Dyonic Born-Infeld black hole in four-dimensional Horndeski gravity
Kun Meng, Lianzhen Cao, Jiaqiang Zhao, Fuyong Qin, Tao Zhou, Meihua Deng
DDyonic Born-Infeld black hole in 4DEinstein-Gauss-Bonnet gravity
Kun Meng, Lianzhen Cao, Jiaqiang Zhao,Tao Zhou, Fuyong Qin, Meihua Deng ∗ School of Physics and Photoelectric Engineering, Weifang University,Weifang 261061, China
Abstract
The action of 4D Einstein-Gauss-Bonnet gravity coupled to Born-Infeld elec-tromagnetic fields is given via the Kaluza-Klein process. Dyonic black hole solutionof the theory is presented. The metric is devoid of singularity at the origin inde-pendent of the parameter selections. Thermodynamics of the black hole is studied,the first law is obtained and the Smarr relation is given. In extended phase space,calculations show that there exists no thermodynamic phase transition for theblack hole.
Lovelock theorem states that, in four dimensions the tensors that satisfy divergence free,symmetric, and concomitant of the metric tensor and its derivatives are no more thanthe metric tensor and the Einstein tensor [1]. That is to say Einstein’s general relativity(GR) is the unique proper theory of gravity in four dimensions. Recently, in order tobypass Lovelock theorem a proposal has been made by adding Gauss-Bonnet term toGR [2]. As we know, the Gauss-Bonnet contribution is a topological invariant in fourdimensions according to Gauss-Bonnet theorem, and does not affect the field equationsof the theory. The authors of Ref. [2] first make the replacement α → αD − to cancel thefactor D − D → S = 14 G (cid:73) d x √ γ (cid:18) αD − R ( γ ) (cid:19) ∗ email: [email protected] a r X i v : . [ g r- q c ] F e b s divergent manifestly in the D → I = − T − F = S − T − M also implies divergence inthe on-shell Euclidean action [11], thus the action cannot account for the Euclideanpath-integral for the topologically nontrivial solutions. Meanwhile, the Gauss-Bonnetcontribution to field equations H µν can be decomposed into two parts H µν = − L µν + Z µν ), where the Z µν part is proportional to D −
4, it is regular after the rescaling α → αD − and taking the D → L µν , which can be expressedin terms of Weyl tensor L µν = C µαβσ C αβσν − g µν C αβρσ C αβρσ , vanishes identically in D ≤
4, thus L µν D − is undefined [12–14].In order to obtain a regularized EGB theory that is well defined in the limit D → D − α → αD − and take the D → D → n − Black hole solution
In general D dimensions, the action of Einstein-Gauss-Bonnet-Born-Infeld gravity isgiven by I D = 116 πG D (cid:90) d D x √− g ( R −
2Λ + α L GB + 16 πG D L ( F )) , (1)where L GB is the Gauss-Bonnet density L GB = R µνρσ R µνρσ − R µν R µν + R , (2)and L ( F ) takes the form L ( F ) = β (cid:113) − det( g µν ) − β (cid:115) − det (cid:18) g µν + F µν β (cid:19) , (3)where F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor. Note that (3) tends to the MaxwellLagrangian − F µν F µν in the limit β → ∞ .In order to obtain the 4D EGB gravity, one considers the Kaluza-Klein diagonalreduction of the action (1), with metric ansatz ds D = ds p + e φ d Σ D − p,λ , (4)where the breathing scalar φ depends only on the external p -dimensional coordinates.The line elements d Σ D − p,λ describe the internal maximally symmetric space, and λ denotes the sign of the Euclidean space curvature. Then the action (1) reduces to the p -dimensional action [15, 16] I p = 116 πG p (cid:90) d p x √− ge ( D − p ) φ (cid:110) R − + 16 πG p L ( F ) + ( D − p )( D − p − (cid:0) ( ∂φ ) + λe − φ (cid:1) + α (cid:16) L GB − D − p )( D − p − (cid:2) G µν ∂ µ φ∂ ν φ − λRe − φ (cid:3) − ( D − p )( D − p − D − p − (cid:2) ∂φ ) (cid:3) φ + ( D − p − ∂φ ) ) (cid:3) +( D − p )( D − p − D − p − D − p − (cid:2) λ ( ∂φ ) e − φ + λ e − φ (cid:3) (cid:17)(cid:111) , (5)where G µν is Einstein tensor. For p ≤
4, it is free to add − α πG p (cid:90) d p x √− g L GB (6)to action (5) without affecting the field equations, since (6) is just a topological invariant.Now rescaling the Gauss-Bonnet coupling as α → αD − p and taking the D → p limit, oneobtains I p = 116 πG p (cid:90) d p x √− g (cid:2) R −
2Λ + 16 πG p L ( F ) − λRe − φ − λ ( ∂φ ) e − φ − λ e − φ + α (cid:0) φ L GB + 4 G µν ∂ µ φ∂ ν φ − ∂φ ) (cid:3) φ + 2(( ∂φ ) ) (cid:1)(cid:3) . (7)3 .2 equations of motion The variation with respect to the electromagnetic field gives rise to the field equation E A = ∇ µ (cid:20) √− h √− g β ( h − ) [ µν ] (cid:21) = 0 (8)where h µν ≡ g µν + F µν β , and h ≡ det( h µν ). The symmetric part and antisymmetric partof h µν are denoted respectively by h ( µν ) and h [ µν ] . ( h − ) µν denotes the inverse of h µν ,similarly, ( h − ) ( µν ) and ( h − ) [ µν ] are respectively the symmetric and antisymmetric partsof ( h − ) µν .The equation of motion of φ is given by [16, 36] E φ = − L GB + 8 G µν ∇ ν ∇ µ φ + 8 R µν ∇ µ φ ∇ ν φ − (cid:3) φ ) + 8( ∇ φ ) (cid:3) φ + 16 ∇ µ φ ∇ ν ∇ ν ∇ µ φ + 8 ∇ µ ∇ ν ∇ µ ∇ ν φ − λ e − φ − λRe − φ + 24 λe − φ (cid:0) ( ∇ φ ) − (cid:3) φ (cid:1) = 0 , (9)The variation with respect to the metric yields E µν =Λ g µν + G µν − πGβ g µν + 8 πGβ √− h √− g h ( µν ) + α (cid:20) φH µν − R [( ∇ µ φ )( ∇ ν φ ) + ∇ ν ∇ µ φ ] + 8 R ρ ( µ ∇ ν ) ∇ ρ φ + 8 R ρ ( µ ( ∇ ν ) φ )( ∇ ρ φ ) − G µν (cid:2) ( ∇ φ ) + 2 (cid:3) φ (cid:3) − ∇ µ φ )( ∇ ν φ ) + ∇ ν ∇ µ φ ] (cid:3) φ + 3 λ e − φ g µν + 8( ∇ ( µ φ )( ∇ ν ) ∇ ρ φ ) ∇ ρ φ − g µν R ρσ [ ∇ ρ ∇ σ φ + ( ∇ ρ φ )( ∇ σ φ )] + 2 g µν ( (cid:3) φ ) − g µν ( ∇ ρ ∇ σ φ )( ∇ ρ ∇ σ φ ) − g µν ( ∇ ρ φ )( ∇ σ φ )( ∇ ρ ∇ σ φ ) + 4( ∇ ρ ∇ ν φ )( ∇ ρ ∇ µ φ )+ 4 R µρνσ [( ∇ ρ φ )( ∇ σ φ ) + ∇ σ ∇ ρ φ ] − (cid:2) g µν ( ∇ φ ) − ∇ µ φ )( ∇ ν φ ) (cid:3) ( ∇ φ ) − λe − φ (cid:0) G µν + 2( ∇ µ φ )( ∇ ν φ ) + 2 ∇ ν ∇ µ φ − g µν (cid:3) φ + g µν ( ∇ φ ) (cid:1) (cid:21) = 0 . (10)Here and in the following we label the gravitational constant G p as G for simplicity.Combining the last two equations in the following manner yields g µν E µν + α E φ = 4Λ − R − α L GB − πGβ + 8 πGβ √− h √− g h ( µν ) g µν = 0 , (11)which is independent of φ and λ . We assume that φ = φ ( r ), and take the metric ansatz and field strength ansatz as ds = − e − χ ( r ) f ( r ) dt + 1 f ( r ) dr + r ( dx + dy ) , (12) F = − a (cid:48) ( r ) dt ∧ dr + pdx ∧ dy. (13)4ubstituting (12) and (13) into (8), and making χ ( r ) to be zero, one has a (cid:48) ( r ) = qβ (cid:112) p + q + β r . (14)Note that, unlike Maxwell theory, the infinity in the intensity at r = 0 has been removednow, hence the infinity in the potential at r = 0 is absent too. Now combining (11)together with (12), one obtains − α ( f (cid:48) ( r ) + f ( r ) f (cid:48)(cid:48) ( r )) r + 4 f (cid:48) ( r ) r + f (cid:48)(cid:48) ( r ) + 2 f ( r ) r + 4Λ − πGβr − (cid:16) βr − ( p + q + 2 β r ) (cid:0) p + q + β r (cid:1) − / (cid:17) = 0 . (15)This equation is not enough for us to find the explicit form of f ( r ). Substituting themetric ansatz (12) into (7), and discarding the total derivative terms, one obtains theeffective Lagrangian [15] L = e − χ ( r ) r (cid:2) − (cid:0) r f (cid:48) ( r ) + 4Λ r + 4 r f ( r ) (cid:1) − αr f ( r ) f (cid:48) ( r ) φ (cid:48) ( r ) (cid:0) r φ (cid:48) ( r ) − rφ (cid:48) ( r ) + 3 (cid:1) +4 αr f ( r ) φ (cid:48) ( r )(4 χ (cid:48) ( r ) (cid:0) r φ (cid:48) ( r ) − rφ (cid:48) ( r ) + 3 (cid:1) + φ (cid:48) ( r ) (cid:0) r φ (cid:48) ( r ) − rφ (cid:48) ( r ) + 6 (cid:1)(cid:1) +96 πG (cid:0) β r − βr ( p + β r )( p + q + β r ) − / (cid:1)(cid:3) , (16)where the internal space is considered to be flat with λ = 0. Taking variation of (16)with respect to f ( r ), and making χ ( r ) to be zero, one has4 αf ( r ) ( rφ (cid:48) ( r ) − (cid:0) φ (cid:48) ( r ) + φ (cid:48)(cid:48) ( r ) (cid:1) = 0 , (17)which implies φ (cid:48) ( r ) = 1 r . (18)Considering (18) the variation with respect to χ yields4 r (cid:0) r − αf ( r ) (cid:1) f (cid:48) ( r ) + 4 f ( r ) (cid:0) αf ( r ) + r (cid:1) + 4Λ r + 16 πG (cid:16) − β r + 2 βr (cid:112) p + q + β r (cid:17) = 0 (19)Combining (15) and (19) one is now able to give the exact black hole solution: f ( r ) = r α (cid:18) − (cid:18) πGαM Σ r − πGαβ α Λ3 + 32 πGαβ · (cid:32)(cid:115) p + q β r − p + q ) β r F (cid:20) , , , − p + q β r (cid:21)(cid:33)(cid:33) / , (20)where Σ ≡ (cid:82) dxdy is the spatial 2-volume.5n order to study the behavior of f ( r ), we expand f ( r ) in the small- r and large- r regions respectively, yielding f ( r ) = − (cid:113) πGM/ ( α Σ ) − G (cid:112) βπ ( p + q ) / Γ(1 / / / (3 α ) r / − πGβ (cid:112) p + q + πGβ (cid:112) p + q Γ(5 / / Γ(1 / (cid:112) πGαM/ Σ − αG √ βπ ( p + q ) / Γ(1 / / / r / + r α + O ( r ) / , (21) f ( r ) = 3 − √ α Λ6 α r − πGM Σ √ α Λ r − + O ( r ) − . (22)From (21) one learns that, f ( r ) is finite when r →
0. This property is specific for BIblack holes, i.e., for Born-Infeld black holes the metric may be free from divergence atthe origin while the curvature invariants definitely diverge there. f ( r ) is finite at theorigin originates partly from the theory model of matter fields, partly from the modelof gravity theory, and partly from the dimensions of spacetime. For the black holesin Gauss-Bonnet-Maxwell gravity, f ( r ) diverges when r →
0. For the black holes inEinstein-Born-Infeld gravity, f ( r ) diverges when r → r → f ( r ) is finite in 5 dimensions while itdiverges in higher than 5 dimensions. For the black holes in 3rd order Lovelock gravitycoupled to BI electromagnetic fields, f ( r ) is finite in 7 dimensions while it diverges inhigher dimensions at the origin. r + M M Figure 1: The mass of the black hole for some parameter selections.One also learns from (21) that, in order to ensure f ( r ) to be well defined, the blackhole mass M is necessary to be larger than some critical one M c , which is given by M c = 2 √ β √ π ( p + q ) / Γ(1 / / . (23)With the black hole horizon r + , which satisfies f ( r + ) = 0, the black hole mass can be6xpressed as M = Σ πGr + (cid:18) πGβr (cid:18) βr − (cid:113) p + q + β r (cid:19) − Λ r +16 πG ( p + q ) F (cid:20) , , , − p + q β r (cid:21)(cid:19) (24)In Fig.1 we plot the black hole mass for some parameter selections. From Fig.1 one seesthat, when r + →
0, the value of M tends to be some finite and positive M . As r + increases, M first decreases and then increases, there exist a minimal value of M . Thisbehavior of M implies that, if the parameter M is taken to be larger than M , M and r + are in one-to-one correspondence, that is, the black hole possesses only one horizon. r f ( r ) M = 0.900 M = 1.026 M = 1.100 M = 1.208 = M c M = 1.300 M = 1.400 Figure 2: The metric function f ( r ) for the parameters selection G = 1 , β = 0 . , p =1 . , q = 1 . , Λ = − , α = 0 . r + → M = lim r + → M = 2 √ β √ π ( p + q ) / Γ(1 / / , (25)which is exactly the critical mass M c , i.e., M = M c . Therefore, M has to satisfies M > M . From Fig.2, one sees that for M > M c , the black hole possesses only onehorizon. The inner (Cauchy) horizon of the BI black hole turns into the curvaturesingularity due to perturbatively instability [38]. In Einstein gravity, the BI black holesmay possess more than one horizon for some parameter selections. For M < M c , theblack hole is not well defined in the whole spacetime [27]. The argument given previouslyimplies that, there exists no extremal black hole in our case. To see this is indeed thecase, we substitute r e ≡ (cid:112) πGβ (cid:18) p + q Λ − πG Λ β (cid:19) / (26)which is obtained by setting the temperature (28) to be zero, into the black hole mass(24), and replace the hypergeometric function with 1 since 0 < F (cid:104) , , , − p + q β r (cid:105) < M e ≡ − ( p + q ) / Σ (64 π G β − πG Λ β + Λ )3 (cid:112) πGβ Λ( − πGβ + Λ) , (27)which is definitely negative for Λ <
0. Therefore, extremal AdS black holes do not existfor the case we discussed.
In this section, we study thermodynamics of the black holes. First Let’s give the ther-modynamic quantities.The temperature of the black hole is T = 8 Gβ (cid:16) βr − (cid:112) p + q + β r (cid:17) − Λ r /π r + (28)With the Iyer-Wald formula, entropy of the black hole is given by S = − π (cid:73) d x √ γY µνρσ (cid:15) µν (cid:15) ρσ = − G (cid:73) d x √ γ (cid:18) − − αφ ˜ R ( γ ) + αδ ρ [ µ ∂ ν ] φ∂ σ φ(cid:15) µν (cid:15) ρσ − α ( ∂φ ) δ ρ [ µ δ σν ] (cid:15) µν (cid:15) ρσ (cid:19) , (29)where the first term in the bracket comes from Einstein gravity, the second term comesfrom αφ L GB in the action (7), while the third and fourth terms come from 4 αG µν ∂ µ φ∂ ν φ in the action (7). The second term in (29) vanishes since ˜ R ( γ ) = 0 for spatially flatblack holes. Meanwhile, straightforward calculations show that the last two terms in(29) cancel out, i.e., the 4 αG µν ∂ µ φ∂ ν φ term does not contribute to the entropy either.Therefore, in our case only the Einstein gravity part contribute to the entropy finally,which reads S = r Σ G . (30)The electric and magnetic charges are given by Q e = Σ √− h (cid:0) h − (cid:1) [ tr ] (cid:12)(cid:12) r →∞ = q Σ , Q m = Σ F xy | r →∞ = p Σ . (31)Note that above electric charge as a conserved quantity follows from the equation ofmotion (8). The electric and magnetic potentials are given byΦ e = qr + 2 F (cid:20) , , , − p + q β r (cid:21) , Φ m = pr + 2 F (cid:20) , , , − p + q β r (cid:21) , (32)8n extended phase space, the thermodynamic pressure of the system is identified as[37, 39] P = − Λ8 πG . (33)The thermodynamic volume conjugate to P is given by V = r Σ δM = T δS + V δP + Φ e δQ e + Φ m δQ m (35)is satisfied. The Smarr formula is given by M = 23 ( T S + Q e Φ e + Q m Φ m ) (36)which agrees with the generalized Smarr relation for AdS planar black holes [40].Now let’s examine if there exist thermal phase transitions of the black hole. Thecritical point is determined by the equations ∂P∂r + (cid:12)(cid:12)(cid:12)(cid:12) r + = r c ,T = T c = ∂ P∂r (cid:12)(cid:12)(cid:12)(cid:12) r + = r c ,T = T c = 0 . (37)From (28) one can solve out Λ in terms of T and substitute Λ into the definition ofpressure (33), yielding P = r + T − Gr β + 2 Gβ (cid:112) p + q + β r Gr . (38)In order for ∂P∂r + = 0 to be satisfied, one finds the temperature T = − βG ( p + q ) r − ( p + q + β r ) − / , which implies there will be no thermal phase transition since the tem-perature is negative. Substituting this T into ∂ P∂r one has ∂ P∂r = 2 β ( p + q )( p + q + 3 β r ) r ( p + q + β r ) / , (39)which can’t be zero. Thus, there exists no thermal phase transition for the black hole(20), the black hole is always thermodynamically stable. In this paper, we construct novel dyonic BI black hole solution of 4D EGB gravity whichis obtained through Kaluza-Klein process. Through the small- r expansion of f ( r ), it is9ound that the metric is devoid of divergence at the origin while the essential singularitystill exists there. This property is specific for BI black holes. This is determined bythe BI theory, the Gauss-Bonnet theory and the dimensions of spacetime together. Thesmall- r expansion of f ( r ) also implies there exist some critical mass M c . In order for theblack hole to be well defined, the black hole mass must be larger than the critical mass.Since M ≡ lim r + → M equals to M c , one finds that a well-defined black hole possessesonly one horizon considering the behavior of M as function of r + .The thermodynamic quantities of the black hole are calculated. It’s found thatthe Wald entropy of the black hole equals to the Bekenstein-Hawking entropy, sincethe Gauss-Bonnet contribution vanishes for spatially flat black holes and the term G µν ∂ µ φ∂ ν φ in the action does not contribute to the entropy. With the thermodynamicquantities, the first law is checked to be satisfied. The Smarr relation is given and it’sfound to agree with the generalized Smarr relation for AdS planar black holes. Finally,we examine whether thermal phase transitions exist or not in extended phase space.Through solving the P - V critical equations analytically, it’s found that the black holeis always thermodynamically stable, no thermodynamic phase transition exists. Acknowledgment
The work of KM and LZC is supported by the National Natural Science Foundation ofChina (No. 62005199), the Key Research and Development Plan of Shandong Province(No. 2019GGX101073), and the Natural Science Foundation of Shandong Province(Nos. ZR2020LLZ001 and ZR2019LLZ006). The work of JQZ is supported by NaturalScience Foundation of Shandong Province (No. ZR2020KF017).
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