℛn, (6) and (11) give the components of Qμν as
(19)Qμν=[κ52Λ52-320(10κ52Λ55-2n)1/n]δμν,
and therefore, its trace is simply
(20)Q=2κ52Λ5-35(10κ52Λ55-2n)1/n.The effective cosmological constant on the brane induced by the tension of the brane and the functionf(ℛ) is
(21)Λeff=κ5412λ2+320(10κ52Λ55-2n)1/n.Note that the particular casen=1 gives the usual result Λeff=Λ, and the black hole reduces to Schwarzschild's metric with cosmological constant and tidal charge.


## 3.2.f(ℛ)=ℛ+(μ4/ℛ) Solution

The functionf(ℛ)=ℛ+(μ4/ℛ) is a good model to explain the positive acceleration of the expanding universe. In this model, a large value of ℛ gives f(ℛ)≈ℛ, so we may expect for those values of the 5-dimensional Ricci scalar a negligible modification of the usual solution. However, for small values of ℛ, gravity is modified.From (6) and (11), we obtain two possible values for components of Qμν as follows:
(22)Qμν=[-21μ420(5Λ5κ52±21μ4+25Λ52κ54)]δμν,
and therefore, its trace is
(23)Q=-21μ45(5Λ5κ52±21μ4+25Λ52κ54).The effective cosmological constant on the brane takes the values as follows:(24)Λeff=Λ+21μ420(5Λ5κ52±21μ4+25Λ52κ54).The resultΛeff=Λ is obtained when μ=0 (i.e., when the modification in f(ℛ) is turned off).


## 4. Thermodynamics

Now we will consider only the case in whichk=1, that is, when the surface of the event horizon is a 2-sphere. The largest root of the depressed quartic equation A(r+)=0, or
(25)r+4+3Λeffr+2-6ΛeffMr++3βΛeff=0,
corresponds to the event horizon radius r=r+. The nature of the roots of this equation is commented in the appendix. However, we will present here a graphical analysis of the characteristic behaviour of this polynomial. In Figure 1, we plot function A(r) for a fixed value of parameter M and different values of Λ and β. The position of the horizons can be seen by looking at the intersections with the r-axis. In Figure 1(a), we set Λ=0, while Figure 1(b) has Λ>0. Note that in both cases, values of β≤0 give one horizon (β=0 corresponds to Schwarzschild's black hole), while values of β>0 present the possibility of two horizons (when β is less than certain critical value βc) and the existence of an extremal configuration (β=βc), resembling the behaviour of Reissner-Nordström or Reissner-Nordström-AdS solutions, respectively. Values of β above the extremal value βc do not show horizons, and therefore, we conclude that they do not correspond to black holes. In Figure 1(c) we consider the Λ<0 case, and the behaviour shows no black hole solutions.A
(
r
) as function of r for different values of Λ and β. The intersections with the r-axis give inner and outer horizons. In (a), we set Λ=0, and (b) has Λ>0. In both cases, values of β≤0 give one horizon, while β>0 give two horizons or an extremal configuration (β=βc), resembling the behaviour of Reissner-Nordström solution. Above this extremal value, there is no black hole. In (c), we consider the Λ<0 case, and the behaviour shows no black hole solution.
(a)
Λ = 0
(b)
Λ > 0
(c)
Λ < 0The extremal black hole is described by the following conditions:(26)A(re)=1-2Mre+βre2+Λeff3re2=0,∂rA(re)=2re[Mre-βre2+Λeff3re2]=0,
from which we obtain the value of the critical parameter βc as
(27)βc=98M2-18[2[(6M+2(9ΛeffM2+2)/Λeff)Λeff2]1/31Λeff[(6M+29ΛeffM2+2Λeff)Λeff2]1/3-2[(6M+2(9ΛeffM2+2)/Λeff)Λeff2]1/3-3M1Λeff[(6M+29ΛeffM2+2Λeff)Λeff2]13]2.The temperature of the black hole is defined in terms of the surface gravity atr+ by
(28)T=κ+2π=14π∂rA(r)|r=r+=12πr+[Mr+-βr+2+Λeff3r+2].In Figure2, we can see the plot of the temperature of the black hole as function of the radius of the event horizon r+ for different values of Λ and β. In Figure 2(a), we set Λ=0, and, is clear that β=0 gives Schwarzschild's Hawking temperature. The behaviour for β<0 reproduces Schwarzschild's divergence for r+→0. However, it is very interesting to note that values of β>0 give a new behaviour: for large values of r+, the temperatures coincide with Schwarzschild's, but for small radii, our solution admits a maximum value Tmax. From this point, the function goes to the zero temperature of the extremal black hole remnant configuration. In Figure 2(b), we set Λ>0. This time, the value of β=0 gives Schwarzschild-AdS temperature, and the behaviour for β<0 is similar to this function. Again, values of β>0 give a function that goes to the zero temperature of the extremal black hole. We do not consider the Λ<0 case because there is no black hole solution and therefore no interesting temperature.Temperature of the black hole as function of the radius of the event horizonr+ for different values of Λ and β. In (a), we set Λ=0. For β=0, we obtain Schwarzschild's temperature and a similar behaviour for β<0. Values of β>0 give a temperature that goes to zero for an extremal black hole remnant configuration. In  (b), we set Λ>0. The value of β=0 gives Schwarzschild-AdS temperature and similar behaviour for β<0. Values of β>0 give again a function that goes to the zero temperature extremal black hole. In both cases, the value β=βc.
(a)
Λ = 0
(b)
Λ > 0


## 5. Conclusions

We have obtained a general black hole solution in the braneworld scenario modified by a bulk action in the formf(ℛ). Due to the complexity of the five-dimensional equations, we have solved the effective field equations for the induced metric on the brane derived by the Shiromizu-Maeda-Sasaki method to obtain a topologically charged solution that generalises the metric presented in [15] to include the effects of the function f(ℛ). Therefore, we have not studied fully the effect of the braneworld black hole on the bulk geometry.In the last section, we also present the temperature associated with the event horizon of the braneworld black hole. The presented analysis of the solution shows that the geometry admits two, one, or no horizon depending on the value of the topological parameterβ with respect to two threshold masses β=0 and β=βc. From the thermodynamical analysis, the possibility of a degenerate horizon gives a temperature that, instead of a divergent behaviour at short scales, admits both a minimum and a maximum before cooling down towards a zero temperature remnant configuration.Finally, it is important to stress that the quantityQμν which originates from the extra geometry terms, is conserved for a conformally flat bulk, and therefore, it can be identified with a new kind of matter [5]. Since this quantity enters in function (16) through the parameter β, our solution could be used to explain the galaxy rotation curves by considering it as the metric describing the central black hole.



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