Einstein-Gauss-Bonnet gravity coupled to bumblebee field in four dimensional spacetime
aa r X i v : . [ g r- q c ] F e b Einstein-Gauss-Bonnet gravity coupled to bumblebee field in four dimensionalspacetime
Chikun Ding , , , ∗ Xiongwen Chen , , † and Xiangyun Fu ‡ Department of Physics, Huaihua University, Huaihua, 418008, P. R. China Department of Physics, Hunan University of Humanities,Science and Technology, Loudi, Hunan 417000, P. R. China Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education,and Synergetic Innovation Center for Quantum Effects and Applications,Hunan Normal University, Changsha, Hunan 410081, P. R. China Institute of Physics, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China
Abstract
We study Einstein-Gauss-Bonnet gravity coupled to a bumblebee field which leads to a sponta-neous Lorentz symmetry breaking in the gravitational sector. We obtain an exactly black hole solu-tion and cosmological solutions in four dimensional spacetime by a regularization scheme. The bum-blebee field doesn’t affect the locations of the black hole horizon. It affects the gravitational poten-tial, decreases the black hole’s Hawking temperature. This black hole is different from Schwarzschildblack hole that the gravitational potential has a minimum in the black hole interior and is positivefinite at short distance.
PACS numbers: 04.50.Kd, 04.20.Jb, 04.70.Dy
I. INTRODUCTION
It is well known that one of the promising theories to unify all interactions is M/string theory [1]. Inthe low energy limit, super-symmetric string theory yields effective field theories in which the Einstein’sgeneral relativity is corrected by terms quadratic and higher in the curvature [2]. The generalization of thesecorrections is Lovelock’s theory [3], in which General Relativity(GR) is merely the first-order approximation,that cannot describe the effect of strong gravity, such as the very early Universe or the interior of a blackhole. The only combination of quadratic terms that leads to a ghost-free nontrivial gravitation interaction isthe Gauss-Bonnet invariant G [4], which can be seen as the second-order approximation of the gravity theory.So the factor α before G can be called as an expansion parameter. However, this invariant is a topologicalinvariant in four dimensions, and hence does not contribute to the gravitational dynamics in D = 4 dimensional ∗ Corresponding author; Electronic address: [email protected] † Corresponding author; Electronic address: [email protected] ‡ Electronic address: [email protected] spacetime.Recently, a novel four-dimensional Einstein-Gauss-Bonnet gravity theory was proposed by Glavan and Lin[6] using a regularization scheme. They rescale the expansion parameter α by α/ ( D −
4) in D dimensions, andthen define the four-dimensional theory in the limit D →
4, at the level of the field equations. Some authorspointed out the drawbacks of this regularized scheme [5]. Some authors give several remedies to overcomethese objections [7] and obtain the same black hole metric as the original version [6] showing that it is still asa valid solution. Therefore, we can still use the original regularization scheme to study Einstein-Gauss-Bonnetgravity coupled to other fields.The string theory also involves considering a spontaneous Lorentz symmetry breaking [8] due to thatLorentz invariance(LI) should not be an exact principle at all energy scales [9]. Both GR and the standardmodel(SM) in particle physics based on LI and the background of spacetime, but they process two differentenergy scales. To unify them at very high energy scales, one builds an effective field theory termed standardmodel extension(SME), that couples the SM to GR, involves extra items embracing information about theLorentz violation(LV) happening at the Plank scale [10]. The simplest case is described by a single vector B µ —bumblebee field, with a nonzero vacuum expectation value and the spontaneous LV triggered by a smoothquadratic potential. Bumblebee gravitational model was first studied by Kostelecky and Samuel in 1989[8, 11, 12] as a specific pattern for unprompted Lorentz violation.So in this paper, we would like to study this effective field theory—Einstein-Gauss-Bonnet gravity coupledto the bumblebee fields, to get some suppressed effects emerging from the underlying unified quantum gravitytheory, on our low energy scale. In 2018, R. Casana et al gave an exact Schwarzschild-like solution in thisbumblebee gravity model and considered its some classical tests [13]. The rotating black hole solutions are themost relational subsets for astrophysics. In 2020, we et al found an exact Kerr-like solution through solvingEinstein-bumblebee gravitational field equations and studied its black hole shadow[14]. Then Li and ¨Ovg¨un[15] study the weak gravitational deflection angle of relativistic massive particles by this Kerr-like black hole.Jha and Rahaman [16] extended this Kerr-like solution to Kerr-sen case.We will study the black hole solutions and cosmological solutions in the theory of Einstein-Gauss-Bonnetgravity coupled to the bumblebee fields in four dimensional spacetime. The rest paper is organized as follows.In Sec. II we give the background for the Einstein-Gauss-Bonnet-bumblebee theory. In Sec. III, we give theblack hole solution by solving the gravitational field equations. In Sec. IV, we study its cosmological solutionsand find some effects of the Lorentz breaking constant ℓ . Sec. V is for a summary. II. EINSTEIN-GAUSS-BONNET-BUMBLEBEE THEORY
In the bumblebee gravity theory, the bumblebee vector field B µ gets a nonzero vacuum expectation value,via a given potential, leading a spontaneous Lorentz symmetry breaking in the gravitational sector. In D − dimensional spacetime, the action of Einstein-Gauss-Bonnet gravity coupled to this bumblebee field is[17], S = Z d D x √− g h R κ + 2 αD − G + ̺ κ B µ B ν R µν − B µν B µν − V ( B µ B µ ∓ b ) + L M i , (2.1)where R is Ricci scalar, κ = 2 G R d Ω D − which is the production between the Newton constant G and thearea of a unit D − α is an expansion parameter and the Gauss-Bonnet invariant G is, G = R µντσ R µντσ − R µν R µν + R , (2.2)which can be seen as the second-order approximation of the gravity theory and may sufficiently describethe strong gravity effects. In four dimensions, it is a topological invariant and has no contribution to thegravitational dynamics. But if one uses a regularization scheme—re-scaling α by α/ ( D −
4) and taking limit D →
4, Glavan and Lin [6] found this Gauss-Bonnet term can make a non-trivial effect.The coupling constant ̺ dominates the non-minimal gravity interaction to bumblebee field B µ . The term L M represents possible interactions with matter or external currents. The constant b is a real positiveconstant. Lorentz and/or CP T (charge, parity and time) violation is opened by the potential V ( B µ B µ ∓ b ).It gives a nonzero vacuum expectation value (VEV) for bumblebee field B µ implying that the vacuum ofthis theory gets a preferential direction in the spacetime. This potential is assumed to have a minimum at B µ B µ ± b = 0 and V ′ ( b µ b µ ) = 0 to assure the breaking of the U (1) symmetry, where the field B µ obtains anonzero VEV, h B µ i = b µ . The vector b µ is a function of the spacetime coordinates and has constant value b µ b µ = ∓ b , where ± signs imply that b µ is timelike or spacelike, respectively. The bumblebee field strengthis B µν = ∂ µ B ν − ∂ ν B µ . (2.3)The action (2.1) yields the gravitational field equation G µν = R µν − g µν R = κT Bµν + 2 ακT
GBµν + κT Mµν , (2.4)where the bumblebee energy momentum tensor T Bµν is T Bµν = B µα B αν − g µν B αβ B αβ − g µν V + 2 B µ B ν V ′ + ̺κ h g µν B α B β R αβ − B µ B α R αν − B ν B α R αµ + 12 ∇ α ∇ µ ( B α B ν ) + 12 ∇ α ∇ ν ( B α B µ ) − ∇ ( B µ B ν ) − g µν ∇ α ∇ β ( B α B β ) i , (2.5)and the Gauss-Bonnet energy momentum tensor T GBµν is, T GBµν = 4 R αβ R α βµ ν − R µαβγ R αβγν + 4 R µα R αν − RR µν + 12 g µν G . (2.6)The prime denotes differentiation with respect to the argument, V ′ = ∂V ( x ) ∂x (cid:12)(cid:12)(cid:12) x = B µ B µ ± b . (2.7)The equation of motion for the bumblebee field is ∇ µ B µν = 2 V ′ B ν − ̺κ B µ R µν . (2.8)In the next sections, we derive the black hole solution and cosmological solution by solving gravitationalequations in this Einstein-Gauss-Bonnet-bumblebee model. III. BLACK HOLE SOLUTION IN EINSTEIN-GAUSS-BONNET-BUMBLEBEE GRAVITY
In this section, we suppose that there is no matter field and the bumblebee field is frosted at its VEV, i.e.,it is B µ = b µ , (3.1)then the specific form of the potential controlling its dynamics is irrelevant. And as a result, we have V =0 , V ′ = 0. Then the first two terms in Eq. (2.5) are like those of the electromagnetic field, the only distinctiveare the coupling items to Ricci tensor. Under this condition, Eq. (2.4) leads to gravitational field equations[17] G µν = 2 ακT GBµν + κ ( b µα b αν − g µν b αβ b αβ ) + ̺ (cid:16) g µν b α b β R αβ − b µ b α R αν − b ν b α R αµ (cid:17) + ¯ B µν , (3.2)with ¯ B µν = ̺ h ∇ α ∇ µ ( b α b ν ) + ∇ α ∇ ν ( b α b µ ) − ∇ ( b µ b ν ) − g µν ∇ α ∇ β ( b α b β ) i . (3.3)The static spherically symmetric black hole metric in a D dimensional spacetime have the general form ds = − e φ ( r ) dt + e ψ ( r ) dr + r d Ω D − , (3.4)where Ω D − is a standard D − b µ b µ = positive constant) and assumed to be b µ = (cid:0) , be ψ ( r ) , , , · · · , (cid:1) , (3.5)where b is a positive constant. Then the bumblebee field strength is b µν = ∂ µ b ν − ∂ ν b µ , (3.6)whose components are all zero. And their divergences are all zero, i.e., ∇ µ b µν = 0 . (3.7)From the equation of motion (2.8), we have b µ R µν = 0 . (3.8)The gravitational field equations (3.2) become G µν = 2 ακT GBµν + ¯ B µν . (3.9)For the metric (3.4), the nonzero components of Einstein tensor G µν , the Gauss-Bonnet momentum tensor T GBµν and the bumblebee tensor ¯ B µν are shown in the appendix. By using the motion equation (3.8) R = ( D − r ψ ′ − ( φ ′′ + φ ′ − φ ′ ψ ′ ) = 0 , (3.10)one can obtain the following three gravitational field equations( D − e ψ −
1) + 2 rψ ′ = − D − ακ h
12 (1 − e − ψ ) ψ ′ + D − r (cid:16) e ψ + e − ψ − (cid:17)i + ℓ (cid:2) ( D − − rψ ′ (cid:3) , (3.11)( D − − e ψ ) + 2 rφ ′ = 4( D − ακ h −
12 (1 − e − ψ ) φ ′ + D − r (cid:16) e ψ + e − ψ − (cid:17)i − ℓ (cid:2) ( D −
3) + 2 rφ ′ (cid:3) , (3.12)( D − D − − e ψ ) + rψ ′ + ( D − rφ ′ = 2( D − ακ n − e − ψ φ ′ ψ ′ + (1 − e − ψ ) · h ( D − D − r ( e ψ − − D − r φ ′ − r ψ ′ io − ℓ h ( D − D − r ( ψ ′ + φ ′ ) i , (3.13)where the prime ′ is the derivative with respect to the corresponding argument, respectively.Adding the Eq. (3.11) to the Eq. (3.12), one can obtain the result φ ′ ( r ) + ψ ′ ( r ) = 0, since the alternativesolution 1 + 2 ℓ + 2( D − ακ (1 − e − ψ ) /r = 0 is incompatible with the remaining field equations. Rewritingthe function e ψ = 1 /f ( r ), then one can change the Eq. (3.11) as the form( D − r (1 − f − ℓf ) + 2 ακ ( D − − f ) ] = rf ′ [(1 + ℓ ) r + 4( D − ακ (1 − f )] . (3.14)It can be written ddr (cid:2) r D − (1 − f − ℓf ) + 2( D − ακr D − (1 − f ) (cid:3) = 0 . (3.15)Its solution is f ( r ) = 1 + (1 + ℓ ) r E " ± s Eℓ (1 + ℓ ) r + 8 EGM (1 + ℓ ) r D − , (3.16)where E = 2( D − ακ , M is the mass of the black hole. Note that if ℓ = 0 and E = 2( D − D − ακ , itis the same as that in [18]. Setting e φ = A ( r ) f ( r ) and using ψ ′ + φ ′ = 0, one can obtain A ( r ) is a constant,here let it to A = 1 + ℓ , i.e., e φ ( r ) = (1 + ℓ ) f ( r ) . (3.17)When D →
4, the black hole metric in the bumblebee gravity is ds = − (1 + ℓ ) f ( r ) dt + 1 f ( r ) dr + r dθ + r sin θ dθ , (3.18)with f ( r ) = 1 + (1 + ℓ ) r παG " ∓ s παGℓ (1 + ℓ ) r + 128 παG M (1 + ℓ ) r . (3.19)There are two branches of solutions for the metric function (3.19) with α >
0, the one corresponding to “ − ”sign; the second to “ + ” sign. Lastly, substituting these quantities into Eqs. (3.4) and (3.5), we can get thebumblebee field b µ = (0 , / p f ( r ) , , ℓ →
0, it recovers the four dimensional Gauss-Bonnet black hole metric [6]. When α →
0, the first branchbecomes ds = − (cid:16) − GMr (cid:17) dt + 1 + ℓ − GM/r dr + r dθ + r sin θ dθ , (3.20)a Schwarzschild-like black hole which is the same as that in Ref. [13]; the second one becomes f ( r ) = 11 + ℓ h ℓ + (1 + ℓ ) r παG + 2 GMr i , (3.21)which is a Schwarzschild-de Sitter like black hole with a negative mass. It has been shown by Boulware andDeser [18] that the second branch is unstable and leads to graviton ghost. The expansion parameter α andcoupling constant ℓ in metric function (3.21) represents a positive cosmological constant. When r → ∞ ,the two branches have similar behaves asymptotically at large distances, i.e., the first is asymptotically to aSchwarzschild-like black hole with positive mass M , the second to a Schwarzschild-de Sitter like black holewith negative mass M .It is easy to see that their horizon, the outer and inner horizon, locates at g ( r ± ) = 0, r ± = GM ± p G M − πGα, (3.22)which doesn’t depend on ℓ , i.e., the bumblebee field doesn’t affect the location of black hole horizon. In theFig. 1, we plot the radial dependence of − g with different α and ℓ . One can see that the gravitationalpotential has only one minimum, and approaches a finite value 1 + ℓ at short distances r →
0, which isdifferent from the Schwarzschild geometry. This minimum locates at r m from f ′ ( r m ) = 0, f ′ ( r m ) = (1 + ℓ ) GM r m − ακ ( ℓr m + GM ) = 0 , (3.23)where κ = 8 πG in D = 4 dimensions. From the Fig. 1, one can see that the LV constant ℓ pushes r m to the (cid:144) - - g Expansion parameter Α= { = { = { = { =- (cid:144) - - g Expansion parameter Α= { = { = { = { =- (cid:144) - g Expansion parameter Α= { = { = { = { =- FIG. 1: Radial dependence of gravitational potential − g with different coupling constant ℓ in the case of differentexpansion parameter α . left point r = 0.From the formula in [19], its Hawking temperature is T = √ ℓ π f ′ ( r + ) = √ ℓ π r + − GM (1 + ℓ ) r + 32 παGM . (3.24) IV. COSMOLOGICAL SOLUTION
In this section, we investigate the cosmological solution. The Friedmann-Robertson-Walker metric is ds = − dt + a ( t ) (cid:16) dr − kr + r d Ω D − (cid:17) , (4.1) Α= Α= Α= Α= { FIG. 2: Variety of Hawking temperature T with different coupling constant ℓ of Einstein-bumblebee black hole fordifferent expansion parameter α . where k = 1 , , −
1, corresponding to closed, flat or open cosmos. The function a ( t ) is the scale factor. Tomaintain the assumption of the large-scale homogeneity and isotropy of Universe, the bumblebee field isassumed as following B µ = ( B ( t ) , , · · · , , (4.2)where there is no spatial orientation. Then the field strength tensor vanishes, B µν = 0, and the bumblebeemotion equation (2.8) becomes 2 V ′ B ν = ̺κ B µ R µν . (4.3) T Bµν becomes T Bµν = − g µν V + ̺κ ˜ B µν , (4.4)where ˜ B µν is ˜ B µν = 12 g µν B α B β R αβ − B µ B α R αν + 12 ∇ α ∇ µ ( B α B ν )+ 12 ∇ α ∇ ν ( B α B µ ) − ∇ ( B µ B ν ) − g µν ∇ α ∇ β ( B α B β ) . (4.5)Supposing that the matter contribution is equivalent to an ideal fluid T Mµν = diag ( ρ, − p, · · · , − p ) with theenergy density of matter ρ and its pressure p . G µν = − κg µν V + κT Mµν + ̺ ˜ B µν + 2 αD − κT GBµν . (4.6)The non-zero components of tensors G µν , ˜ B µν and T GBµν are also shown in the appendix. When D →
4, the tt and rr components of Eq. (4.6) are3 h ka + H + 2 ακ ( ka + H ) − ̺B H i = κ ( ρ + V ) + 3 ̺HB ˙ B (4.7)2 ¨ aa h ακ ( ka + H ) + 1 − ̺B i + h ka + H (1 − ̺B ) i = κ ( p + V ) − ̺ h HB ˙ B + ˙ B + B ¨ B i . (4.8)If one lets k = 0 , ̺ = V = 0, the above both equations can reduce to those in Ref. [6]; if one lets α = k =0 , p = 0, they are the same as those in Ref. [20]. If k = 0 and ˙ B = ¨ B = 0, i.e., the bumblebee field B ( t )doesn’t evolve with time and its potential rest at a constant V = V , they become3(1 − ̺B ) H + 6 ακH = κ ( ρ + V ) (4.9)2 ¨ aa h ακH + 1 − ̺B i + H (1 − ̺B ) = κ ( p + V ) . (4.10) V. SUMMARY
In this paper, we have studied Einstein-Gauss-Bonnet gravity coupled to a bumblebee field. We obtainan exactly black hole solution and cosmological solutions in four dimensional spacetime by a regularizationscheme. The bumblebee field doesn’t affect the locations of the black hole horizon. This black hole is differentfrom Schwarzschild black hole that the gravitational potential has a minimum in the black hole interior andis positive finite at short distance. The bumblebee fields affect the gravitational potential: decrease theminimum of the gravitational potential in the black hole interior and increase the positive finite value at shortdistance. At large distance, this effects will become weaken. The bumblebee fields also decrease the blackhole’s Hawking temperature.Note that one can obtain a D -dimensional Schwarzschild-like solution in bumblebee gravity. In the firstbranch of the metric function (3.16), one lets α → ds = − (cid:16) − GMr D − (cid:17) dt + 1 + ℓ (1 − GMr D − ) dr + r d Ω D − . (5.1) Acknowledgments
This work was supported by the Scientific Research Fund of the Hunan Provincial Education Departmentunder No. 19A257, the National Natural Science Foundation (NNSFC) of China (grant No. 11247013), HunanProvincial Natural Science Foundation of China (grant No. 2015JJ2085).0
Appendix A: Some quantities I
In this appendix, we showed the nonezero components of Einstein’s tensor for the metric (3.4). They areas following (cf. [2]) G = ( D − e φ − ψ r h ( D − e ψ −
1) + 2 rψ ′ i , (A1) G = ( D − r h ( D − − e ψ ) + 2 rψ ′ i , (A2) G = e − ψ h ( D − D − − e ψ ) + ( D − r ( φ ′ − ψ ′ ) + r ( φ ′′ + φ ′ − φ ′ ψ ′ ) i , (A3) G ii = G i − Y j =1 sin θ j , (A4) R = ( D − r ψ ′ − ( φ ′′ + φ ′ − φ ′ ψ ′ ) , (A5)where ( D − ≥ i ≥
3. ¯ B µν are¯ B = b e φ − ψ r h ( D − D − − ( D − rψ ′ − r ( φ ′′ + φ ′ − φ ′ ψ ′ ) i , (A6)¯ B = − b r h ( D − D − − ( D − r ( ψ ′ − φ ′ ) + r ( φ ′′ + φ ′ − φ ′ ψ ′ ) i , (A7)¯ B = − b e − ψ h ( D − D − − ( D − rψ ′ + 2( D − rφ ′ + r ( φ ′′ + φ ′ − φ ′ ψ ′ ) i , (A8)¯ B ii = ¯ B i − Y j =1 sin θ j . (A9)The nonzero components of the Gauss-Bonnet term T GBµν are T GB = ( D − D − D − r e φ − ψ h r (1 − e − ψ ) ψ ′ + ( D − r ( e ψ + e − ψ − i , (A10) T GB = ( D − D − D − r h − r (1 − e − ψ ) φ ′ + ( D − r ( e ψ + e − ψ − i , (A11) T GB = ( D − D − e − ψ n − e − ψ φ ′ ψ ′ + (1 − e − ψ ) h ( D − D − r ( e ψ − − D − r ( φ ′ − ψ ′ ) − φ ′′ + φ ′ − φ ′ ψ ′ ) io , (A12) T GBii = T GB i − Y j =1 sin θ j . (A13) Appendix B: Some quantities II
We showed the nonezero components of Einstein’s tensor for the metric (4.1). They are as following (cf.[1]) G = ( D − D − k + ˙ a a , (B1) G = − ( D − − kr h a ¨ a + D −
32 ( k + ˙ a ) i . (B2)1˜ B µν are ˜ B = ( D − B ˙ aa h a ˙ B + ( D − B ˙ a i , (B3)˜ B = − − kr h ( D − D − B ˙ a + ( D − Ba (2 ˙ B ˙ a + B ¨ a ) + a ( ˙ B + B ¨ B ) i . (B4)The nonzero components of the Gauss-Bonnet term T GBµν are T GB = − ( D − D − D − D −
4) ( k + ˙ a ) a , (B5) T GB = ( D − D − D − k + ˙ a − kr h aa + ( D − k + ˙ a )2 a i . (B6) [1] S. A. Pavluchenko, Phys. Rev. D , 024046 (2016).[2] D. L. Willtsire, Phys. Lett. B , 36 (1986).[3] D. Lovelock, J. Math. Phys. , 498 (1971); ibid , 874 (1972).[4] B. Zwiebach, Phys. Lett. B , 315 (1985).[5] F. Shu, Phys. Lett. B , 135907 (2020); W. Ai, Commun. Theor. Phys. , 095402 (2020); M. Gurses, T. C.Sisman and B. Tekin, Eur. Phys. J. C , 647 (2020); S. Mahapatra, Eur. Phys. J. C , 992 (2020); J. Arrechea,A. Delhom and A. Jim´enez-Cano, Chin. Phys. C , 149001 (2020); J. Arrechea, A. Delhom and A. Jim´enez-Cano, Phys. Rev. Lett. , 149002(2020).[6] D. Glavan and Chunshan Lin, Phys. Rev. Lett. , 081301 (2020).[7] R. A. Hennigar, D. Kubiznak, R. B. Mann and C. Pollack, J. High Energy Phys. , 027 (2020); A. Casalino, A.Colleaux, M. Rinaldi and S. Vicentini, Phys. Dark Universe, , 100770 (2021); H. Lu and Y. Pang, Phys. Lett.B , 135717 (2020); T. Kobayashi, J. Cos. Astro. Phys. , 013 (2020).[8] V. A. Kosteleck´y and S. Samuel, Phys. Rev. D , 683 (1989).[9] D. Mattingly, Living Rev. Rel. , 5 (2005).[10] V. A. Kosteleck´y, Phys. Rev. D , 105009 (2004).[11] M. H. Dickinson, F. O. Lehmann and S. P. Sane, Science , 1954 (1999).[12] V. A. Kosteleck´y and S. Samuel, Phys. Rev. D , 1886 (1989).[13] R. Casana and A. Cavalcante, Phys. Rev. D , 104001 (2018).[14] C. Ding, C. Liu, R. Casana and A. Cavalcate, Eur. Phys. C , 178 (2020).[15] Z. Li and A. ¨Ovg¨un, Phys. Rev. D , 024040 (2020).[16] S. K. Jha and A. Rahaman, arXiv: 2011.14916, (2021).[17] C. Ding and X. Chen, Chin. Phys. C , 025106 (2021).[18] D. G. Boulware and S. Deser, Phys. Rev. Lett. , 2656 (1985).[19] C. Ding and J. Jing, Class. Quantum Grav. , 145015 (2008).[20] D. Capelo, Phys. Rev. D91