A Topologically Charged Black Hole in the 𝑓(ℛ) Braneworld

Abstract

We find a new general black hole solution in the braneworld scenario, considering a modified 5-dimensional 𝑓(ℛ) 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.

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 with 𝑓(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 𝜋_𝜇𝜈, which depend on the energy-momentum tensor on the brane, the projection of the Weyl tensor, 𝐸_𝜇𝜈, and a quantity 𝑄_𝜇𝜈 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 𝑄_𝜇𝜈 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, 𝑆=𝑑⁵𝑥√−𝔤𝑓(ℛ)+𝑆_𝑀, (1) where 𝔤 is the determinant of the bulk metric, ℛ is the bulk Ricci scalar, and 𝑆_𝑀 is the matter action. This gives the 5-dimensional field equations: 𝒢_𝐴𝐵=ℛ_𝐴𝐵−12𝔤_𝐴𝐵ℛ=𝒯_𝐴𝐵, (2) where 𝒯_𝐴𝐵=1𝑓^′(ℛ)𝜅²₅𝑇_𝐴𝐵−12ℛ𝑓^′(ℛ)−12𝑓(ℛ)+𝑓^′(ℛ)𝔤_𝐴𝐵+∇_𝐴∇_𝐵𝑓^′(ℛ), (3) and 𝑓^′(ℛ)=(𝑑𝑓/𝑑ℛ). Assuming that the brane is located at 𝑦=0, we write the energy-momentum tensor as 𝑇_𝐴𝐵=−Λ₅𝔤_𝐴𝐵+𝑆_𝐴𝐵𝛿𝑦, (4) where Λ₅ is the cosmological constant in the bulk and 𝑆_𝜇𝜈=−𝜆𝑔_𝜇𝜈+𝜏_𝜇𝜈 (5) with 𝜆 being the brane tension and 𝜏_𝜇𝜈 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 ℛ=−5𝜅²₅Λ₅𝑓^′(ℛ)+52𝑓(ℛ)𝑓^′(ℛ)−4□𝑓^′(ℛ)𝑓^′(ℛ). (6)

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]: 𝐺_𝜇𝜈=−Λ𝑔_𝜇𝜈+8𝜋𝐺𝜏_𝜇𝜈+𝜅²₅𝜋_𝜇𝜈+𝑄_𝜇𝜈−𝐸_𝜇𝜈, (7) where 𝐺_𝜇𝜈=𝑅_𝜇𝜈−(1/2)𝑔_𝜇𝜈𝑅, is the 4-dimensional Einstein tensor, and 𝜅₅ is the 5-dimensional gravity coupling constant, 𝜅⁴₅=8𝜋𝐺₅² (8) 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 𝜅Λ=²⁵2Λ₅+𝜅²₅6𝜆². (9)

𝜋_𝜇𝜈 is a quadratic tensor in the energy-momentum tensor given by 𝜋_𝜇𝜈=112𝜏𝜏_𝜇𝜈−14𝜏_𝜇𝜎𝜏^𝜎_𝜈+18𝑔_𝜇𝜈𝜏_𝛼𝛽𝜏^𝛼𝛽−13𝜏², (10) with 𝜏=𝜏^𝜎_𝜎, while 𝐸_𝜇𝜈 is the projection of the 5-dimensional bulk Weyl tensor 𝐶_{𝐴𝐵𝐶𝐷} on the brane and can be written as 𝐸_𝜇𝜈=𝛿^𝐴_𝜇𝛿^𝐵_𝜈𝐶_{𝐴𝐵𝐶𝐷}𝑛^𝐴𝑛^𝐵 with 𝑛^𝐴 the unit normal to the brane. 𝐸_𝜇𝜈 represents the nonlocal bulk effect and its only known property is that it is traceless, 𝐸^𝜎_𝜎=0.

Finally, the term 𝑄_𝜇𝜈 encompasses the 𝑓(ℛ) effects because it completely depends on this function and its derivatives. It is given by 𝑄_𝜇𝜈=ℱ(ℛ)𝑔_𝜇𝜈+23∇_𝐴∇_𝐵𝑓^′(ℛ)𝑓^′(ℛ)𝛿^𝐴_𝜇𝛿^𝐵_𝜈+𝑛^𝐴𝑛^𝐵𝑔_{𝜇𝜈𝑦=0}, (11) where ℱ(ℛ)4=−15□𝑓^′(ℛ)𝑓^′(ℛ)−1ℛ3102+𝑓^′(ℛ)+14𝑓(ℛ)−25□𝑓^′(ℛ). (12)

For simplicity, in order to obtain the black hole solution, we will consider the following traced field equation: 𝜅𝑅=4Λ−8𝜋𝐺𝜏−⁴⁵4𝜏_𝛼𝛽𝜏^𝛼𝛽−13𝜏²−𝑄 (13) with 𝑄=𝑄^𝜎_𝜎.

3. Black Hole on the Brane

We will propose a solution on the brane with the following form: 𝑑𝑠²=−𝐴(𝑟)𝑑𝑡²+𝑑𝑟²𝐴(𝑟)+𝑟²𝑑Ω²_𝑘, (14) where 𝑑Ω²_𝑘 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 𝑑Ω²_𝑘=𝑑𝜃²+sin²𝜃𝑑𝜑²if𝑘=1𝑑𝜃²+𝜃²𝑑𝜑²if𝑘=0𝑑𝜃²+sinh²𝜃𝑑𝜑²if𝑘=−1. (15)

We are interested primarily on the 𝑘=1 solution, and we impose the condition 𝑔_𝑟𝑟=−𝑔⁻¹_𝑡𝑡 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 (𝜏_𝜇𝜈=0), we obtain the line element coefficient as follows: 𝐴(𝑟)=𝑘−2𝑀𝑟+𝛽𝑟²+Λ_eff3𝑟², (16) where 𝑀 and 𝛽 are integration constants, and Λ_eff𝑄=Λ−4 (17) 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: 𝐸^𝑡_𝑡=𝐸^𝑟_𝑟=−𝐸^𝜃_𝜃=−𝐸^𝜑_𝜑𝛽=−𝑟⁴. (18)

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 curvature 𝑘.

In the special case 𝑘=1 and 𝑄=4Λ (i.e., Λ_eff=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)=ℛ^𝑛 Solution

Considering constant curvature and the particular function 𝑓(ℛ)=ℛ^𝑛, (6) and (11) give the components of 𝑄_𝜇𝜈 as 𝑄^𝜈_𝜇=𝜅²₅Λ₅2−32010𝜅²₅Λ₅5−2𝑛^1/𝑛𝛿^𝜈_𝜇, (19) and therefore, its trace is simply 𝑄=2𝜅²₅Λ₅−3510𝜅²₅Λ₅5−2𝑛^1/𝑛. (20)

The effective cosmological constant on the brane induced by the tension of the brane and the function 𝑓(ℛ) is Λ_eff=𝜅⁴₅𝜆12²+32010𝜅²₅Λ₅5−2𝑛^1/𝑛. (21)

Note that the particular case 𝑛=1 gives the usual result Λ_eff=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 function 𝑓(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 𝑄_𝜇𝜈 as follows: 𝑄^𝜈_𝜇=−21𝜇⁴205Λ₅𝜅²₅±21𝜇⁴+25Λ²₅𝜅⁴₅𝛿^𝜈_𝜇, (22) and therefore, its trace is 𝑄=−21𝜇⁴55Λ₅𝜅²₅±21𝜇⁴+25Λ²₅𝜅⁴₅. (23)

The effective cosmological constant on the brane takes the values as follows: Λ_eff=Λ+21𝜇⁴205Λ₅𝜅²₅±21𝜇⁴+25Λ²₅𝜅⁴₅. (24)

The result Λ_eff=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 which 𝑘=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 𝑟⁴₊+3Λ_eff𝑟²₊−6Λ_eff𝑀𝑟₊+3𝛽Λ_eff=0, (25) 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 𝛽_𝑐) and the existence of an extremal configuration (𝛽=𝛽_𝑐), resembling the behaviour of Reissner-Nordström or Reissner-Nordström-AdS solutions, respectively. Values of 𝛽 above the extremal value 𝛽_𝑐 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)  Λ = 0

(b) Λ > 0

(c) Λ < 0

Figure 1 

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.

The extremal black hole is described by the following conditions: 𝐴𝑟_𝑒=1−2𝑀𝑟_𝑒+𝛽𝑟²_𝑒+Λ_eff3𝑟²_𝑒𝜕=0,_𝑟𝐴𝑟_𝑒=2𝑟_𝑒𝑀𝑟_𝑒−𝛽𝑟²_𝑒+Λ_eff3𝑟²_𝑒=0, (26) from which we obtain the value of the critical parameter 𝛽_𝑐 as 𝛽_𝑐=98𝑀²−181Λ_eff6𝑀+29Λ_eff𝑀²+2Λ_effΛ²_eff^1/3−26𝑀+29Λ_eff𝑀²+2/Λ_effΛ²_eff^1/3−3𝑀². (27)

The temperature of the black hole is defined in terms of the surface gravity at 𝑟₊ by 𝜅𝑇=⁺=12𝜋𝜕4𝜋^𝑟𝐴(𝑟)_𝑟=𝑟+=12𝜋𝑟₊𝑀𝑟₊−𝛽𝑟²₊+Λ_eff3𝑟²₊. (28)

In Figure 2, 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 𝑇_max. 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.

(a) Λ = 0

(b) Λ > 0

Figure 2 

Temperature of the black hole as function of the radius of the event horizon 𝑟+ 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.

5. Conclusions

We have obtained a general black hole solution in the braneworld scenario modified by a bulk action in the form 𝑓(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 parameter 𝛽 with respect to two threshold masses 𝛽=0 and 𝛽=𝛽_𝑐. 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 quantity 𝑄_𝜇𝜈 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.

Appendix

The nature of the roots of the depressed quartic polynomial 𝑓(𝑥)=𝑥⁴+𝑎₃𝑥³+𝑎₁𝑥+𝑎₀ (A.1) with 𝑎₀,𝑎₁, and 𝑎₃ being real can be determined by its discriminant Δ [24]. Denoting by 𝛼_𝑗 the roots of 𝑓(𝑥)=0, we define the discriminant as the product of the squares of the differences of the roots: Δ=_{𝑖𝑗>𝑖}𝛼_𝑖−𝛼_𝑗². (A.2) The criteria regarding the nature of the roots are as follows.

Case I (𝑎₃<0)(i)𝑎₀>𝑎²₃/4;  Δ<0: two roots real and distinct, Δ=0: two roots real and equal, Δ>0: no real roots. (ii)𝑎₀=𝑎²₃/4;  Δ<0: two roots real and distinct, Δ=0: two pairs of equal real roots. (iii)−𝑎²₃/12<𝑎₀<𝑎²₃/4;  Δ<0: two roots real and distinct, Δ=0: all roots real, two equal, Δ>0: all roots real and distinct. (iv)𝑎₀=−𝑎²₃/12;  Δ<0: two roots real and distinct,  Δ=0: all roots real, three equal. (v)𝑎₀<−𝑎²₃/12;  Δ<0: two roots real and distinct. Case II (𝑎₃=0)(i)𝑎₀>0;  Δ<0: two roots real and distinct, Δ=0: two roots real and equal, Δ>0: no real roots.(ii)𝑎₀=0;  Δ<0: two roots real and distinct, Δ=0: four equal real roots. (iii)𝑎₀<0;  Δ<0: two roots real and distinct. Case III (𝑎₃>0)(i)𝑎₀>0; Δ<0: two roots real and distinct, Δ=0: two roots real and equal, Δ>0: no real roots. (ii)𝑎₀=0; Δ<0: two roots real and distinct, Δ=0: two roots real and equal. (iii)𝑎₀<0;  Δ<0: two roots real and distinct.

In particular, for the horizon equation (25), the coefficients of the quartic equation in terms of the parameters 𝑀,𝛽, and Λ_eff≠0 are 𝑎₃=3Λ_eff,𝑎₁=−6𝑀Λ_eff,𝑎₀=3𝛽Λ_eff, (A.3) and following the condition for the parameters, the criteria for the nature of the possible roots become as follows.

Case I (Λ_eff<0) No black holes; therefore, it is not considered.

Case II   (1/Λ_eff→0)(i)𝛽>0; Δ<0: two roots real and distinct, Δ=0: two roots real and equal, Δ>0: no real roots.(ii)𝛽=0; Δ<0: two roots real and distinct, Δ=0: four equal real roots.(iii)𝛽<0; Δ<0: two roots real and distinct. Case III  (Λ_eff>0)(i)𝛽>0;  Δ<0: two roots real and distinct, Δ=0: two roots real and equal, Δ>0: no real roots.(ii)𝛽=0;  Δ<0: two roots real and distinct, Δ=0: two roots real and equal. (iii)𝛽<0;  Δ<0: two roots real and distinct.In order to solve the depressed polynomial (25), we may note that it can be factorized by solving the following resolvent cubic [25]: 𝑧³−3Λ_eff𝑧²−12𝛽Λ_eff𝑧+36Λ²_eff𝛽−𝑀²=0. (A.4) The nature of the roots of this equation depends on the discriminant as follows: 1𝛿=Λ⁴_eff12𝛽−18𝑀²−1Λ_eff²−1Λ³_eff14𝛽+Λ_eff³. (A.5) If 𝛿>0, there is one real root and two conjugates imaginary roots. If 𝛿=0, there are three real roots of which at least two are equal. If 𝛿<0, there are three real and unequal roots.

In any case, we can obtain one real root 𝑧=𝑧₁ of the resolvent cubic from which we build the four roots of the original quartic equation as 𝑟₁=12(),𝑟𝑅+𝐷₂=12(),𝑟𝑅−𝐷₃1=−2(),𝑟𝑅−𝐸₄1=−2(𝑅+𝐸), (A.6) where 𝑅=𝑧₁−3Λ_eff,𝐷=12𝑀Λ_eff𝑅−𝑧₁−3Λ_effif𝑅≠0,−6Λ_eff+2𝑧²₁−12𝛽Λ_effif𝑅=0,𝐸=−12𝑀Λ_eff𝑅−𝑧₁−3Λ_effif𝑅≠0,−6Λ_eff−2𝑧²₁−12𝛽Λ_effif𝑅=0. (A.7)

Acknowledgment

This work was supported by the Universidad Nacional de Colombia, Hermes Project Code 17318.