A new exact solution of black-strings-like with a dS core
aa r X i v : . [ g r- q c ] F e b A new exact solution of black-strings-like with a dS core
Milko Estrada ∗ Departmento de f´ısica, Universidad de Antofagasta, 1240000 Antofagasta, Chile (Dated: February 17, 2021)
Abstract
We provide a new five dimensional black string–like solution by means of the embedding of a fourdimensional regular black hole into a compact extra dimension. We enunciate a list of constraintsin order to that the five dimensional black string–like solution to be regular, and, following theseconstraints we construct our solution. Instead of the usual singularity, it is formed a core whosetopology corresponds to the product between the de–Sitter core of the four dimensional Haywardsolution and S . The horizon has topology S × S . At infinity of the radial coordinate the regularfour dimensional geometry is asymptotically flat, i.e , at this place the topology of the completesolution corresponds to the product between Minkowski and S . At the induced four dimensionalgeometry we compute the correct values of temperature and entropy. ∗ [email protected] . INTRODUCTION In the last decades, several branches of theoretical physics have predicted the existence ofextra dimensions, examples of this latter are the string theory, brane world models, etc. Inthis context, the study of higher dimensional black hole solutions have showed some differentfeatures respect to the well known four dimensional solutions. Examples of this latter arehigher dimensional black holes in modified theories of gravity, black holes in brane worldmodels, black strings (branes) and black rings [1].Black string are the simplest extension of the black hole solutions. The simplest versionof black string is given by the line element : ds D = ds D + dz , (1)where ds D corresponds to the four dimensional Schwarzschild space time. The line element(1) corresponds to the Kaluza-Klein black string [2]. So, the last equation corresponds to asolution of the Einstein field equations G MN = 0 and it was obtained by the oxidation of theSchwarzschild solution [3]. This solution has S × R horizon topology for z non compact, i.e has a cylindrical event horizon [2]. If the extra coordinate z is compact the topology ofthe horizon is S × S . The horizon topology differs from the usual S topology of the fivedimensional Schwarzschild-Tangherlini solution [4]. On the other hand, the topology of thecentral singularity corresponds to the product between the Schwarzschild singularity and R ( S ) for a non compact (compact) extra coordinate.Black string have been widely studied in literature for different contexts. For examplesolutions in GR with cosmological constant [5], with axionic scalar fields in Horndeski gravity[6], in f ( R ) gravity [7], Lovelock gravity [8] and in brane world models [9–11]. Otherexamples of recent applications in references [12–14].Black strings with a central singularity are easily constructed by embedding singular blackhole solutions into an extra dimension. However, in the literature there are also non singularblack holes solutions that could be embedded into an extra dimension, which clearly wouldnot lead to a central singularity. These latter solutions are known as regular black holes . Inthese models, instead the formation of a central singularity, it is formed a de–Sitter core , i.e , the solution behaves as a de–Sitter space time near the origin. Thus, all the curvatureinvariants are regular everywhere. One very interesting and intriguing model is the Hayward2etric [15]. Using this metric, the formation of the Sitter core is associated to quantumfluctuations, where the energy density is of order of Planck units near the origin. This lattermodel is called Planck stars [16, 17]. Although these models are theoretical, in reference[18], using radio astronomy data, is conjetured that Planck Stars represent a speculative butrealistic possibility of testing quantum gravity effects.Thus, motivated by the number of applications of black strings, where are embeddedsingular black holes into an extra dimension, is of physical interest the embedding of regularblack hole solutions also into an extra dimension. In this work, we will study this lattermentioned problem. For this, we will use the Hayward metric as toy model. We will computethe five dimensional solution and analyze the way how this solution modifies locally the fourdimensional geometry and the information of the energy momentum tensor. Furthermore,we will discuss the first law of thermodynamics at the induced four dimensional geometryunder some assumptions below discussed. II. OUR MODEL
The Einstein field equation in 5 D are given by: G MN + Λ D δ MN = 8 π ˜ G D T MN , (2)where M, N = 0 , , , ,
5, and where ˜ G D is the five dimensional Newton constant. Forsimplicity, we consider arbitrarily that this constant has a magnitude equal to unity.We will study the following space time: ds D = W ( z ) · d ¯ s D + dz (3)= exp( − A ( z )) · g (4 D ) µν dx µ dx ν + dz (4)= exp( − A ( z )) · (cid:0) − f ( r ) dt + f ( r ) − dr + r d Ω (cid:1) + dz (5)with µ, ν = 0 , , , z . Thus, the 4 D geometry, d ¯ s D , is deformed by the warp factor,and thus, our solution corresponds to a non-uniform string-like [4]. The energy momentumtensor is: T MN = diag( − ρ, p r , p θ , p φ , p z ) , (6)3here ρ, p r , p θ , p φ , p z represent the energy density and the r, θ, φ, z pressure components. Wewill impose that the energy momentum tensor has the form: T MN = f (cid:0) W ( z ) (cid:1) · diag( − ¯ ρ ( r ) , ¯ p r ( r ) , ¯ p θ ( r ) , ¯ p φ ( r ) , ˜ p z ( r )) , (7)where f (cid:0) W ( z ) (cid:1) represents a function of the warp factor. For convenience, we will write theenergy momentum tensor as follow: T MN = δ Mµ δ νN · f (cid:0) W ( z ) (cid:1) · ¯ T µν + T , (8)where ¯ T µν represents to the energy momentum tensor associated with the four dimensionalgeometry: ¯ T µν = diag( − ¯ ρ ( r ) , ¯ p r ( r ) , ¯ p θ ( r ) , ¯ p φ ( r )) (9)Thus ¯ ρ ( r ) , ¯ p r ( r ) , ¯ p θ ( r ) , ¯ p φ ( r ) represent the energy density and the r, θ, φ pressure compo-nents of the four dimensional geometry. On the other hand T corresponds to the energymomentum tensor along the extra dimension: T = δ f (cid:0) W ( z ) (cid:1) ˜ p z ( r ) . (10)where ˜ p z ( r ) represents the radial dependent part of T .Thus, we can point out that the warp factor locally modifies both the geometry as thecontent of the energy momentum tensor. Examples in 5 D , where the energy momentumtensor depends on the extra coordinate we can see in reference [19] for brane world models,and, where the energy momentum tensor depends on the radial coordinate in reference [20]for black strings surrounded by quintessence matter. A. Constraints for regularity of the solution
For the line element (5), the corresponding expressions for the Ricci scalar is: R D = −
20( ˙ A ) + 8 ¨ A + W − ¯ R D , (11)where the dot indicates derivation with respect to the extra coordinate ( z ) , andwhere ¯ R D corresponds to the Ricci scalar of the four dimensional geometry whose lineelement is d ¯ s D in equation (3):¯ R D = − f ′′ − r f ′ + 2 r (1 − f ) . (12)4here ′ represents derivation with respect to the radial coordinate ( r ).On the other hand, the Kretschmann scalar is given by: K D = 40( ˙ A ) + 16( ¨ A ) −
32 ¨ A ( ˙ A ) − W − ( ˙ A ) ¯ R D + W − ¯ K D , (13)where the four dimensional Kretschmann scalar ¯ K D , which also corresponds to the 4 D geometry d ¯ s D , is given by: ¯ K D = ( f ′′ ) + 4 r ( f ′ ) + 4 r (1 − f ) . (14)Thus, in order to describe a regular solution, i.e both invariants R D and K D must beregular, we will impose the following constraints: • The function A ( z ) must be continuous in its first and second derivatives. So, ourmodel does not have Z symmetry as brane world models [21–23], because the secondderivative of the absolute value behaves as Dirac delta and is not regular in z = 0+ N π ,with z compact. • The function A ( z ) must be such that the Warp factor W ( A ( z )) can not be zero. • The four dimensional geometry must be regular, therefore the four dimensional invari-ants, ¯ R D and ¯ K D must be regular. B. Equations of motion
We will use the following ansatz for the f ( r ) function: f ( r ) = 1 − m ( r ) r , (15)where m ( r ) is the so called mass function and, as it was above mentioned, must be suchthat the four dimensional invariants must be regular.For the line element (5) with the ansatz (15), using the coinstraints T = T and T = T ,the components of the Einstein equations (2) are:( t − t ) and ( r − r ) components:6( ˙ A ) − A + Λ D − r W − dmdr = − πf ( W ) ¯ ρ (16)5 θ − θ ) and ( φ − φ ) components:6( ˙ A ) − A + Λ D − r W − d mdr = 8 πf ( W )¯ p θ (17)and the ( z − z ) component:6( ˙ A ) + Λ D − r W − dmdr − r W − d mdr = 8 πf ( W )˜ p . (18)For solving these equations we will use the following steps: • In the ( z − z ) component we assume that6( ˙ A ) = − Λ D . (19)From this latter equation it is direct to check that ¨ A = 0. Thus, using the condition(19), the two first terms of the equation (18) and the three first terms of the equations(16) and (17) are equal to zero.Thus, for determining the form of A ( z ), solving the equation (19) is enough. Belowwe will discuss about the solution. • From equation (16) it is direct to check that:2 r W − dmdr = 8 πf ( W ) ¯ ρ, (20)thus, it is fulfilled that: f ( W ( z )) = W − = exp(2 A ( z )) (21)and m ( r ) = 4 π Z r x ¯ ρ ( x ) dx, (22)The equation (22) corresponds to the usual mass function for regular black holes[24, 25]. Below will be described the form for the four dimensional energy density ¯ ρ . • From equation (17) it is direct to check that: − r d mdr = 8 π ¯ p θ (23) • The radial and tangential pressures are easily computed by the conditions T = T , i.e − ¯ ρ = ¯ p r , and T = T , i.e ¯ p θ = ¯ p φ .6 From equation (18) it is direct to check that: − ¯ ρ + ¯ p θ = ˜ p . (24)From equation (21) we can see that the warp factor modifies the four dimensional ge-ometry (3). Furthermore, from equations (21) and (8), we note that the warp factor alsomodifies the information of the four dimensional energy momentum tensor (9) and the energymomentum tensor along the extra dimension (10). C. The four dimensional geometry
We will choose the Hayward [15] metric as example. This model has a de–Sitter core,and thus, both the Ricci and the Kretschmann are regular everywhere. The energy densityis given by [17]: ¯ ρ = 32 π LM (2 LM + r ) , (25)where L is a constant parameter. Replacing into equation (22): m ( r ) = M r LM + r , (26)where the M parameter corresponds to the total mass [25]. The mass function behaves as m ( r ) | r ≈ ≈ (1 / L ) r near the origin. Thus, the four dimensional solution (15) behaves asa de–Siter space time near the origin, i.e. the solution has a de–Sitter core. A completeanalysis of the solution can be viewed in reference [17]. See also [24, 26] for a thermodynamicsanalysis .It is worth to stress that both the energy density and the mass function ensure a wellasymptotic behavior of the four dimensional solution [27]:lim r →∞ ¯ ρ = 0 (27)lim r →∞ m ( r ) = constant = M. (28)From equation (28), it is easy to check that in equation (23):lim r →∞ ¯ p θ = 0 (29)Thus, the four dimensional solution is asymptotically flat. Below we will discuss theconsequences of this latter in the complete five dimensional solution.7 . Complete five dimensional solution We choose the following cosmological constant:Λ D = − l , (30)where l corresponds to the AdS radius when the space time represents to an AdS space.From equation (19) it is easy to check that the line element (3) is: ds D = C exp (cid:18) ± l z (cid:19) · d ¯ s D + dz . (31)where d ¯ s D is given by the Hayward space time described in the previous subsection and C is a positive constant.As it was above mentioned for the regularity of the solution, the warp factor W = C exp (cid:0) ± l z (cid:1) must be non zero. So, we need that the extra coordinate, z , to be compactsuch that z = z + 2 N π . This latter is because, for the positive (negative) branch, the warpfactor tends to zero at −∞ ( ∞ ).The warp factor differs from the brane world models [21, 22] because our solution doesnot have Z symmetry. This latter is because in models with Z symmetry the secondderivative ¨ A behave as Dirac Delta, i.e are singular for z = 0 + N π , with z compact, seeequations (11) and (13). Although the four dimensional solution behaves asymptotically asthe Schwarzschild space time, at infinity (of the radial coordinate) our solution does notcoincide with the solution of reference [10] , since our solution does not have Z symmetryand in our case the extra coordinate z is compact.As it was above mentioned, the four dimensional solution is regular. Thus, insteadthe formation of a black string where the radial singularity is represented by the productbetween the Schwarzschild singularity and S , it is formed a black string–like where the coreis represented by the product between the de–Sitter core of the four dimensional Haywardsolution and S .From equations (27) and (29) it is direct to check that in equation (24):lim r →∞ ˜ p = 0 . (32)So, the energy momentum tensor tends to zero at the asymptote of the radial coordinate.Thus, due that the four dimensional solution is asymptotically flat, our complete solution8ehaves at this place as an Anti de–Sitter five dimensional space time. On the other hand,also at the infinity of the radial coordinate the topology corresponds to the product betweenMinkowski and S . Furthermore at this asymptote, for zero cosmological constant, l → ∞ ,our solution behaves as the Kaluza-Klein black string [2]. III. THERMODYNAMICS ANALYSIS
It is direct to check that for an embedding z = z , where z represents an arbitrarylocation z ∈ z , the induced metric is given by [23]: h µν = g (4 D ) µν W ( z ) . (33)In order to compute the first law of thermodynamics we will use the conditions h tt ( r + , M, z ) =0 and δh tt ( r + , M, z ) = 0, where h tt and r + represent the temporal component of the inducedmetric and the horizon radius. These latter conditions can be viewed as constraints on theevolution along the space parameters [24, 28]. From equations (33) and (5), the temporalcomponent of the induced metric is: h tt = − W ( z ) f ( r ) δ tt . (34)Following the above mentioned conditions: δW ( z ) f ( r + , M ) + W ( z ) δf ( r + , M ) = 0 . (35)The first term is zero due that f ( r + , M ) = 0. On the other hand, since, as it wasabove mentioned W ( z ) = 0, therefore W ( z ) = 0. Thus, our problem is reduced to solve δf ( r + , M ) = 0: 0 = ∂f∂r + dr + + ∂f∂M dM. (36)Following equation (36), for our four dimensional solution (15), where the mass functionis given by the equation (26), the first law takes the form: ∂m∂M dM = (cid:18) π f ′ (cid:12)(cid:12) r = r + (cid:19) d (cid:18) πr (cid:19) . (37)The above equation can be rewritten as: du = T d (cid:18) A (cid:19) , (38)9here A corresponds to the area of a three dimensional sphere and the temperature andentropy are easily computed as: T = 14 π f ′ (cid:12)(cid:12) r = r + (39) S = A πr . (40)Thus, our computed definition of entropy follow the area’s law and so the existence ofextra dimensions does not modify the correct values of temperature and entropy at theinduced four dimensional geometry. The term du corresponds to a local definition of thevariation of the energy defined in [24, 27]. In this definition the factor dm/dM in equation(37) is always positive, and thus, the sign of the variation of du always coincides with thesign of the variation of the total energy dM . Furthermore at infinity it is fulfilled thatlim r →∞ dm/dM = 1, therefore the variation of the local and total energy are similar at theasymptotically region, and thus, the first law is reduced to the usual form dM = T dS [27].
IV. CONCLUSION AND SUMMARIZE
We have provided a new five dimensional black string–like solution. This is based on theembedding of a four dimensional regular black hole solution into a compact extra dimension.The warp factor depends only on the extra coordinate W ( z ) and modifies the four di-mensional geometry d ¯ s D , see equation (3). On the other hand, we have imposed the formof the energy momentum tensor (8), such that the warp factor also modifies the informationof the four dimensional energy momentum tensor (9) through the function (21).In subsection (II A) we have found relations between the four and five dimensional invari-ants of curvature. Using this latter, we have enunciated a list of constrains in order to thatthe complete five dimensional solution to be regular. Following these constraints we havecomputed the complete five dimensional solution. As example, we have used the Haywardmetric as four dimensional toy model. However, following our list, could be constructedanother regular black strings-like using other models of four dimensional regular black holeswith a de–Sitter core, see for example [29] . This could be analyzed in elsewhere.Due that the four dimensional geometry is regular, instead the formation of a blackstring where the radial singularity is represented by the product between the Schwarzschildsingularity and S , it is formed a black string-like where the core is represented by the10roduct between the de–Sitter core of the four dimensional Hayward solution and S . Onthe other hand, due that the extra coordinate is compact the horizon has topology S × S .Due that the four dimensional solution is asymptotically flat, our complete solution be-haves at the infinity of the radial coordinate as an Anti de–Sitter five dimensional space time.On the other hand, also at this asymptote the topology corresponds to the product betweenMinkowski and S . Furthermore at this place for zero cosmological constant, l → ∞ , oursolution behaves as the Kaluza-Klein black string [2].The induced geometry is characterized by the induced metric at an arbitrary location z = z . Following local relations, based on the evolution along the space of parameters[24, 28], we have computed the first law of thermodynamics on the induced geometry. Sothe existence of extra dimensions does not modify the correct values of temperature andentropy at the induced four dimensional geometry.It is wort to mention that small perturbations around black strings show that these latterare unstable [30] and this instability could be stabilized if the extra dimension is compactifiedto a scale smaller than a minimum value. This issue is outside of the scope of our work,however should be tested in a future work. [1] Roberto Emparan and Harvey S. Reall, “Black Holes in Higher Dimensions,”Living Rev. Rel. , 6 (2008), arXiv:0801.3471 [hep-th].[2] Ruth Gregory, “Braneworld black holes,” Lect. Notes Phys. , 259–298 (2009),arXiv:0804.2595 [hep-th].[3] Adolfo Cisterna, Sebasti´an Fuenzalida, Marcela Lagos, and Julio Oliva, “Homo-geneous black strings in Einstein–Gauss–Bonnet with Horndeski hair and beyond,”Eur. Phys. J. C , 982 (2018), arXiv:1810.02798 [hep-th].[4] Barak Kol, “Topology change in general relativity, and the black hole black string transition,”JHEP , 049 (2005), arXiv:hep-th/0206220.[5] Adolfo Cisterna and Julio Oliva, “Exact black strings and p-branes in general relativity,”Class. Quant. Grav. , 035012 (2018), arXiv:1708.02916 [hep-th].[6] Adolfo Cisterna, Mokhtar Hassaine, Julio Oliva, and Massimiliano Rinaldi, “Axionic blackbranes in the k-essence sector of the Horndeski model,” Phys. Rev. D , 124033 (2017), rXiv:1708.07194 [hep-th].[7] A. Sheykhi, S.H. Hendi, and Y. Bahrampour, “Rotating black strings in f ( R )-Maxwell the-ory,” Phys. Src. , 045004 (2013), arXiv:1304.3057 [gr-qc].[8] Alex Giacomini, Marcela Lagos, Julio Oliva, and Aldo Vera, “Charged black strings and blackbranes in Lovelock theories,” Phys. Rev. D98 , 044019 (2018), arXiv:1804.03130 [hep-th].[9] A. Chamblin, S.W. Hawking, and H.S. Reall, “Brane world black holes,”Phys. Rev. D , 065007 (2000), arXiv:hep-th/9909205.[10] P. Kanti, I. Olasagasti, and K. Tamvakis, “Schwarzschild black branes and strings in higherdimensional brane worlds,” Phys. Rev. D , 104026 (2002), arXiv:hep-th/0207283.[11] Theodoros Nakas, Nikolaos Pappas, and Panagiota Kanti, “New Black-String Solutionsfor an Anti-de Sitter Brane in Scalar-Tensor Gravity,” Phys. Rev. D99 , 124040 (2019),arXiv:1904.00216 [hep-th].[12] Theodoros Nakas, Panagiota Kanti, and Nikolaos Pappas, “Incorporating Physical Con-straints in Braneworld Black-String Solutions for a Minkowski Brane in Scalar-Tensor Grav-ity,” Phys. Rev.
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