Brown-York quasilocal energy in Lanczos-Lovelock gravity and black hole horizons
aa r X i v : . [ g r- q c ] D ec Brown-York quasilocal energy in Lanczos-Lovelock gravityand black hole horizons
Sumanta Chakraborty ∗ IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India andNaresh Dadhich † Center for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, IndiaandIUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune 411 007, India
December 7, 2015
Abstract
A standard candidate for quasilocal energy in general relativity is the Brown-York energy,which is essentially a two dimensional surface integral of the extrinsic curvature on the two-boundary of a spacelike hypersurface referenced to flat spacetime. Several years back one ofus had conjectured that the black hole horizon is defined by equipartition of gravitational andnon-gravitational energy. By employing the above definition of quasilocal Brown-York energy,we have verified the equipartition conjecture for static charged and charged axi-symmetricblack holes in general relativity. We have further generalized the Brown-York formalism toall orders in Lanczos-Lovelock theories of gravity and have verified the conjecture for pureLovelock charged black hole in all even d = 2 m + 2 dimensions, where m is the degree ofLovelock action. It turns out that the equipartition conjecture works only for pure Lovelock,and not for Einstein-Lovelock black holes. Defining quasilocal energy for general relativity is an extremely important but a long eluding prob-lem. Initial attempts in this direction involved pseudotensor methods, leading to coordinate de-pendent expressions, identification of certain symmetries and defining the Noether charge or somemathematical constructs from the Cauchy data showing similar physical properties associated withenergy. Most of these definitions are quite useful and have physical implications in one case or theother, but no unified description has emerged (for a partial set of references see [1–7]).The main reason for this ambiguity in defining an energy for gravitational field is due to itsnon-linear nature and the fact that gravitational energy is non-localizable. This in turn implies not ∗ [email protected]; [email protected] † [email protected] D >
D >
3. For D = 3, gravity is kinematic, as Riemanntensor is determined entirely by Ricci tensor. Then requiring the kinematic property of gravity tohold in all odd dimensions singles out pure Lanczos-Lovelock gravity, i.e., one particular order outof the full Lanczos-Lovelock Lagrangian [19–22]. Moreover, from thermodynamic perspectives aswell Lanczos-Lovelock gravity has unique features [23]. Most of the thermodynamic results whichhold for null surfaces in general relativity can be generalized to hold in Lanczos-Lovelock gravity aswell [24–28]. Thus even from the thermodynamic perspective, Lanczos-Lovelock gravity has a spe-cial status. As argued earlier, the quasilocal energy is an important measure of gravitational field,it would therefore be pertinent to study the Brown-York quasilocal energy for Lanczos-Lovelockgravity as well.In general, location of black hole horizon is defined as a limit for timelike world lines to exist(when 4-velocity turns null) and when a spatial surface turns one way membrane (i.e., it can becrossed in one direction only). It has been argued in [10] that at the black hole horizon timelikeparticles (that are pulled by gradient of potential produced by matter (non-gravitational) energy)tend to photons that can feel no gravitational pull but only follow curvature of space produced bygravitational field energy. The horizon should therefore be defined when their respective sourcesare equal in magnitude. That is equipartition between gravitational and non-gravitational (matter)energy. By using Brown-York energy, gravitational field energy was computed and the equipartitionwas shown to exist for static black hole horizons. This is the conjecture for location of black holehorizon which we would like to examine for stationary black holes in general relativity as well aswith the appropriate generalization of Brown-York energy for static black holes in Lanczos-Lovelockgravity. It has been previously tested for static black holes in general relativity [10]. In this workour main motivation, besides obtaining the quasilocal energy for Lanczos-Lovelock gravity, is totest the veracity of this equipartition conjecture for stationary black holes in general relativity andstatic black holes in Lanczos-Lovelock gravity.The paper is organized as follows, in Section 2 we discuss the Brown-York energy in the context2f general relativity and then we have applied it to Reissner-Nordstr¨om and Kerr-Newmann blackholes to obtain the location of the horizon through equipartition. Then in the subsequent section,i.e., in Section 3 we have discussed extensively the quasilocal energy for Lanczos-Lovelock gravityand equipartition conjecture in it. Finally we conclude with a short discussion on our results.In this paper we will work with the ( − , + , + , . . . ) signature for the spacetime metric and shallset the fundamental constants G and c to unity. We will start with a spacetime region M , which is topologically Σ times a real line interval, whereΣ is the three-space. The boundary of the three-space Σ, is denoted by B (not necessarily simplyconnected), which is two dimensional. The product of B with timelike world lines normal to Σ isdenoted by B and the end points of the timelike world lines are denoted by Σ and Σ . Hence B ,Σ and Σ form the three-boundary of the full spacetime region M .The full spacetime metric is g ab , u a is the future pointing timelike normal to the hypersurfaceΣ, and n a is the spacelike normal to the three-boundary B . The metric and extrinsic curvatureon Σ are denoted by h ab and K ab respectively, with h ab acting as the projection tensor on Σ.From hypersurface orthogonality between Σ and B it immediately follows that u a n a = 0. Similarprojection tensors and extrinsic curvatures can be defined on B as well. This can be extended tofinally introduce the extrinsic curvature k AB on the two-boundary B of Σ with the induced metric, q ab = δ ab + u a u b − n a n b . Given this spacetime foliation we can introduce the ADM decomposition[13, 29, 30] and thus the action can be written as a bulk term which includes intrinsic quantitiesdefined on Σ and a surface term on the three-boundary. On the two end points Σ and Sigma the surface term equals 2 K , where K = K ab h ab is the trace of the extrinsic curvature on Σ. On theother surface, namely B , the surface term equals the trace of the extrinsic curvature defined on B .Then under variation we will obtain the gravitational momentum conjugate to both h ab on Σ andthe respective one on B . Treating the gravitational action analogously to a matter action and fromthe Hamilton-Jacobi method the quasilocal gravitational energy contained within the two-surface B turns out to be [8] E BY = 18 π Z B d x √ q ( k − k ) (1)where q is the two-metric defined on the two-surface B and k stands for the trace of extrinsiccurvature for some reference spacetime. As the two-surface B tends to infinity, the Brown-Yorkenergy would approach the ADM mass (the positivity of Brown-York energy and its relation to hoopconjecture have been explored in [31–34]). We will be interested in asymptotically flat solutions inwhich k is the trace of extrinsic curvature of B as embedded in a flat spacetime. In this section wewill evaluate the quasilocal energy for two cases: (a) static spherically symmetric and (b) stationaryaxially symmetric spacetimes, in particular for Reissner-Nordstr¨om and Kerr-Newmann black holes.In both these cases we will explicitly see that in the asymptotic limit the Brown-York energy goes tothe ADM energy [29], which is a crucial check for the validity of any definition of energy in generalrelativity. 3 .1 Brown-York quasilocal energy in an arbitrary static spherically sym-metric spacetime We will first derive the expression for Brown-York energy for the most general static sphericallysymmetric spacetime [35–37] with the following line element ds = − f ( r ) dt + dr g ( r ) + r d Ω (2)The metric is assumed to be asymptotically flat, i.e., in the limit r → ∞ both f ( r ) and g ( r ) goto unity giving the flat Minkowskian metric. The hypersurface Σ is taken to be a t = constanthypersurface and B as a r = constant surface. The two-surface B is a r = constant hypersurfacewithin Σ. Then the unit timelike normal u a to Σ and unit spacelike normal n a to B turn out to be u a = − p f (1 , , ,
0) ; u a = 1 √ f (1 , , ,
0) (3) n a = 1 √ g (0 , , ,
0) ; n a = √ g (0 , , ,
0) (4)Having obtained the normal to Σ and B we can now compute the corresponding extrinsic curvatures,and in particular for the latter we have, k = − √ h ∂ µ (cid:16) √ hn µ (cid:17) = − ∂ r n r − n r ∂ r ln √ h = − ∂ r √ g − √ g∂ r ln (cid:18) r sin θ √ g (cid:19) = − r p g ( r ) (5)The embedding of B in a flat spacetime is trivial and the trace of extrinsic curvature k is simply − /r . Then using the expression for k from Eq. (5) the Brown-York energy as defined in Eq. (1)yields, E BY = 14 π Z dθdφ r sin θ r (cid:16) − p g ( r ) (cid:17) = r (cid:16) − p g ( r ) (cid:17) (6)This expression for Brown-York energy is completely general. Given any static, spherically symmet-ric spacetime in general relativity with line element given by Eq. (2) the above result holds. Thisalso shows that the Brown-York energy singles out the coefficient of g rr , rather than that of g tt .The implication of which is straightforward — the gravitational energy really resides in the spatialcurvature, i.e. it curves the three space. This is in complete accord with some of the earlier studiesby one of us [9].Having derived the general expression, let us now consider an application of this result. For thatwe pick up the Reissner-Nordstr¨om black hole, whose metric elements are given by f ( r ) = g ( r ) = 1 − Mr + Q r (7)The Brown-York quasilocal energy within a sphere of radius r turns out to be, E ( r ≤ r ) = r " − s − Mr + Q r (8)4here M and Q respectively stand for mass and charge of the black hole. Let us expand it for large r to write E ( r ≤ r ) ≈ M − Q r + M r (9)Clearly it goes to the desired limit, the ADM mass M, asymptotically. It is also evident from theexpression of quasi-local energy that, E is the sum of matter energy and potential energy associatedwith building a charged fluid ball by bringing together individual particles from some initial radius.It can also have the following understanding: M being the total energy including rest mass and allkinds of interaction energies. The energy lying exterior to the radius r will be Q / r , arising fromthe energy momentum tensor component T = Q / r due to electric field. The contribution ofgravitational potential energy corresponds to the second term in the approximation, i.e., − M / r .Thus the energy within radius R will correspond to M − ( Q / r − M / r ), exactly coincidingwith Eq. (9).For extremal black hole M = Q , the two energies cancel out each other exactly, not only inthe asymptotic expansion but everywhere. Thus alike the Komar mass for the Schwarzschild blackhole, the Brown-York energy is conserved, and is equal to ADM mass, for the extremal Reissner-Nordstr¨om black hole. The energy within radius r is given by the Brown-York expression aspresented in Eq. (8), while the energy outside r has two parts — (a) the energy contained withinthe gravitational field E grav and (b) the energy contained within the electric field Q / r [10]. Thisseparation of energy into two parts can be written as, E ( r ≥ r ) = E grav + Q r (10)The total energy in the full spacetime manifold corresponds to the ADM mass. This implies, E ( r ≤ r ) + E ( r ≥ r ) = M , the ADM mass. Using which the gravitational energy turns out to be E grav = M − Q r − r " − s − Mr + Q r (11)On the other hand the non-gravitational energy outside radius r arises from the energy density Q / r due to electric field. Integrating the electric field energy over the range of radial distancefrom ∞ to r , we get − Q / r . Adding M to it we obtain the non-gravitational energy to be, E non − grav = M − Q / r . Now according to the equipartition conjecture, the horizon is definedwhen energy is equally divided between the matter fields and the gravitational field, leading to, E grav + E non − grav = 0, which implies s − Mr + Q r " − s − Mr + Q r = 0 . (12)This clearly has two solutions, namely, (a) r = r + = M + p M − Q , the larger root of r − M r + Q = 0, yielding the location of the event horizon and (b) r = Q / M , the hard core radius fornaked singularity for the parameter space Q > M . In the latter case, note that the hard coreradius marks vanishing of non-gravitational (matter) energy which consequently implies vanishingof gravitational energy as well because the latter is created by the former. Then energy contained5nside the hard core radius also vanishes and the entire mass M lies outside. The hard core radiusshould rather be looked upon as the radius where non-gravitational energy goes to zero ratherthan equipartition (because it implies zero equal to zero). Thus equipartition of energy betweengravitational and non-gravitational energy characterizes horizon . Following on the Reissner-Nordstr¨om black hole, we will now compute the Brown-York energy forthe Kerr-Newman black hole and verify the veracity of the equipartition conjecture for the locationof its horizon. It is described by the metric ds = − (cid:18) ∆ − a sin θρ (cid:19) dt − a sin θ (cid:0) r + a − ∆ (cid:1) ρ dtdφ + ρ ∆ dr + ρ dθ + Σ ρ sin θdφ (13)where we have defined the following quantities,∆ = r + a + Q − M r ; ρ = r + a cos θ ; Σ = (cid:0) r + a (cid:1) − ∆ a sin θ (14)where M , Q and a have the usual meaning as mass, charge and specific angular momentum ofthe black hole. The normalized normal to r = constant surface within the t = constant surface isspacelike, i.e., n a n a = 1, which for the spacetime described by the above metric ansatz turns out tobe, n µ = r ρ ∆ , , ! ; n µ = s ∆ ρ , , ! (15)Thus the extrinsic curvature of any r = constant surface within the t = constant hypersurface canbe obtained as k = − √ h ∂ µ (cid:16) √ hn µ (cid:17) = − ∂ r n r − n r ∂ r ln √ h (16)The above expression nicely breaks into two parts, ∂ r n r and ∂ r ln √ h . Each of them can be evaluatedindividually leading to ∂ r n r = 12 s ∆ ρ (cid:18) ∂ r ∆∆ − ∂ r ρ ρ (cid:19) ; ∂ r ln √ h = ∂ r ρ ρ + ∂ r Σ2Σ − ∂ r ∆2∆ (17)This immediately leads to the following expression for extrinsic curvature k = − s ∆ ρ ∂ r Σ2Σ= − r r − Mr + a + Q r r (cid:0) r + a (cid:1) − ( r − M ) a sin θ √ r + a cos θ h ( r + a ) − a ∆ sin θ i (18)6ow the metric on the two-surface is √ q = √ Σ sin θ . Hence the unreferenced Brown-York energyfor the Kerr-Newmann black hole within a sphere of radius r turns out to be E = 18 π Z d x √ q k = 18 π Z dφdθ sin θ √ Σ k = − r r − Mr + a + Q r Z π dθ sin θ r (cid:0) r + a (cid:1) − ( r − M ) a sin θ r ( r + a cos θ ) h ( r + a ) − a ∆ sin θ i (19)which in general cannot be integrated to obtain a closed form expression. However a closed formexpression is indeed possible to obtain in the slow rotation limit i.e., a/r ≪
1. Then the aboveexpression for unreferenced Brown-York energy reduces to [12] E = − r r − Mr + a + Q r Z π dθ sin θ (cid:20) − a r (cid:26) cos θ + (cid:18) Mr − Q r (cid:19) sin θ (cid:27)(cid:21) = − r r − Mr + a + Q r (cid:20) − a r (cid:18) Mr − Q r (cid:19)(cid:21) (20)Getting the reference term is more difficult, for that we have to map the two dimensional surfaceto a flat three dimensional spacetime. If the flat three dimensional surface is described by thethree coordinates R, ϑ,
Φ, then the matching with two-surface would provide the following relations, R = R ( θ ) , ϑ = ϑ ( θ ) and Φ = φ . These relations have to be obtained through their substitution inthe flat space line element and comparing with two-surface and Kerr-Newmann metric. This leadsto the following differential equations for R and ϑ (cid:18) dRdθ (cid:19) + R (cid:18) dϑdθ (cid:19) = ρ (21) R sin ϑ = Σ ρ sin θ (22)We need to solve these two coupled differential equations to get both R and ϑ in terms of θ .Eq. (22) can be solved to get R in terms of ϑ and θ . This when substituted in Eq. (21) would yielda differential equation of ϑ . However the differential equation being complicated, in general (i.e., forarbitrary choices of the rotation parameter a ) would not posses any analytic closed form solution.Thus to get analytic expression we need to use slow rotation limit, in which the solutions for R ( θ )and ϑ ( θ ) can be obtained assin ϑ = sin θ (cid:20) a r (cid:18) Mr − Q r (cid:19) cos θ (cid:21) (23) R ( θ ) = r (cid:20) a r sin θ − a r (cid:18) Mr − Q r (cid:19) cos θ (cid:21) (24)From these two relations we could obtain an expression for the extrinsic curvature k of a two surfaceas embedded in flat space. Then this extrinsic curvature can be used to compute the reference term,which yields [12] E = − r (cid:20) a r (cid:18) Mr − Q r (cid:19)(cid:21) (25)7t is interesting to note that there also appears the contribution of rotational energy in the flat spacereferenced energy. On expansion of E in powers of 1 /r , the first order correction due to rotationalenergy is − a / r , we will comment on it later on.Using the expression for E from Eq. (20) and the reference term as in Eq. (25), the referencedBrown-York energy turns out to be [12] E ( r ≤ r ) ≡ E − E = r " − s − Mr + a + Q r + a r h (cid:18) Mr − Q r (cid:19) + (cid:18) Mr − Q r (cid:19) s − Mr + Q r i (26)which gives the quasilocal energy within a sphere of radius r . If we had taken the sphere of radius r to infinity, it would as expected go to the ADM mass M . Expanding terms within square rootin powers of 1 /r we arrive at E ( r ≤ r ) ≈ M − a + Q − M r − M ( a + Q )2 r + a r h Mr − M r − Q r + Q − M r i (27)Note that there is no contribution of rotational energy at order 1 /r . This is precisely becauserotational energy along with Kerr-Newmann spacetime, is also shared by the referenced spacetime.Thus as the reference term E is subtracted in order to get the Brown-York energy, the contributionfrom rotational part is exactly canceled. That is why the Brown-York energy for Kerr-Newmannspacetime is free of pure rotational terms and rotation only contributes through coupling with massand charge. This is also the reason behind the fact that for the case of extremal black hole (i.e., M = a + Q ) the Brown-York energy reduces to M + ( a / r ). Thus in addition to the ADMmass the rotational energy a / r also contributes to the Brown-York energy at large distance forextremal Kerr-Newmann black hole. Note that for a = 0, we get back the result for extremalReissner-Nordstr¨om black hole.Now let’s turn to our main aim of obtaining the location of horizon at the equipartition ofgravitational and non-gravitational energy. The energy outside r has the expression, E ( r ≥ r ) = E grav + ( Q / r ). Since total energy in the spacetime is the ADM mass M , we obtain the gravita-tional energy to be E grav = M − Q r − r " − s − Mr + a + Q r − a r " (cid:18) Mr − Q r (cid:19) + (cid:18) Mr − Q r (cid:19) s − Mr + Q r (28)On the other hand the non-gravitational energy is E non − grav = M − ( a + Q / r ). Requiringequipartition of the two, i.e., gravitational and non-gravitational to be equal (with proper sign to8nsure attractive nature of gravity), we finally obtain s − Mr + a + Q r "s − Mr + a + Q r − − a r s − Mr + a + Q r + a r (cid:18) Mr − Q r (cid:19) = 0 (29)This algebraic equation has two solutions: (a) r = r + = M + p M − a − Q , the larger root of r − M r + ( Q + a ) = 0 defining the horizon, and (b) r = ( a + Q ) / M , the hard core radiusfor naked singularity, a + Q > M . Thus the equipartition conjecture is also verified for theKerr-Newman black hole albeit in the slow rotation limit.Having succeeded in deriving the horizon (or hard radius) location, starting from equipartitionin the case of both charged and charged axi-symmetric configurations in general relativity it wouldbe interesting to ask, what happens in higher dimensions, still in the premise of general relativityor more importantly, what happens to this equipartition conjecture when higher order curvatureinvariants are present in the action. That is what we take up in the next section. Before addressing the Brown-York energy for Lanczos-Lovelock gravity, as a warm up as well asto show the complexities involved, we will first discuss the case of Brown-York energy in higherdimension but within the context of general relativity. The Brown-York energy for a D-dimensionalspacetime in general relativity is still given by the difference k − k but this time evaluated for a( D − M is D -dimensional,in which we have constant time hypersurfaces, namely Σ, which are now ( D − B . Hence the boundary of Σ as intersected by B forms the desired co-dimensiontwo-surface B , which is now ( D − E BY = 18 πγ Z dA D − ( k − k ) (30)where γ is a numerical factor to be fixed later. Any static spherically symmetric spacetime can bepresented by the metric Eq. (2). The extrinsic curvature of the ( D − B is k = − [( D − /r ] p g ( r ) and k = − ( D − /r . On substitution in Eq. (30) and integration over the( D − E BY = D − πγ π ( D − / Γ( D − ) r D − (cid:16) − p g ( r ) (cid:17) (31)By setting the constant γ to the value γ = D − D − ) π ( D − / E BY = r D − (cid:16) − p g ( r ) (cid:17) (33)In D -dimensions static spherically symmetric solution of general relativity corresponds to, f = g − = 1 − ( M/r D − ), which when substituted in the above expression, leads to lim r →∞ E BY = M ,i.e., the above expression agrees as required with ADM mass in the asymptotic limit.However we cannot use this for Lanczos-Lovelock gravity. This can be seen directly as in m thorder Lanczos-Lovelock gravity the static spherically symmetric black hole solution is given by f ( r ) = g ( r ) = 1 − ( γ/r ( D − (2 m +1)) /m ). In analogy with the Einstein case we write from Eq. (33) theBrown-York energy as E BY = r D − m − m (cid:20) − (1 − γr D − m − m ) / (cid:21) r →∞ = γ r goes as γ/
2. It does not agree with the ADM mass which should really be ∝ γ m [38].However it is interesting that we get almost the right result but for mismatch in dimension of mass.This is of course not the right expression for quasi-local Brown-York energy, and we need to re-derive the Brown-York energy expression which would be appropriate for the Lanczos-Lovelocktheory. That is what we will do next. The Brown-York energy for Lanczos-Lovelock gravity can be obtained in a straightforward mannerfollowing the general relativity prescription. The derivation essentially amounts to calculate thecounter term for gravitational action on a t = constant surface (e.g., Σ) and the correspondingterm for the ( D − B . Thus we can use the counter term for Lanczos-Lovelock gravity[14, 39–44] on the t = constant hypersurface Σ and then take it to the ( D − B . Thisleads to the following generalization of Brown-York energy to m th order pure Lanczos-Lovelockgravity as, E ( m ) BY = c ( m ) Z dA D − E (35)where c ( m ) is a numerical factor which can be adjusted to get E BY = M at infinity. In generalrelativity the quantity E was just ( k − k ) and c (1) = (Γ( D − )) / (2( D − π ( D − / ), however for m th order Lanczos-Lovelock gravity E turns out to be E = m !2 m +1 m − X s =0 c s Π ( s ) (36)where,Π ( s ) = δ A A ...A m − B B ...B m − R B B A A · · · R B s − B s A s − A s (cid:16) k B s +1 A s +1 − k B s +1 (0) A s +1 (cid:17) · · · (cid:16) k B m − A m − − k B m − (0) A m − (cid:17) (37)10ere k A (0) B is the extrinsic curvature of the two surface as embedded in flat spacetime. Also thearbitrary constants c s appearing in Eq. (36) has the following expression: c s = m − X q = s ( − q − s m − q ( q C s ) q !(2 m − q − q C s is the combination symbol and stands for q ! / { ( q − s )! s ! } . Before proceeding further letus pause for a while, and consider the case of static vacuum solution for m th order pure Lanczos-Lovelock gravity which is given by [38, 45–47] ds = − f ( r ) dt + dr f ( r ) + r d Ω D − ; f ( r ) = 1 − (cid:16) β/r D − (2 m +1) m (cid:17) (39)It is clear that β is a dimension full constant related to the ADM mass and D is the dimension ofthe spacetime such that D ≥ m + 2. Note that D ≥ m + 1 ensures that the Lagrangian is nota topological term and D = 2 m + 1 is excluded because it represents a solid angle deficit that canonly describe a global monopole [48] without black hole. The later computations would involve thecurvature tensor components which are R klij = 1 r (1 − f ) δ klij (40)However the ADM mass defined in the context of m th order Lanczos-Lovelock gravity turns outto be M = ˆ cβ m , where ˆ c is just a numerical constant. This numerical constant depends explicitlyon the order of Lanczos-Lovelock Lagrangian and also on the coefficient of m-th Lanczos-LovelockLagrangian, c ( m ) . The exact value of ˆ c in this and subsequent examples would not hit us, since wewill be concerned primarily with the structure of the results. Thus the acid test for our prescriptionof Brown-York energy as presented in Eq. (35) would be to see whether it matches with the ADMmass M as defined above.In order to show the validity of our result, let us first see that it includes the general relativitycase for m = 1. We evaluate Eq. (35) for m = 1, which immediately leads to s = 0. Using this inEq. (36) and Eq. (37) we readily obtain, Eq. (30), the correct quasilocal energy for general relativity.Next we take up the Gauss-Bonnet action, which corresponds to m = 2. Thus we have twopossibilities s = 1 and s = 0 respectively. For s = 0, we obtainΠ (0) = δ A A A B B B (cid:16) k B A − k B (0) A (cid:17) (cid:16) k B A − k B (0) A (cid:17) (cid:16) k B A − k B (0) A (cid:17) ∼ β r r D − / (41)while for s = 1, we get Π (1) = δ A A A B B B R B B A A (cid:16) k B A − k B (0) A (cid:17) ∼ r β r D − (42)Then the Brown-York energy at large r takes the form E BY = c (2) r D − (cid:20) c β r r D − / + c r β r D − (cid:21) = c (2) c β + c (2) c β r − D − r →∞ = ˆ cβ = M (43)11hich exactly agrees with the ADM mass. Hence for pure Gauss-Bonnet gravity, the Brown-Yorkenergy at infinity exactly matches with the ADM mass.It is now time to consider m th order pure Lanczos-Lovelock gravity in which s can take values0 , , . . . , ( m −
1) respectively. This immediately leads to the following expressionΠ (0) = δ A A ...A m − B B ...B m − (cid:16) k B A − k B (0) A (cid:17) (cid:16) k B A − k B (0) A (cid:17) · · · (cid:16) k B m − A m − − k B m − (0) A m − (cid:17) ∼ r m − (cid:16) − p f (cid:17) m − r →∞ −−−→ r m − β m − r (2 m − D − m − /m (44)Then, Π ( s ) = δ A A ...A m − B B ...B m − R B B A A · · · R B s − B s A s − A s (cid:16) k B s +1 A s +1 − k B s +1 (0) A s +1 (cid:17) · · · (cid:16) k B m − A m − − k B m − (0) A m − (cid:17) ∼ r m − (1 − f ) s (cid:16) − p f (cid:17) m − s − r →∞ −−−→ r m − β m − s − r (2 m − s − D − m − /m (45)and finally, Π ( m − = δ A A ...A m − B B ...B m − R B B A A · · · R B m − B m − A m − A m − (cid:16) k B m − A m − − k B m − (0) A m − (cid:17) ∼ r m − (1 − f ) m − (cid:16) − p f (cid:17) r →∞ −−−→ r m − β m r D − m − (46)Hence the Brown-York energy at large r turns out to have the following expressionlim r →∞ E BY = c ( m ) r D − h c (0) r m − β m − r (2 m − D − m − /m + · · · + c ( s ) r m − β m − s − r (2 m − s − D − m − /m + · · · + c ( m − r m − β m r D − m − i = ˆ cβ m + · · · + ¯ c ( s ) (cid:16) β m − s − /r ( D − m − m − s − /m (cid:17) + · · · + ¯ c (0) (cid:16) β m − /r ( D − m − m − /m (cid:17) r →∞ = M (47)Thus we have proved that the Brown-York energy as defined by Eq. (35) in the asymptotic limitleads to the ADM mass. Hence the definition of Brown-York energy works perfectly well and itpasses the acid test of matching with ADM mass asymptotically for static black hole in Lanczos-Lovelock gravity. We have thus generalized the Brown-York energy for Lanczos-Lovelock gravity. Having derived the quasilocal energy for Lanczos-Lovelock theories of gravity, we would now liketo apply the equipartition conjecture to Lanczos-Lovelock gravity and verify its veracity. For anystatic spherically symmetric spacetime given by Eq. (39) describing an m th order Lovelock staticblack hole, the Brown-York energy reads as E BY = r D − m − h ¯ c (0) (cid:16) − p f (cid:17) m − + · · · + ¯ c ( s ) (cid:16) − p f (cid:17) m − s − (1 − f ) s + · · · + ¯ c m − (1 − f ) m − (cid:16) − p f (cid:17) i (48)12his is the energy, E BY ( r ≤ r ), lying inside radius r = r . Let us consider a charged black holewith f ( r ) = 1 − (cid:18) β m r D − m − − ¯ Q r D − m − (cid:19) /m (49)where β m = 2 M/ ¯ c and ¯ c = ¯ c (0) + · · · + ¯ c ( s ) + · · · + ¯ c ( m ) . For Maxwell field the energy outside r = r is as before E ( r ≥ r ) = E grav + ( Q / r D − ) while non-gravitational component is E non − grav = M − ( Q / r D − ). Requiring the total energy in spacetime to be equal to the ADM mass, we readilyobtain the gravitational contribution to be, E grav = M − Q r D − − r D − m − " ¯ c (0) (cid:16) − p f (cid:17) m − + · · · + ¯ c ( s ) (cid:16) − p f (cid:17) m − s − (1 − f ) s + · · · + ¯ c m − (1 − f ) m − (cid:16) − p f (cid:17) (50)where f = f ( r = r ). Now the equipartition of gravitational and non-gravitational energy leads tothe following algebraic equation1 − f = " ¯ c (0) (cid:0) − √ f (cid:1) m − + · · · + ¯ c ( s ) (cid:0) − √ f (cid:1) m − s − (1 − f ) s + · · · + ¯ c m − (1 − f ) m − (cid:0) − √ f (cid:1) ¯ c (0) + · · · + ¯ c ( s ) + · · · + ¯ c ( m ) /m (51)Note that the above equation has various powers of f upto order m, which is also the order ofpure Lanczos-Lovelock Lagrangian. In general this is a complicated algebraic equation to solvefor f . However we will avoid this difficulty by performing the following trick: we will substitute f = 0 and f = 1 in the above equation and see whether it is satisfied. As before it turns outthat the above two indeed satisfy Eq. (51). These two conditions have the two familiar solutions,one defining the horizon r + , the larger root of r D − m − − β m r D − + ¯ Q = 0 and the other, thehard core radius r = ( ¯ Q /β m ) /D − for naked singularity, with M < a + Q . This is for the pureLovelock analogue of Reissner-Nordstr¨om black hole which includes for Q = 0 the pure Lovelockanalogue of the Schwarzschild black hole. Thus we verify that the equipartition conjecture continuesto hold good for pure Lanczos-Lovelock static black holes.The important point to note is that the conjecture turns out to hold good only for pure Lovelock(for a fixed m ) black holes but not for Einstein-Lovelock (with sum over m ) black holes. This iswhat we show next for the case of Einstein-Gauss-Bonnet black hole. In the previous section we have concentrated on the pure Lovelock theories and the validity ofequipartition conjecture. In this section we will illustrate that Equipartition conjecture does nothold for the general Lovelock theories. For that we will use an action, which is sum of the Einstein-Hilbert and Gauss-Bonnet terms. The static black hole solution corresponding to the Einstein-13auss-Bonnet action, is known as the Boulware-Deser solution [49, 50], it reads as follows: ds = − f ( r ) dt + dr f ( r ) + r d Ω ; f ( r ) = 1 − r α " − ± r αMr (52)Here α is the Gauss-Bonnet coupling constant and M is the mass term, and both of them are ofdimension L . There are two branches of the solution, the one with +ve sign has the right Einsteinlimit with attractive gravity while the other (i.e., the one with − ve sign) is repulsive. We wouldtherefore choose the former. At r = 0 we have a curvature singularity, which is cloaked by an eventhorizon for the +ve branch for M ≥ α and is located at, r = r h = M − α . Otherwise the abovesolution would represent a naked singularity. Let us now compute the Brown-York energy for theEinstein-Gauss-Bonnet solution. For which the quasi-local energy for D = 5 involves both m = 1,the Einstein-Hilbert term and the m = 2, the Gauss-Bonnet term. For this, from Eq. (33) andEq. (48) we write E ( r ≤ r ) = r (cid:16) − p f (cid:17) + α (cid:20) c (cid:16) − p f (cid:17) + c (1 − f ) (cid:16) − p f (cid:17)(cid:21) (53)where c and c are two constants whose values can be obtained from Eq. (38) and f = f ( r = r ).As an illustration we can consider the asymptotic limit of the Brown-York energy defined in Eq. (53).For which the first term yields r × ( M/r ), while the second term leads to, M /r + M /r . Thusin the asymptotic limit, i.e., r → ∞ Brown-York energy exactly equals the ADM mass M .Given the Brown-York energy, the gravitational energy is just the difference, M − E ( r ≤ r ),and non-gravitational energy is anyway the ADM mass M . Then the equipartition demands2 M = r (cid:16) − p f (cid:17) + α (cid:20) c (cid:16) − p f (cid:17) + c (1 − f ) (cid:16) − p f (cid:17)(cid:21) (54)Clearly this does not define horizon for the Einstein-Gauss-Bonnet black hole. For horizon f ( r ) = 0which gives 2 M = r h + α ( c + c ). This cannot be satisfied because c , c are numerical factors.The equipartition conjecture therefore does not work for Einstein-Gauss-Bonnet and in general forEinstein-Lovelock black holes. It works only for pure Lovelock black holes.This explicitly shows the special status of pure Lanczos-Lovelock gravity as also exposed in[19–21]. In this case as well only for pure Lovelock black holes, the equipartition conjecture holdsgood and defines the horizon. Thus equipartition conjecture discerns the pure Lanczos-Lovelockgravity from the general Einstein-Lanczos-Lovelock gravity. One of the promising candidates for obtaining quasilocal energy in general relativity is the Brown-York energy. This definition is based on the Hamilton-Jacobi treatment of the gravitational actionwritten in terms of ADM variables and then identifying the correct expression for energy. Thesalient feature of this prescription is that, at asymptotic infinity it correctly reproduces the ADMmass, which acts as an acid test for any definition of energy in general relativity. In this workwe have generalized the Brown-York energy for Lanczos-Lovelock gravity and have employed it toverify the veracity of the equipartition conjecture for defining black hole horizon. That is horizonmarks equality of gravitational and non-gravitational energy.14t is envisioned that when a configuration is infinitely dispersed, it has energy equal to ADMmass M as it begins collapsing under its own gravity, it picks up gravitational energy which isnegative, and electric field and rotational energy for a charged and rotating black hole. Thus atany finite r , there are gravitational and non-gravitational components of energy. The Brown-Yorkenergy gives energy contained inside a given radius, from which if we subtract non-gravitationalpart, we can compute gravitational energy. This could be done for static black holes and foraxially symmetric Kerr-Newmann black hole it can be evaluated in the slow rotation limit. Bymeans of the Brown-York energy expression, we can tame the notorious gravitational field energyto obtain a quantitative expression. The interesting application of which was made by one of us inproposing the equipartition conjecture [10] for characterization of black hole horizon by equality ofgravitational and non-gravitational energy. It is motivated by the fact that as horizon is approachedtimelike particles tend to null particles. Motion of the former is governed by ∇ Φ produced by non-gravitational energy while that of the latter by spatial curvature caused by gravitational energy [10].Thus as the former approaches the latter at the horizon so should be their sources. Thus gravitationaland non-gravitational energy must be equal at the horizon.We have verified the equipartition conjecture for static and axially symmetric (in the slow rota-tion limit) black holes, in particular Kerr-Newmann black hole that includes static black holes. It isremarkable that for extremal Reissner-Nordstr¨om black hole gravitational and electric field energyexactly cancel out each-other everywhere so that the Brown-York energy is conserved mass M .It turns out that the definition of Brown-York energy cannot straightway be taken over toLanczos-Lovelock gravity but it needs to be supplemented with the counter terms. With this mod-ification, we generalize the Brown-York energy expression for Lanczos-Lovelock gravity and againseparate out gravitational and non-gravitational parts. We use that to establish the equipartitionconjecture for static pure Lovelock black holes. It is interesting that the conjecture holds good onlyfor pure (for a fixed m ) Lovelock but not for general (sum over m ) Einstein-Lovelock black holes.Like some other features [19–21], the equipartition of gravitational and non-gravitational energydefining the black hole horizon is also yet another discriminator of pure Lovelock gravity. Acknowledgements
Research of S.C. is funded by a SPM fellowship from CSIR, Government of India. A part of thework was done while ND was visiting Albert Einstein Institute, Golm and University ofKwaZulu-Natal, Durban, and for that he thanks respectively Hermann Nicolai and Sunil Maharaj.
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