Black branes in four-dimensional conformal equivalent theories
aa r X i v : . [ g r- q c ] J u l Black branes in four-dimensional conformal equivalent theories
N. Dimakis, ∗ Alex Giacomini, † and Andronikos Paliathanasis
2, 3, 4, ‡ Center for Theoretical Physics, College of Physical Science and Technology Sichuan University, Chengdu 6100064, China Instituto de Ciencias F´ısicas y Matem´aticas, Universidad Austral de Chile, Valdivia, Chile Department of Mathematics and Natural Sciences, Core Curriculum Program,Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia Institute of Systems Science, Durban University of Technology,PO Box 1334, Durban 4000, Republic of South Africa (Dated: July 3, 2018)The physical properties of static analytic solutions which describe black brane geometries arediscussed. In particular we study the similarities and differences of analytic black brane/stringsolutions in the Einstein and Jordan frames. The comparison is made between vacuum power law f ( R ) gravity solutions and their conformal equivalents in the Einstein frame. In our analysis weexamine how the geometrical and physical properties of these analytic axisymmetric solutions - suchas singularities, the temperature and the entropy - are affected as we pass from one frame to theother. PACS numbers:Keywords:
1. INTRODUCTION
The need to determine a mechanism which provides a theoretical prediction on the recent observable phenomenain cosmology, such as the late-time acceleration phase of the universe [1–5], has led cosmologists to introduce (al-ternative/modified) gravitational theories which are defined in the Jordan frame [6] instead of the Einstein frame ofGeneral Relativity [7–10].One of the first proposed theories in the literature, which is formulated in the Jordan frame, is the Brand-Dickegravitational theory [11]. More specifically, Brans-Dicke theory is based on Mach’s principle [12]; the gravitationalfield is described by the metric tensor and a scalar field nonminimally coupled to gravity. Moreover, the gravitationalconstant G depends on the value of nonminimally scalar field, and on a constant ω BD which is called Brans-Dickeparameter. Hence, G is actually an effective function rather a constant, for a discussion see also [13, 14]. In whatregards the limits of parameter ω BD , while someone expects General Relativity to be recovered when ω BD reachesinfinity, it was actually demonstrated that this is not true. This implies that the two theories are fundamentallydifferent [15]. On the other hand, in the limit where the Brans-Dicke parameter vanishes, i.e. ω BD = 0, the theory isequivalent with that of a massive Dilaton theory [16], as it has been shown by O’Hanlon [17]. Moreover, Brans-Dicketheory provides a radiation epoch in the evolution of the universe as also a de Sitter phase [18, 19], which can berelated with the late-time acceleration phase of the universe [20]. It has also been proposed that the Brans-Dickeparameter should not be a constant but rather depend on the nonminimally coupled field. This led to a zoology oftheories known as Scalar-tensor theories [21]. For other generalizations of Brans-Dicke action see for instance [22–31]and references therein.Scalar-tensor theories are conformally equivalent to General Relativity [21, 32–34]. However the latter is just amathematical equivalence between the Jordan and the Einstein frame on the solution space of the field equations.What is more, the latter is true only in the case where there are not any specific conditions imposed from an extramatter source, that is except of the scalar field; we refer the reader to a recent discussion in [35]. In particular, if thereexists an additional matter source which is minimally coupled to the scalar field in one of the frames, then under theconformal transformation there will be an interaction between the matter source and the scalar field, which meansthat the extra matter source will not be minimally coupled to the latter. Moreover, while the energy conditions for thescalar field are satisfied in the Einstein frame, they can be violated in the Jordan frame [36, 37]. From cosmologicalstudies, it has been found that while a theory can provide an acceleration phase in one frame it may not be able to ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] The Brans-Dicke action with ω BD = 0 is also known as O’Hanlon gravity. do so in the other. From this it is clear the conformal equivalence transformation between the two frames does notnecessarily mean physical equivalence; see also the discussions in [34, 38–45] and references therein. The study of thephysical properties between the Jordan and the Einstein frame has been extended to astrophysical objects, such asblack-holes [46–50] or to even more exotic configurations like the wormholes [51–53] and the black branes or strings[54, 55].There is a family of higher-order theories which can also be described in the form of second order scalar-tensortheories [56–62]. In these cases new degrees of freedom, which are introduced by the higher-order derivatives, canbe attributed to nonminimally scalar fields. The standard method to do so is with the use of Lagrange multipliers[63–67]. Consequently, higher-order theories can provide the mechanics for the scalar field description, an approachwhich has been applied to describe inflation [68–71].One of the simplest higher-order theories which has drawn the attention of the scientific society the last years is f ( R )-gravity. More specifically, it is a fourth-order theory in which the Einstein-Hilbert action is modified so asto become a general function, f , of the Ricci scalar, R , of the underlying geometry [72]. Its equivalent descriptionas a scalar-tensor theory is in the form of Brans-Dicke theory with parameter ω BD zero. That is, f ( R )-gravity isequivalent with the so-called O’Hanlon theory [56]. In the family of f ( R )-gravity there belongs one of the mostpopular inflationary models, the Starobinsky model for inflation [68]. There are other cosmological tests that haveshown that the theory can also describe the late-time acceleration phase of the universe [73–75]. Some astrophysicalapplications of f ( R )-gravity can be found in [76–85] and a black hole topology theorem has been presented lately in[86].Recently, in the context of f ( R )-gravity, a new family of analytical solutions in a four-dimensional static spacetimewas found in [87]. That new family of solutions can support black branes or strings in vacuum when axisymmetryis introduced in the line element. These solutions are not asymptotically maximally symmetric. The study of thethermodynamic parameters for the black brane/string solution extracted in the context of the power-law theory f ( R ) = R k , has shown that the entropy function has positive values only when 0 < k < or the power k is anegative, even integer.The purpose of this work is to study the effects of the conformal transformation between the Jordan and the Einsteinframes for the black brane solution determined in [87]. Black branes/string solutions can be easily constructed inEinstein’s gravity and in some modified theories of gravity by trivially embedding black hole solutions in higherdimensions [88–94]. However, the results of [87] are not based on this technique and the resulting spacetime is notconstructed by the embedding of a lower dimensional solution in four dimensions. In this work we use the resultsobtained in [87] so as to construct the analytical solution of General Relativity with a minimally coupled scalar fieldfor a four-dimensional static axisymmetric spacetime. Furthermore, we shall study the differences of the specificsolutions; an important analysis in order the investigate the physical (in-)equivalence of the two frames for staticaxisymmetric solutions.The plan of the paper is as follows: In Section 2 we briefly discuss the conformal transformation which connectsthe Jordan and the Einstein theory. In Section 3 we present the gravitational field equations that correspond to thesystem under study. The analytical solution which was determined in [87] is discussed in Section 4. The black branesolution in the Einstein frame is determined in Section 5. The temperature and the entropy for the solution in theEinstein frame are calculated in Section 6. Finally, in Section 7 we draw our conclusions.
2. CONFORMAL TRANSFORMATIONS
By definition conformal transformations generalize the concept of isometries; in particular, conformal transforma-tions are angle-preserving transformations compared to the isometries which preserve both the length and the anglesbetween lines. Two metric tensors are conformally related when there exists a scalar N ( x ) such that¯ g µν ( x ) = N ( x ) g µν ( x ) (1)while the Ricci scalar ¯ R (¯ g ) of ¯ g µν is given in terms of the Ricci scalar R ( g ) for the metric g µν and N ( x ) as [95]¯ R (¯ g ) = N − R ( g ) − N − g µν N ; µ ; ν , (2)whenever g µν denotes a four-dimension space, that is, dim g µν = 4. The covariant derivatives on the right hand sideof (2) are defined with respect to the metric g µν . Two Action integrals, or more specifically Lagrangians, are calledconformally equivalent iff the solution of the Euler-Lagrange equations survives under the action of the conformaltransformation [35].Let us now consider the scalar-tensor theory in the four-dimensional space with metric ¯ g µν , that is given by S ( ϕ, ¯ g ) = Z d x √− ¯ g (cid:16) F ( ϕ ) ¯ R (¯ g ) − α g µν ϕ ; µ ϕ ; ν − V ( ϕ ) (cid:17) , (3)where ϕ denotes the nonminimally coupled scalar field and V ( ϕ ) is the potential which drives the dynamics for thisfield. The parameter α is the analogue of the Brans-Dicke parameter, α = ω BD , when the function F ( ϕ ) is quadratic,that is , F ( φ ) ≃ ϕ . On the other hand, when α = 0, the Action integral (3) describes the O’Hanlon gravity whichis equivalent to the f ( R )-gravity.For the conformally related metric g µν , the Action Integral (3) with the use of expression (2) becomes S ( ϕ, g ) = Z d x √− g (cid:16) F ( ϕ ) N R ( g ) − α N g µν ϕ ; µ ϕ ; ν − N V ( ϕ ) − F ( ϕ ) N g µν N ; µν (cid:17) . (4)while integration by parts simplify the latter to [96] S ( ϕ, g ) = Z d x √− g (cid:16) F ( ϕ ) N R ( g ) − α N g µν ϕ ; µ ϕ ; ν − N V ( ϕ ) − N F ,ϕ g µν N ; µ φ ; ν − F ( ϕ ) g µν N ; µ N ; ν (cid:17) . (5)It is straightforward to see that scalar N ( x ) does not provide any new degrees of freedom and actually is a degeneratefunction. Thus, without loss of generality, we can remove it from the Action integral by choosing N = N ( φ ( x )). Insuch a case, the Action Integral (5) is written as S ( ϕ, g ) = Z d x √− g (cid:16) ˆ F ( ϕ ) R ( g ) − α H ( ϕ ) g µν ϕ ; µ ϕ ; ν − ˆ V ( ϕ ) (cid:17) . (6)where H ( ϕ ) = (cid:18) N + 12 α (cid:16) N F ,ϕ N ,ϕ + F ( ϕ ) ( N ,ϕ ) (cid:17)(cid:19) , ˆ F ( ϕ ) = F ( ϕ ) N , and ˆ V ( ϕ ) = N V ( ϕ ) , (7)or equivalently as follows S ( φ, R ) = Z d x √− γ (cid:16) ˆ F ( φ ) R ( g ) − α g µν φ ; µ φ ; ν − ˆ V ( φ ) (cid:17) (8)in which the scalar field φ ( x ) is related with ϕ ( x ) with the transformation dφ = p H ( φ ) dϕ . Recall that the lattertransformation is a coordinate transformation and does not affect the dynamics of the system.The Action Integral (8) still describes a scalar-tensor theory in the Jordan frame. However, when ˆ F ( φ ) = const. ,that is when F ( ϕ ) N ( ϕ ) = const. , the Action Integral takes the form of the one of General Relativity with aminimally coupled scalar field, i.e. S ( φ, R ) = Z d x √− g (cid:18) R ( g ) − g µν φ ; µ φ ; ν − ˆ V ( φ ) (cid:19) . (9)where without loss of generality we have chosen α = 1.Indeed, the two Action integrals (3) and (9) are conformally related but it is important to mention that g and ¯ g arein principle two different line elements. In the sense that there does not necessarily exist a space-time diffeomorphismlinking the two metrics. Hence, while the above transformation can be used to map solutions between the two frames,one should be very careful about the possible physical (in-)equivalence of the two theories.At this point, just for the convenience of the reader, we briefly want to demonstrate the equivalence of f ( R )-gravitywith Brans-Dicke theory with ω BD = 0, that is the O’Hanlon theory. The Action Integral in f ( R )-gravity, for theline element ¯ g µν , is given as follows S f ( R ) = Z d x √− gf (cid:0) ¯ R (¯ g ) (cid:1) , (10)with the field equations resulting from the variation with respect to the metric being f ,R R µν − f g µν − ( ∇ µ ∇ ν − g µν ∇ σ ∇ σ ) f R = 0 . (11) When F ( ϕ ) ≃ ϕ , then under the reparametrizations ϕ = √ φ for the scalar-field, the Action integral (3) is written in the usualBrans-Dicke form [11]. Without loss of generality we introduce the Lagrange multiplier λ , such that S f ( R ) = Z d x √− g (cid:0) f (cid:0) ¯ R (¯ g ) (cid:1) + λ (cid:0) ¯ R (¯ g ) − ¯ R ex (cid:1)(cid:1) . (12)where ¯ R ex denotes the expansion of the Ricci scalar in terms of the functions of the metric ¯ g µν . Variation with respectto the Ricci scalar ¯ R at the Action Integral (12) gives the Euler-Lagrange equation δS f ( R ) δ ¯ R = 0, or λ = − f , ¯ R (cid:0) ¯ R (cid:1) ,which defines the Lagrange multiplier λ . By substituting the latter in (12) it follows that [62] S f ( R ) = Z d x √− g (cid:0) f , ¯ R (cid:0) ¯ R (cid:1) ¯ R ex + (cid:0) f (cid:0) ¯ R (cid:1) − f , ¯ R ¯ R (cid:1)(cid:1) (13)and by defining the new scalar ϕ = f , ¯ R (cid:0) ¯ R (cid:1) we obtain the scalar-tensor theory S f ( R ) ,φ = Z d x √− g (cid:0) ϕ ¯ R ex (¯ g ) − V f ( ϕ ) (cid:1) , (14)in which V f ( ϕ ) = (cid:0) ϕ ¯ R ( ϕ ) − f (cid:0) ¯ R ( ϕ ) (cid:1)(cid:1) . This latter action describes the Brans-Dicke theory with ω BD = 0. Thecorresponding equations for the gravitation and scalar field are ϕ (cid:18) R µν − g µν R (cid:19) + ( g µν ∇ σ ∇ σ − ∇ µ ∇ ν ) ϕ + 12 g µν V ( ϕ ) = 0 (15a) R − V ′ ( ϕ ) = 0 (15b)respectively. It is also true that f ( R )-gravity defined in the Jordan frame is conformally equivalent to GeneralRelativity induced by a minimally coupled scalar field. The conformal transformation which relates the ActionIntegrals (9), (14) is [56] g µν = ϕ ¯ g µν , φ = √ ϕ. (16)In the following section, we proceed with the derivation of the field equations for the static axisymmetric spacetimein the Einstein and Jordan frames, in particular that of f ( R ) − gravity. Moreover, we assume that there is not anyextra matter source in the gravitational system except of the scalar field.
3. FIELD EQUATIONS IN STATIC AXISYMMETRIC SPACETIME
Consider the four-dimensional static axisymmetric spacetime with line element d ¯ s = − a ( r ) dt + n ( r ) dr + b ( r ) (cid:0) dy + dz (cid:1) , (17)which in general admits a four dimensional Killing algebra, consisted by the vector fields { ∂ t , ∂ y , ∂z, z∂ y − y∂ z } .In f ( R )-gravity, the gravitational field equations (11) for metric (17) are equivalent to those produced by thevariation, with respect to the variables { N ( r ) , a ( r ) , b ( r ) , R ( r ) } , of the Lagrangian function L f ( R ) = 2 n (cid:2) f ,R (cid:0) ab ′ + 2 ba ′ b ′ (cid:1) + f ,RR (cid:0) b a ′ R ′ + 2 abb ′ R ′ (cid:1)(cid:3) + nab ( f − Rf ,R ) , (18)where the prime denotes a total derivative with respect to the variable r , that is x ′ ( r ) = dx ( r ) dr .On the other hand, in the scalar-tensor equivalent description, Lagrangian (18) takes the equivalent form L ϕ = 2 n (cid:2) ϕ (cid:0) ab ′ + 2 ba ′ b ′ (cid:1) + (cid:0) b a ′ ϕ ′ + 2 abb ′ ϕ ′ (cid:1)(cid:3) − nab V ( ϕ ) , (19)while the corresponding Euler - Lagrange equations yield2 n (cid:2) ϕ (cid:0) ab ′ + 2 ba ′ b ′ (cid:1) + (cid:0) b a ′ ϕ ′ + 2 abb ′ ϕ ′ (cid:1)(cid:3) + ab V ( ϕ ) = 0 , (20) Recall that by definition ¯ R ex ≡ ¯ R (¯ g ) . a ′′ = 2 ab b ′ + 6 n a ′ n ′ − an V ,ϕ − ϕ a ′ ϕ ′ + 4 b ′ bϕ ( aϕ ′ − ϕa ′ ) , (21)6 b ′′ = − b b ′ − bn V ,φ + 2 b ′ (cid:18) n ′ n − ϕ ′ ϕ (cid:19) + 2 aϕ a ′ ( bφ ′ − φb ′ ) , (22)6 ϕ ′′ = n (2 ϕV ,ϕ − V ( ϕ )) + 6 n n ′ ϕ ′ + 2 ϕb ′ (2 ba ′ + ab ′ ) − b ( ba ′ + 2 ab ′ ) φ ′ ab . (23)It can be straightforwardly verified that the above system is also equivalent to field equations (15) under the ansatz(17) for the line element.The conformal transformation (16) reads a ( r ) → ϕ ( r ) − A ( r ) , b ( r ) → ϕ ( r ) − A ( r ) , n ( r ) → ϕ ( r ) − N ( r ) , ϕ → e √ φ (24)and under its use the field equations (20)-(23) become1 N (cid:18) AB ′ + 4 BA ′ B ′ − AB φ ′ (cid:19) + AB N ˆ V ( φ ) = 0 , (25)8 A ′′ = 8 A ′ (cid:18) N ′ N − B ′ B (cid:19) + 4 AB B ′ − Aφ ′ − AN ˆ V ( φ ) , (26)8 B ′′ = − B B ′ + 8 N N ′ B ′ − Bφ ′ − BN ˆ V ( φ ) , (27)and φ ′′ = − (cid:18) A ′ A + 2 B ′ B (cid:19) φ ′ + N ′ N φ ′ + N ˆ V ,φ , (28)in which ˆ V ( φ ) = e − √ φ V ( φ ). Finally, under the transformation (24) the line element (17) can be written as d ¯ s = ϕ − ds , where the new line element ds is now ds = − A ( r ) dt + n ( r ) dr + B ( r ) (cid:0) dy + dz (cid:1) . (29)The set of equations (25)-(28) describe the Einstein field equations with a minimally coupled scalar field, i.e. thosegenerated from Action Integral (9), but for the line element (29), which is different from that of (17). Although thesolutions of the conformal equivalent theories commute thought the conformal transformation (24), the physical andgeometrical properties of the solutions can be extremely different due fact that a conformal transformation cannot ingeneral correspond to a diffeomorphism linking the two line elements.
4. BLACK BRANES IN THE JORDAN FRAME
In this section, we briefly discuss the analytical solution for the gravitational field equations (20)-(23) of f ( R )-gravity which was derived recently in [87], and can describe four-dimensional black branes (or strings depending onthe topology on which one endows the two dimensional surface). It was found in [87] that the line elements d ¯ s = − r r − kk − m − ! dt + r − k k +2( k − k − r − kk − m − ! − dr + r (cid:0) dy + dz (cid:1) , for 1 < k <
54 (30)and d ¯ s = − r − r − kk − m ! dt + r − k k +2( k − k − − r − kk − m ! − dr + r (cid:0) dy + dz (cid:1) , for k < k > , k = 12 . (31)solve field equations (11) for a power-law f ( R ) = R k theory, with k = 0 , , , . Given the form of the transformationthat maps f ( R )-gravity to the O’Hanlon theory we expect that the corresponding potential V f ( ϕ ) also has a powerlaw form. Specifically, we can see that the aforementioned potential is V f ( ϕ ) = ± k − k − ϕ kk − . (32)and the corresponding scalar field is given by ϕ ( r ) = r k − k − . (33)Expressions (32) and (33) satisfy the field equations (15) for line element (30) when we have the plus sign in (32). Onthe contrary, when we consider line element (31), we need to take into account the minus sign in potential (32).The gravitational properties of the solutions in f ( R )-gravity and the O’Hanlon theory are identical since they referto the same metrics. For these two line elements the Ricci scalar and the Kretschmann scalar are derived to be R = ± k (4 k − r k − k − k − ( k − k − , (34)and K = 4(1 − k ) ( k − m h (cid:0) k − k + 468 k − k + 111 (cid:1) r − ( k − k +9 ) (2 k − k − + − m ( k − (2 k − r − k k − k − k − + 3 m (cid:0) k − k + 13 k − k + 2 (cid:1) r k − k − k − i . (35)From the latter expressions it is clear that the spacetimes admit curvature singularity at r = 0, when k < / < k ≤ . Moreover, it is straightforward to observe that a coordinate singularity exists at r = r h = m − k k − , when m > x and y as periodic variables). While the resulting spacetime isasymptotically Ricci flat, we can see that the curvature tensor R κλµν does not tend to zero as r reaches infinity, whichmeans that the spacetime cannot be asymptotically deformed to a maximally symmetric space. For a discussion onthis kind of topological black holes (especially for a toroidal case two-surface) we refer the reader to [97].In this context we can also study the type of an effective fluid corresponds two the line elements (30) and (31).By considering the anisotropic energy-momentum tensor T ( ϕ ) µν = R µν − g µν R generated by these geometries we mayderive the energy density to be ρ ϕ ( r ) = ± r k − k − k − m (2 k −
1) ( k − (cid:16) r − kk − (2 k −
3) ( k −
2) + m (cid:0) k − k − (cid:1)(cid:17) , (36)where the plus sign corresponds to metric (30) and the minus to (31) (together with the appropriate range of valuesfor k ). For the case where 1 < k < we can see that ρ ϕ is positive as long as r > A k r h , where A k = (cid:18) k − k + 6 − k + 7 k + 1 (cid:19) k − k − . (37)The multiplicative constant A k in front of the horizon radius r h assumes values in the segment (1 , e ) as k varies from1 to . In the other range for k corresponding to solution (31) the situation is more complicated. In order to have apositive energy density (36) one of the following conditions has to be met: • < k <
1, with r < A k r h • (cid:0) − √ (cid:1) < k < or < k < (cid:0) √
73 + 7 (cid:1) , with r > A k r h • < k <
2, with r < A k r h .A positive energy density in a region that rests outside the horizon is only possible in the second of the above cases.The effective energy-momentum tensor is anisotropic. The radial pressure p ϕr ( r ) in the r direction is given by p ϕr ( r ) = ± r − k k − k − k − (1 − k ) m (cid:16) k − mr k − k − + k − (cid:17) , (38)while the isotropic tangential pressure in the x and y directions is p ϕT = p ϕx ( r ) = p ϕy ( r ) = ± r k − k − k − (2 k − k − m (cid:16)(cid:0) − k + 7 k + 1 (cid:1) m + 2( k − k − r − kk − (cid:17) . (39)We can define the mean pressure p ϕ = (cid:0) p ϕr + p ϕx + p ϕy (cid:1) and with respect to the latter write a “mean” equation ofstate parameter w ϕ ( r ) = p ϕ ( r ) ρ ϕ ( r ) .What we may observe is that near the curvature singularity r → k corresponding to solution(31) the equation of state parameter has the limit w ϕ ( r →
0) = . That “anisotropic” radiation fluid is provided bythe extra terms of f ( R )-gravity [98]. We can now proceed with the study of the analytical solution in the Einsteinframe.
5. BLACK BRANES IN THE EINSTEIN FRAME
At this point we can map the results expressed in the previous section to the Einstein frame by performing trans-formation (16). After a reparametrization with respect to the r variable the resulting spacetime becomes ds = − r r − (2 k − k − k − m − dt + (cid:18) − k k − (cid:19) r − k k +23( k − r − (2 k − k − k − m − r (cid:0) dy + dz (cid:1) , (40)for 1 < k < or ds = − r − r − (2 k − k − k − m dt + (cid:18) − k k − (cid:19) r − k k +23( k − − r − (2 k − k − k − m + r (cid:0) dy + dz (cid:1) (41)for k ∈ ( −∞ , S ( , + ∞ ) − { } . The corresponding minimally coupled scalar field that satisfies Einstein’s equations R µν − Rg µν = T µν , where T µν = 12 (cid:20) φ ,µ φ ,ν − g µν (cid:16) φ ,µ φ ,µ + 2 ˆ V ( φ ) (cid:17)(cid:21) , (42)is φ ( r ) = 2( k − √ k −
1) ln r (43)with a potential given by ˆ V ( φ ) = ± k − e − ( k − φ ( r ) √ k − k − . (44)Again the plus sign corresponds to solution (40), while the minus is reserved for (41).Unlike to the previous case, where we have a curvature singularity hidden by an horizon (thus a black brane/string)for a specific range of the parameter k , here we can see that a singularity is always present at r = 0. The Ricci scalarof the resulting spacetimes is R = ± − k ) m r − k − k +133( k − (cid:20) k − k + 2) mr (2 k − k − k − + ( k − (cid:21) (45)and it can be seen that − k − k + 133( k − < , and (2 k − k − k − ≤ k − k + 133( k − (46) The r appearing in this section is not the same as the one appearing in the previous. hold for all k = 1, with the equality in the latter expression being true for k = 2. Thus, we can derive that R issingular at r = 0 for all values, but k = 1 or k = 2. The first is already excluded from our analysis, while the latterleads to a constant R . However, this does not prevent a curvature singularity for k = 2; As we can observe from theKretschmann scalar K = 36(1 − k ) m r − ( k − k +13 ) k − h (cid:0) k − k + 54 k − k + 10 (cid:1) m r k − k − k − k − ( k ( k + 2) − mr (2 k − k − k − + 23 k − k + 132 k − k + 35 i , (47)there exists a curvature singularity for all admissible k in solutions (40) and (41), even when k = 2. This latter casecorresponds to having a pure cosmological constant since φ = 0 from (43) and as a result the potential ˆ V becomes aconstant. We can write in this case the line element as ds = − (cid:18) r − r (cid:19) dt + 12 R (cid:18) r − r (cid:19) − dr + r (cid:0) dy + dz (cid:1) (48)with ˆ V = R , where R is the now constant Ricci scalar. Solution (48) is a known black string solution in the case ofa pure cosmological constant and a special case of a more general solution presented in [99]. The horizon that hidesthe singularity is at r = r H , where r H = m − k − k − k − (49)(of course, for the aforementioned solutions, we consider m > k for which the solution holds. Something which did not happen in the f ( R ) or the O’Hanlon theory. Although the two frames are conformally equivalent this does not mean that they arephysically equivalent. Solutions can be mapped at a purely mathematical level, but their study produces in principledifferent results.We can also write the basic components of the effective fluid generated by the scalar field in this case. The energydensity ρ = − T is given by ρ = ± r − k k − k − (1 − k ) m (cid:20) r (2 k − k − k − (7 k − k + 1) m + ( k − (cid:21) . (50)The radial pressure p r = T is expressed as p r = ± r − k k − k − (1 − k ) m (cid:20) ( k − − r (2 k − k − k − ( k − m (cid:21) , (51)while the tangential pressure in this case is p T = p x = p y = T = T = − ρ . Thus, we can see that in this case a“partial” equation of state w i = p i ρ = − x and y directions. It can be easily verified that ρ ispositive in the region 1 < k < , if the radius r satisfies r > B k r H , where B k = (cid:18) − ( k − k − k + 1 (cid:19) k − k − k − . (52)In the second region of the solution k < k > , k = 12, the energy density can only be positive in the subsets (cid:0) − √ (cid:1) < k < < k < (cid:0) √ (cid:1) given that the radius is again greater than B k r H . It is also nowinteresting to study the basic thermodynamic quantities to compare with those derived for (30) and (31) in [87].
6. TEMPERATURE AND ENTROPY
In this section we calculate some basic thermodynamic quantities like the temperature and the entropy. At first,we can perform a transformation t τ to bring the metric into an ingoing Eddington-Finkelstein coordinate system.Thus, if we impose t = τ ± Z r g rr − g tt dr, (53)the line elements (40) and (41) are transformed into ds = ± r − r − (2 k − k − k − m dτ ± k − k − r ( k − k − dτ dr + r (cid:0) dy + dz (cid:1) . (54)with the plus sign corresponding to solution (40) and the minus to (41). In these coordinates, the surface gravitycan either be given by κ = Γ τττ | r = r H or - with the help of the timelike Killing vector X µ = ∂ τ that exhibits a Killinghorizon at r = r H - by κ = ∇ µ ( X ν X ν ) | r = r H = − κX µ | r = r H . In any case, the result is κ = ± (5 − k )2( k − m − k k +1(2 k − k − . (55)It is rather impressive that the surface gravity is exactly the same with the one derived in [87] for a different setof spacetimes, namely (32) and (30) that correspond to the relative solutions in f ( R ) = R k gravity and which areconnected to (40) and (41) through a conformal transformation. We note that once again the plus sign correspondsto the case 1 < k < , while the minus is for k < k > , k = . As a result of κ being the same in the two frames,the resulting temperatures at the horizon are also identical. We can see that the latter is given in terms of the singleessential constant of the geometry m and the parameter of the theory k as: T = κ π = (5 − k )4 π ( k − m − k k +1(2 k − k − , < k < k − π ( k − m − k k +1(2 k − k − , k ∈ ( −∞ , S ( , + ∞ ) − { } (56)with each branch corresponding to the appropriate solution for the assumed values of k . The exponent of m in theexpressions appearing in (56) is negative in the region < k < and when k > (1 + √
3) or k < (1 − √ (1 − √ < k < and < k < (1 + √ σ = 2 πr H = 2 πm − k − k − k − . In this manner, we candefine an entropy per unit length as S ∼ σ π m − k − k − k − . (57)We can see that the power of m is positive when k ∈ ( , ) − { } , while it is negative for all the other values (excludedof course k = , ). The entropy is always well defined and positive for all admissible values of k , in contrast to whathappens to the corresponding solution in the Jordan frame. i.e. the one expressed previously in the context of the f ( R ) theory. In the latter, the entropy per unit length can be found to be [87] S f ( R ) ∼ k (cid:18) ± k (4 k − k − k − (cid:19) k − m − k − k − k − . (58)The S f ( R ) of f ( R ) = R k gravity is positive only in the cases where 0 < k < or when k is a negative even integer.This is another difference in the physics of the system owed to the conformal transformation linking the two theories.In Einstein’s gravity with the minimally coupled scalar field the solution results in a positive entropy for all valuesof the parameter characterizing the potential of the scalar field. We can note however, that the dependence in whatregards the constant m appearing in the line element remains the same in the two theories.In both frames the corresponding solution in not asymptotically maximally symmetric. This results in complicationsin what regards a possible calculation of the mass of the system. Even in the Einstein frame, an indirect computationthrough the first law of thermodynamics is not trivial due to the contributing work due to the matter source which ischaracterized by a non constant pressure. This is why we restrict ourselves to carry on the comparison only over thewell defined thermodynamic quantities of the temperature and the entropy.0
7. CONCLUSIONS
In this work we have studied analytical solutions of black branes which are related through conformal transforma-tions. According to our knowledge this is the first examples where black brane solutions are compared between theJordan and the Einstein frame. At the Jordan frame we consider an analytic solution which was derived recentlyfor f ( R )-gravity in the metric formalism; more specifically in the context of a power law f ( R ) = R k theory over anaxisymmetric spacetime in vacuum. The conformal equivalent theory in the Einstein frame corresponds to a scalarfield with an exponential potential.In principle it is expected that not only the physical but also the geometrical properties are different when one passesfrom one frame to the other. This can be attributed to the fact that conformal transformations do not necessarilycorrespond to a general coordinate transformation. The latter constitute the gauge freedom transformations ofgravitational theories and hence map different line elements describing the same geometry. As a result, when aconformal transformation is performed, the resulting space-time may in principle possess extremely different properties.In our case we where able to witness several striking differences but also a few similarities.The analytic solution in the Jordan frame was found to describe black branes (solutions with a singularity at r = 0that is encompassed by a horizon r = r h ) only for specific values of the free parameter k . Moreover, it has been shownthat the entropy of these black brane solutions is positive only for specific values of k , and more specifically when0 < k < or when k is a negative, even integer. The parameter k in the f ( R ) = R k theory being the power in whichthe Ricci scalar is raised, while in the Einstein theory is the exponent in the exponential potential characterizing thescalar field.In contrast to what happened in the Jordan frame, the conformally equivalent solution in the Einstein frame wasfound to describe black branes for all the admissible real values of the parameter k , while the entropy was calculatedto be well defined, i.e. positive, for all the values of this parameter. We can also note that the horizon r H in theEinstein frame is shifted r H = r h when we make the comparison by keeping the same constant m appearing in thetwo geometries.A similarity that may be noticed in the two solutions is that the black objects that they describe are in neithercase asymptotically maximally symmetric. Additionally, the surface gravity on the horizon ends up to be the samefor both cases. A property which results in the same relation for the thermodynamic temperature as well.When seeing the solution in the Jordan frame as a result of the O’Hanlon theory, we calculated the energy density andanisotropic pressure of the respective effective fluid. It was interesting to note that, in this situation, by approachingthe curvature singularity, the solution looks like the effect of a radiation fluid. We also derived the relative expressionsin context of Einstein’s gravity and we investigated for which ranges of the parameter k characterising the two models,the weak energy condition is satisfied.From all of the above results it follows that the basic physical properties of the aforementioned axisymmetricsolutions do not survive through the conformal transformation connecting them. Indeed, horizons, the entropy andeven singularities change between the two systems. The effect of the latter being the alteration of the conditions overthe parameter k needed in order to identify the solution as a black brane. In a forthcoming work we want to comparethe two theories by studying dynamical evolution of test particles among the two different solutions. Acknowledgments
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