The relation between F(R) gravity and Einstein-conformally invariant Maxwell source
aa r X i v : . [ g r- q c ] A ug The relation between F ( R ) gravity andEinstein–conformally invariant Maxwell source S. H. Hendi ∗ Physics Department, College of Sciences, Yasouj University, Yasouj 75914, IranResearch Institute for Astrophysics and Astronomy ofMaragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
In this paper, we consider the special case of F ( R ) gravity, in which F ( R ) = R N andobtain its topological black hole solutions in higher dimensions. We show that, the sameas higher dimensional charged black hole, these solutions may be interpreted as black holesolutions with two event horizons, extreme black hole and naked singularity provided theparameters of the solutions are chosen suitably. But, the presented black hole is differentfrom the standard higher-dimensional Reissner-Nordstr¨om solutions. Next, we present theconformally invariant Maxwell field coupled to Einstein gravity and discuss about its blackhole solutions. Comparing these two class of solutions, shows that there is a correspondencebetween the Einstein-conformally invariant Maxwell solutions and the solutions of F ( R )gravity without matter field in arbitrary dimensions. I. INTRODUCTION
Various proposals of diverse characters have been suggested by the physicists during the pastdecades for going beyond, or modifying, Einstein general relativity and often for few viable reasons.The most important motivation coming from high-energy physics for adding higher order invariantsto the gravitational action, as well as a general motivation coming from cosmology and astrophysics[1] for seeking generalizations of Einstein gravity. The so-called modified gravities constructed byadding correction terms to the usual Einstein-Hilbert action (e.g: [2, 3, 4]), is only one endeavoramong others to go beyond Einstein general relativity and have opened a new window to studythe origin of the current accelerated expansion of the Universe [5, 6, 7]. For example in Lovelockgravity, there have been some attempts for understanding the role of the higher curvature termsfrom various points of view [4]. As an another example, a special case of considering the effect ofhigher curvature corrections, we will deal with in this paper the so-called F ( R ) gravity [2] (and fora review, see, [8]), whose action is an arbitrary function of curvature scalar R . When F ( R ) = R ,the Einstein’s general relativity is recovered. Some of the main reasons to consider F ( R ) gravity ∗ [email protected] are as follows: First of all, there is simplicity: F ( R ) actions are sufficiently general to encapsulatesome of the basic characteristics of higher-order gravity, but at the same time they are simpleenough to be easy to handle. Second, there are serious reasons to believe that F ( R ) theories areunique among higher-order gravity theories, in the sense that they seem to be the only ones whichcan avoid the long known and fatal Ostrogradski instability [9]. Third, F ( R ) theories have noghosts [10], and the stability condition F ′′ ( R ) ≥ F ( R ) gravity, Einstein equations posses extra terms induced from geometry which, whenmoved to the right hand side, may be interpreted as a matter source represented by the energy-momentum tensor T Curvµν , see equation (3,4). In a similar fashion, the Space-Time-Matter (
ST M )theory, discussed below, results in Einstein equations in d − dimension with some extra matterterms showed by the energy-momentum tensor T mattµν . It therefore seems plausible to make acorrespondence between the matter terms in ST M theory and geometrical terms in T Curvµν resultingin F ( R ) gravity.From the other point of view, straightforward generalization of the electromagnetic field tohigher dimensions one essential property of it is lost, namely, conformal invariance. In Ref. [12],it has been shown that the conformal excitations of the extra-dimensional space components havethe form of massive scalar fields living in the external (our) spacetime. Maxwell theory can bestudied in a gauge which is invariant under conformal rescalings of the metric, and firstly, has beenproposed by Eastwood and Singer [13]. Also, there exists a conformally invariant extension of theMaxwell action in higher dimensions (Generalized Maxwell Field, GMF), if one uses the lagrangianof the U (1) gauge field in the form [14, 15, 16, 17, 18] I GMF = κ Z d n +1 x √− g ( F µν F µν ) s , (1)where F µν = ∂ µ A ν − ∂ ν A µ is the Maxwell tensor and κ is an arbitrary constant. It is straightforwardto show that for s = ( n + 1) /
4, the action (1) is invariant under conformal transformation ( g µν −→ Ω g µν and A µ −→ A µ ) and for n = 3, the action (1) reduces to the Maxwell action as it shouldbe. The idea is to take advantage of the conformal symmetry to construct the analogues of thefour-dimensional Reissner-Nordstr¨om black hole solutions in higher dimensions.The main scope of this work is to present the correspondence between F ( R ) gravity and Einstein-conformally invariant Maxwell theory. As we show later, these solutions have some interestingproperties, specially in the electromagnetic fields, which do not occur in Einstein gravity in thepresent of ordinary Maxwell field.The outline of our paper is as follows. In section II we present a short review of field equations of( n +1)-dimensional F ( R ) gravity. In section III the field equations of F ( R ) = R N gravity are solvedin the absence of matter field and the resulting solutions are interpreted as black hole. The solutionsof Einstein-conformally invariant Maxwell source are considered in section IV. Conclusions aredrawn in the last section. II. BASIC FIELD EQUATIONS OF F ( R ) GRAVITY:
The action of F ( R ) gravity, in the presence of matter field has the form of I G = − π Z d n +1 x √− g [ F ( R ) + L matt ] , (2)where R is the scalar curvature and F ( R ) is an arbitrary function of R , and L matt is the Lagrangianof matter fields. Variation with respect to metric g µν , leads to the field equations G µν = T Curvµν + T mattµν F ′ ( R ) (3)where G µν is the Einstein tensor and the gravitational stress-energy tensor isT Curvµν = 1 F ′ ( R ) ( 12 g µν ( F ( R ) − RF ′ ( R )) + F ′ ( R ) ; αβ ( g αµ g βν − g µν g αβ )) (4)with F ′ ( R ) ≡ dF ( R ) /dR and T mattµν the standard matter stress-energy tensor derived from thematter Lagrangian L matt in the action (2). One can consider geometrical terms in the left handside of the field equation, and therefore Eq. (3) reduces to R µν F ′ ( R ) − ∇ µ ∇ ν F ′ ( R ) + (cid:18) ✷ F ′ ( R ) − F ( R ) (cid:19) g µν = T mattµν . (5)The trace of Eq. (5) reduces to n ✷ F ′ ( R ) + RF ′ ( R ) − n + 12 F ( R ) = T (6)Here, we consider the special case of F ( R ) gravity, namely, R N gravity and therefore the fieldequation (5) reduces to (cid:20) N R µν − g µν R (cid:21) R N − + N [ g µν ✷ − ∇ µ ∇ ν ] R N − = T mattµν (7)It is easy to show that for N = 1, Eq. (7) reduces to familiar Einstein gravity. III. BLACK HOLE SOLUTIONS OF R N GRAVITY WITHOUT MATTER FIELD:
Here we want to obtain the ( n + 1)-dimensional static solutions of Eqs. (7) without any matterfield ( L m = 0, and then T mattµν = 0) with N ∈ N . We assume that the metric has the following form ds = − g ( r ) dt + dr g ( r ) + r d Ω k , (8)where d Ω k = dθ + n − P i =2 i − Q j =1 sin θ j dθ i k = 1 dθ + sinh θ dθ + sinh θ n − P i =3 i − Q j =2 sin θ j dθ i k = − n − P i =1 dφ i k = 0 , (9)which represents the line element of an ( n − n − n − k and volume V n − . To find the function g ( r ), one may use any components of Eq.(7). The solution which satisfies all the components of the gravitational field equations (7), can bewritten as g ( r ) = k − mr n − + λ ( N − r n − (10)where m and λ are integration constants which proportional to the mass and charge parameterrespectively. In order to study the general structure of this solution, we first look for the curvaturesingularities. It is easy to show that the Kretschmann scalar R µναβ R µναβ diverges at r = 0, it isfinite for r > r → ∞ . Thus, there is an essential singularity located at r = 0.The event horizon(s), if there exists any, is (are) located at the root(s) of g rr = g ( r ) = 0 kr n − − mr + + λ = 0 . The temperature may be obtained through the use of the definition of surface gravity. Oneobtains T = g ′ ( r + )4 π = ( n − mr + − ( n − λ πr n + = ( n − r n − − λ πr n + (11)Using the fact that the temperature of the extreme black hole is zero, it is easy to show that thecondition for having an extreme black hole is that the mass parameter is equal to m ext , where m ext is given as m ext = ( n − (cid:18) λn − (cid:19) ( n − / ( n − . (12) –3–2–1012345 1 2 3 4 5 6 7 8 FIG. 1: g ( r ) versus r with for N = 2, λ = 2, k = 1, n = 4 and m = 2 < m ext (continuous line), m = m ext = 3(doashed line) and m = 4 > m ext (bold line). The metric of Eqs. (8), (9) and (10) presents a black hole solution with inner and outer horizons,provided the mass parameter m is greater than m ext , an extreme black hole for m = m ext , and anaked singularity otherwise (see Fig(1) for more details). This behavior is the same as Reissner-Nordstr¨om black holes, but, the metric function g ( r ), presented here, differ from the standardhigher-dimensional Reissner-Nordstr¨om solutions since the electric charge term in the metric func-tion is proportional to r − ( n − and in the standard higher dimensional charged black hole solutionsis proportional to r − n − . It is notable that for N = 1 ( F ( R ) = R : Einstein gravity), the thirdterm in Eq. (10) vanishes and the solutions reduce to Schwarzschild like solutions. IV. THE SOLUTIONS OF EINSTEIN GRAVITY IN THE PRESENCE OF NONLINEARMAXWELL SOURCE:
In this section, we consider the ( n +1)-dimensional action in which gravity is coupled to nonlinearelectrodynamics field with an action I ′ G = − π Z ∂M d n +1 x √− g h R − α ( F αβ F αβ ) s i , (13)where α is a coupling constant and the exponent s represented the nonlinear power of the elec-tromagnetic field. Varying the action (13) with respect to the metric tensor g µν and the elec-tromagnetic field A µ , the equations of gravitational and electromagnetic fields may be obtainedas G µν = T mattµν , (14) ∂ µ h √− gF µν ( F αβ F αβ ) s − i = 0 , (15)where T mattµν = 2 α (cid:20) sF µρ F ρν ( F αβ F αβ ) s − − g µν ( F αβ F αβ ) s (cid:21) (16)It is easy to show that when s goes to 1, the Eqs. (13)–(15), reduce to the in Einstein-standardMaxwell gravity in higher dimensions. The Maxwell equation (15) with metric (8) can be integratedimmediately to give F tr = , s = 0 , − qr , s = n n − s ) q (2 s − r ( n − / (2 s − , otherwise , (17)where q , an integration constant where the electric charge of the spacetime is related to thisconstant. Inserting the Maxwell fields (17) and the metric (8) in the field equation (14), one canshow that these equations have the following solutions g ( r ) = k − mr n − + α × , s = 0 , − n/ n/ q n ln rr n − , s = n − s (2 s − ( n − s − n ) r ns − s +1) / (2 s − h s − n ) q (2 s − i s , Otherwise , (18)where m is the integration constant which is related to mass parameter. In the linear case ( s = 1),the solutions reduce to the higher dimensional Reissner-Nordstr¨om solutions with linear Maxwellsource as they should be. Straightforward calculation of Kretschmann scalar shows that there isan curvature singularity located at r = 0.Before studying in details the spacetime, we first specify the sign of the coupling constant α interm of the exponent s in order to ensure a physical interpretation of our future solutions. In fact,the sign of the coupling constant in the action (13) can be chosen such that the energy density, i.e.the T b b component of the energy-momentum tensor in the ortonormal frame, is positive T b b = ( − s +1 α (2 s − (cid:0) F tr (cid:1) s > . (19)This condition selects two branches depending on the range of the nonlinear parameter s , sgn ( α ) = ( − − s , s > ( − − s , s < (20)while the cases s = 0 , / F tr (and charge term in (18)) vanishes.Now, we want to investigate the special case, such that the electromagnetic field equationbe invariant under conformal transformation ( g µν −→ Ω g µν and A µ −→ A µ ). The idea is totake advantage of the conformal symmetry to construct the analogues of the four dimensionalReissner-Nordstr¨om solutions in higher dimensions. It is easy to show that for Lagrangian in theform L ( F = F αβ F αβ ) in ( n + 1)-dimensions, T µµ ∝ (cid:2) F dLdF − n +14 L (cid:3) ; so T µµ = 0 implies L ( F ) = Constant × F ( n +1) / . Indeed, in our case ( L ( F ) ∝ F s ), for s = ( n + 1) /
4, the Maxwell action (14 )enjoys the conformal symmetry in arbitrary dimensions. In this case the metric function f ( r ) andelectromagnetic field F tr reduce to [17] g ( r ) = k − mr n − + 2 ( n − / q ( n +1) / r n − , (21) F tr = qr . (22)Here, we calculate the electric charge of the Einstein-conformally invariant Maxwell solutions. Todetermine the electric field, we should consider the projections of the electromagnetic field tensoron special hypersurfaces. The electric charge per unit volume of the black hole can be found bycalculating the flux of the electromagnetic field at infinity, obtaining Q = 2 ( n − / ( n + 1) q ( n − / π , (23)which confirm that q is related to the electrical charge of the spacetime. Comparing Eq. (10) withEq. (21) (and let λ = [2 ( n − / q ( n +1) / ] / ( N − F ( R ) gravity withoutmatter field. One can think about the same effects of T Curvµν in F ( R ) gravity (Eq. (3) withoutT mattµν ) and T mattµν in Einstein-conformally invariant Maxwell gravity (Eq.(14)) and therefore λ isrelated to the charge parameter q . V. CONCLUSIONS
In the presented paper, we considered the special case of F ( R ) gravity, so-called R N gravity andobtained its topological black hole solutions in higher dimensions. We foud that, such as chargedblack hole solutions, these solutions may be interpreted as black hole solutions with two eventhorizons, extreme black hole and naked singularity provide that the mass parameter m is greaterthan an extremal value m ext , m = m ext and m < m ext respectively. But the presented solutionsdiffer from the standard higher-dimensional Reissner-Nordstr¨om solutions since the electric chargeterm in obtained metric function is proportional to r − ( n − and in the standard higher dimensionalcharged black hole solutions is proportional to r − n − .Next, we presented topological black hole solutions in Einstein gravity with nonlinear elec-tromagnetic field. Then, we restricted ourself to the special case, namely, the conformally in-variant Maxwell field coupled to Einstein gravity and discuss about the correspondence betweenthe Einstein-conformally invariant Maxwell solutions and the solutions of F ( R ) gravity withoutmatter field in arbitrary dimensions. It is easy to show that presented solutions reduce to Reissner-Nordstr¨om black hole in four dimension. Acknowledgments
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