Gauss-Bonnet Gravity with Scalar Field in Four Dimensions

Transcription

1 arxiv: v3 [gr-qc] 13 Jul 2007 Gauss-Bonnet Gravity with Scalar Field in Four Dimensions Metin Gürses Department of Mathematics, Faculty of Sciences Bilkent University, Ankara - Turkey March 18, 2008 Abstract We give all exact solutions of the Einstein-Gauss-Bonnet Field Equations coupled with a scalar field in four dimensions under certain assumptions. 0

2 Recently there is an increasing interest in the Gauss-Bonnet theory. Spherically symmetric solutions of this theory were first studied in [1], [2]. It has been observed that the Post-Newtonian approximation does not give any new contribution in addition to the post-newtonian parameters of the general relativity [3]. Black hole solutions in the framework of the GB gravity are investigated recently in [4] (see also [5], [6]). Accelerated cosmological solutions were first suggested in [7], [8] and also discussed in [9], [10]. There are also attempts to find exact solutions and to study the stability of the Gauss-Bonnet theory in various dimensions with actions containing higher derivative scalar field couplings [11], [12], [13]. Since the Gauss-Bonnet term is a topological invariant in four dimensions it does not contribute to the Einstein field equations. On the other hand it contributes to the field equations if it couples to a spin-0 zero field. In this work we consider a four dimensional action containing the Einstein-Hilbert part, massless scalar field and the Gauss-Bonnet term coupled with the scalar field. The corresponding action is given by [3] S = where κ 2 = 8πG (c = = 1) and d 4 x g [ R 2κ µφ µ φ V (φ) + f(φ)gb] (1) GB = R 2 4R αβ R αβ + R αβσγ R αβσγ (2) and f is an arbitrary function of the scalar filed φ (coupling function). Here V is potential term for the scalar field. The field equations are given by R µν = κ 2 [ 1 2 µ φ ν φ V (φ) g µν + 2( µ ν f)r g µν ( ρ ρ f)r 1

3 4( ρ µ f)r νρ 4( ρ ν f)r µρ + 4( ρ ρ f)r µν +2g µν ( ρ σ f)r ρσ 4( ρ σ f)r µρνσ ] (3) ρ ρ φ V (φ) + f GB = 0 (4) 1) We now assume the spacetime geometry (M, g) is such that (assumption µ ν f = Λ 1 g µν + Λ 2 l µ l ν (5) where Λ 1 and Λ 2 are scalar functions and l µ is a vector field. In the sequel we will assume that Λ 2 = 0 (assumption 2). Eq.(5) restricts the space-time (M, g). Among these space-times admitting (5) we have conformally flat space-times (assumption 3). g µν = ψ 2 η µν (6) where ψ is a scalar function. In such space-times the conformal tensor vanishes identically. Hence Then the field equations (3) reduce to GB = 2R αβ R αβ R2 (7) (1 4Λ 1 κ 2 ) R µν = κ 2 [ 1 2 µ φ ν φ V (φ) g µν] (8) We have now a last assumption: All functions depend on z = k µ x µ where k µ is a constant vector, µ k ν = 0. Then from (5) 2

4 we get R µν = 2 ψ,µν ψ + [ 1 ψ ηαβ ψ,αβ 3 ψ 2 ηαβ ψ,α ψ,β ] η µν, (9) f = Cψ 2, Λ 1 = Ck 2ψ ψ (10) where C is an arbitrary constant and k 2 = η µν k µ k ν. By using (8) and the Ricci tensor R µν = 2 ψ,µν ψ + [ 1 ψ ηαβ ψ,αβ 3 ψ 2 ηαβ ψ,α ψ,β ] η µν, (11) for the metric (6) we obtain the following equations (1 4Λ 1 κ 2 )ψ 1 ψ = κ2 4 (φ ) 2, (12) V = 2k2 κ (1 4Λ 1κ 2 )[3(ψ ) 2 ψψ ], 2 (13) k 2 ψ 4 (ψ 2 φ ) V + fgb = 0, (14) f = Cψ 2, Λ 1 = Ck 2ψ ψ (15) where GB = 72(k 2 ) 2 ψ 4 [(ψ 1 ψ ) 2 ψ 1 ψ ](ψ 1 ψ ) 2 (16) and a dot over a letter denotes derivative with respect to the scalar field φ. Eqs(12) and (14) give coupled ODEs for the functions ψ and φ. Letting ψ /ψ = u and φ = v then these equations become (1 + 4Ck 2 κ 2 u)(u + u 2 ) = κ2 4 v2, (17) k 2 ψ 2 [(v 2uv)v 27Ck 2 u u 2 ] = V (18) 3

5 where V is given by (from (13)) V = 2k2 κ 2 (1 + 4Ck2 κ 2 u)(2u 2 u ) ψ 2 (19) Inserting V from (19) into Eq. (18) (and using (17) in (18)) we obtain simply 3C(k 2 ) 2 u 2 u = 0 (20) Hence we have the following solutions. (A) C = 0: This corresponds to pure Einstein filed equations with a massless scalar field. The effect of the Gauss Bonnet term disappears. Solutions of these field equations have been given in [14] (B) k 2 = 0: The vector field k µ is null. Then the only field equation is u + u 2 = κ2 4 v2 (21) and V becomes zero. There is a single equation for the two fields u and v. This means that, if one of the fields u or v is given then the other one is determined directly. The metric takes the form ds 2 = ψ(p) 2 [2dpdq + dx 2 + dy 2 ] (22) where p and q are null coordinates and k µ = δ p µ becomes and the above equation (21) ψ pp = κ2 4 (φ ) 2 ψ (23) and the Einstein tensor represents a null fluid with zero pressure. 4

6 G µν = κ2 2 (φ ) 2 k µ k ν (24) Although the coupling function f is nonzero the effect of the GB term is absent in this type. Such a class of solutions belongs to class (A). (C) k 2 0: The vector field k µ is non-null. Then u = m a real constant which leads to the following solution. where ψ 0 and φ 0 are arbitrary constants and ψ = ψ 0 e m z, φ = φ 0 + φ 1 z (25) (1 + 4Ck 2 κ 2 m)m 2 = κ2 4 φ2 1, V = k 2 φ 2 1 ψ 2 (26) where φ 1 0. The potential function V takes the form V (φ) = V 0 e ± φ ξ, V0 = k 2 φ 2 1 ψ2 0 e φ 0 ξ (27) where ξ = 1 + 4Ck 2 κ 2 m and coupling function f takes the form f = f 0 f 1 e φ ξ, f1 = (C/ξ ψ0 2 ) e φ ξ (28) The solution we obtained here is free of singularities but not asymptotically flat. On the other hand, by using this solution it is possible to obtain an asymptotically flat cosmological solution. This solution is well understood in a new coordinate chart {x a, t} where the line element takes the following form (after a scaling) ds 2 = u2 t 2 0 η ab dx a dx b + ǫdt 2 (29) 5

7 where t 0 is a nonzero constant. If t is a spacelike coordinate ǫ = 1 and Latin indices take values a = 0, 1, 2. If t is a timelike coordinate the ǫ = 1 and Latin indices take values a = 1, 2, 3. η ab is the metric of the flat three dimensional geometry orthogonal to the u-direction. The Ricci tensor of the four dimensional metric Hence the solution takes the form R tt = 0, R ta = 0, R ab = 2ǫ t 2 0 η ab (30) φ = ± 2 ξ, V (φ) = 4ǫξ (31) t t 2 where ξ = 1 + 4κ 2 C, Λ 1 = C a constant, and f = f 0 + ǫc 2 t2, f 0 is an arbitrary constant. The curvature scalars are given by R = 6 t 2, R µν R µν = 12 t 4 (32) and the Gauss-Bonnet scalar density GB = 0. It clear that t = 0 is the spacetime singularity. Letting u α = δ t α, the Einstein tensor becomes G αβ = 2 t u 2 α u β + ǫ t g 2 αβ (33) This tensor has a physical meaning when ǫ = 1 in which case the Gauss- Bonnet gravity produces a singular cosmological model. The Einstein tensor represents a perfect fluid with an energy density ρ = 3/t 2 and a negative pressure p = 1/t 2. Both of them are singular at t = 0. We have found the most general solutions of the Gauss-Bonnet gravity coupled to a scalar field under the assumptions stated in the text. One solution (B) depends on a null coordinate whose Einstein tensor corresponds 6

8 to the energy momentum tensor of a null fluid with zero pressure. The other solution (C) depends on variable t whose curvature invariants are all singular at t = 0. When t represents the time coordinate then GB gravity gives a cosmological model with a negative pressure. The solution is singular on the 3-surface t = 0. This work is partially supported by the Scientific and Technological Research Council of Turkey (TUBITAK) and Turkish Academy of Sciences (TUBA). References [1] D.G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). [2] D.G. Boulware and S. Deser, Phys. Lett. 175B, (1986). [3] Thomas Sotiriou and Enrico Barausse, Phys.Rev. D75, (2007). [4] Chiang-Mei Chen, Dmitri V. Galtsov, and Dmitry G. Orlov, Extremal black goles in D = 4 Gauss-Bonnet gravity (hep-th/ ) [5] S. Mignemi and N.R. Stewart, Phys. Rev.D47, 5259 (1993). [6] P. Kanti, N.E. Mavromatos, J. Rizos, K, Tamvakis, and E. Winstanley, Phys. Rev. D54, 5049 (1996). [7] S. Nojiri, S.D. Odintsov and M. Sasaki, Phys. Rev. D71, (2004). [8] G. Cognola,E. Elizalde, S. Nojiri, S.D. Odintsov and S.Zerbini, Phys. Rev. D73, , (2006). [9] T. Kolvisto and D. Mota, Phys. lett B644, (2007). 7

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