Generalized Vaidya Spacetime in Lovelock Gravity and Thermodynamics on Apparent Horizon
aa r X i v : . [ h e p - t h ] D ec Generalized Vaidya Spacetime in Lovelock Gravity andThermodynamics on Apparent Horizon
Rong-Gen Cai a, ∗ , Li-Ming Cao b, † , Ya-Peng Hu a,c, ‡ , Sang Pyo Kim d,b § a Institute of Theoretical Physics, Chinese Academy of Sciences,P.O. Box 2735, Beijing 100190, China b Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea c Graduate School of the Chinese Academy of Sciences, Beijing 100039, China and d Department of Physics, Kunsan National University, Kunsan 573-701, Korea
We present a kind of generalized Vaidya solutions in a generic Lovelock gravity.This solution generalizes the simple case in Gauss-Bonnet gravity reported recentlyby some authors. We study the thermodynamics of apparent horizon in this gen-eralized Vaidya spacetime. Treating those terms except for the Einstein tensor asan effective energy-momentum tensor in the gravitational field equations, and usingthe unified first law in Einstein gravity theory, we obtain an entropy expression forthe apparent horizon. We also obtain an energy expression of this spacetime, whichcoincides with the generalized Misner-Sharp energy proposed by Maeda and Nozawain Lovelock gravity. ∗ e-mail address: [email protected] † e-mail address: [email protected] ‡ e-mail address: [email protected] § e-mail address: [email protected] I. INTRODUCTION
The Lovelock gravity [1] is a natural generalization of general relativity (Einstein gravitytheory) in higher dimensional spacetimes. Its action is a sum of some dimensionally extendedEuler densities, where there are no more than second order derivatives with respect to metricin equations of motion. Also it is known that the Lovelock gravity is free of ghost. Over thepast years, due to the development of some theories in higher dimensional ( >
4) spacetimessuch as string theories, brane world scenarios etc, the Lovelock gravity has attracted a lotof attention, and some new features and properties have been revealed.The action of Lovelock gravity can be written as L = p X i =0 c i L i , (1.1)where p ≤ [( n − / N ] denotes the integer part of the number N ), c i are arbitraryconstants with dimension of [Length] i − , n is the spacetime dimension and L i are the Eulerdensities L i = 12 i √− gδ a ··· a i b ··· b i c ··· c i d ··· d i R c d a b · · · R c i d i a i b i , (1.2)where the generalized delta function is totally antisymmetric in both sets of indices. Fromthe Lagrangian (1.1), one can get the equations of motion, G ab = 0, where G ab = p X i =0 i +1 c i δ aa ··· a i b ··· b i bc ··· c i d ··· d i R c d a b · · · R c i d i a i b i . (1.3)After introducing matter into the theory, we can get the equations of motion including theenergy-momentum tensor of matter G ab = 8 πGT ab . (1.4)From (1.3), it can be shown that the equations of motion do not have more than secondorder derivatives with respect to metric and that the Lovelock theory has the same degreesof freedom as the ordinary Einstein gravity theory. Just because of this fact, the Lovelockgravity is free of ghost when expanded on a flat spacetime, avoiding any problem withunitarity [2]. Note that, L denotes the unity, and L gives us the usual curvature scalarterm, while L is just the Gauss-Bonnet term. Usually in order for the Einstein gravity tobe recovered in the low energy limit, the constant c should be identified as the cosmologicalconstant up to a constant and c should be positive (for simplicity one may take c = 1).For example, in n = 4, we have12 c g ab + c ( R ab − Rg ab ) = 8 πGT ab . (1.5)This is just the Einstein equation with the cosmological constant Λ = − c / c tobe one. When n = 5 and p = 2, after expanding the Kroneker-Delta, one gets the equationsof motion for Gauss-Bonnet gravity in the usual manner.In the literature on the Lovelock gravity, the most extensively studied theory is the so-called Einstein-Gauss-Bonnet (EGB) gravity. The EGB gravity is a special case of Lovelock’stheory of gravitation, whose Lagrangian just contains the first three terms in (1.1). TheGauss-Bonnet term naturally appears in the low energy effective action of heterotic stingtheory [3]. Spherically symmetric black hole solutions in the Gauss-Bonnet gravity have beenfound and discussed in [4, 5, 6], and topological nontrivial black holes have been studied in[7]. Rotating Gauss-Bonnet black holes have been discussed in [8]. Some other extensionssuch as including the perturbative AdS black hole solutions in gravity theories with secondorder curvature corrections could be seen in [9]. In addition, the references in [10] haveinvestigated the holographic properties associated with the Gauss-Bonnet theory. And thepapers in [11, 12, 13, 14] gave some exact solutions for Vaidya-like solution in the Einstein-Gauss-Bonnet gravity.For a generic case, although the Lagrangian (1.1) looks complicated, some exact blackhole solutions have been found and their associated thermodynamics was investigated in [15,16, 17, 18, 19, 20]. In the so-called third-order Lovelock gravity, that is, containing the firstfour terms in (1.1), some exact solutions have been found in [21]. Furthermore, it is alsoknown that those higher derivative terms in the Lagrangian (1.1) with positive coefficientsarise as higher order corrections in superstring theories, and their cosmological implicationhas been studied [22].In this paper, we are mainly interested in dynamical black hole solutions in Lovelockgravity by generalizing those discussions in [11, 12, 13, 14]. The organization of the paperis as follows. In Sec. II, we present a kind of generalized Vaidya spacetime in the generalLovelock gravity (1.1). In Sec. III, we study thermodynamics of apparent horizon of thegeneralized Vaidya spacetime, and give corresponding entropy expression associated withthe apparent horizon. In addition, we propose a way to obtain the generalized Misner-Sharpenergy in the Lovelock gravity. Section. IV. is devoted to conclusion and discussion. II. GENERALIZED VAIDYA SOLUTION IN LOVELOCK THEORY
In the four-dimensional general relativity, the Vaidya spacetime is a typical dynamicalone. In this spacetime there exists a pure radiation matter. The metric of the spacetimecan be written as ds = − f ( r, v ) dv + 2 dvdr + r d Ω . (2.1)where f = 1 − m ( v ) /r , and d Ω is the line element of a two-dimensional unit sphere. In thisspacetime, the apparent horizon is given by f = 0, or r = 2 m ( v ). The energy-momentumtensor for the radiation matter in the spacetime is given by T ab = µl a l b , where l a = (1 , , , v, r, θ, φ ). The quantity µ is the energy density of the radiation matter. Now,in an n -dimensional spacetime, assume a similar form of spherically symmetric metric ds = − f ( r, v ) dv + 2 dvdr + r d Ω n − . (2.2)The energy-momentum tensor of radiation matter has a similar form as that in four dimen-sions. For this metric, it is not hard to calculate the nonvanishing components of Riemanntensor given by R vrvr = − f ′′ , R vivj = − f ′ r δ ij , R rirj = − f ′ r δ ij ,R rivj = − ˙ f r δ ij , R virj = 0 , R ij kl = 1 − fr δ ijkl . (2.3)Here a prime/overdot denotes the derivative with respect to r/v . Substituting these resultsinto the equations of motion (1.4), and using identities δ a ··· a m b ··· b m δ b m a m = [ n − ( m − δ a ··· a m − b ··· b m − and δ a ··· a m − a m b ··· b m − b m δ b m − b m a m − a m = 2[ n − ( m − n − ( m − δ a ··· a m − b ··· b m − , we find the equations of motion G vv = p X i c i ( n − n − i − " i (cid:18) − f ′ r (cid:19) (cid:18) − fr (cid:19) i − + 12 ( n − i − (cid:18) − fr (cid:19) i = 0 , (2.4) G rv = p X i c i ( n − n − i − " i − ˙ f r ! (cid:18) − fr (cid:19) i − = 8 πGµ , (2.5)and G jk = 0 = δ jk p X i c i ( ( n − n − i − (cid:18) − fr (cid:19) i + 4 i ( n − n − i − (cid:18) − f ′ r (cid:19) (cid:18) − fr (cid:19) i − + 2 i ( n − n − i − (cid:18) − f ′′ (cid:19) (cid:18) − fr (cid:19) i − + 4 i ( i −
1) ( n − n − i − (cid:18) − f ′ r (cid:19) (cid:18) − fr (cid:19) i − ) . (2.6)Other components of G ab are given by G rr = G vv , G vr = 0. The components G ij are notindependent, because they are linearly expressed in terms of ∂ r G vv and G vv , as will be shownbelow. Defining a new function F ( v, r ) F ( v, r ) = 1 − f ( v, r ) r , (2.7)we can put the equations (2.4) and (2.6) into the forms p X i c i ( n − n − i − r n − (cid:2) r n − F i (cid:3) ′ = 0 , (2.8)and G jk = δ jk p X i c i ( n − n − i − r n − (cid:2) r n − F i (cid:3) ′′ = 0 . (2.9)¿From these two equations, it is easy to show G ij = δ ij [ r∂ r G vv / ( n −
2) + G vv ], so G ij = 0 do notyield independent equations. Integrating the equation (2.8) leads to an order- p algebraicequation for F ( v, r ) or f ( v, r ) p X i c i ( n − n − i − F i = 16 πGm ( v )Ω n − r n − , (2.10)where m ( v ) is an arbitrary function of v , which appears as an integration constant. (Cer-tainly, to ensure some energy condition, this mass function should be positive.) The coeffi-cient 16 πG/ Ω n − is chosen such that m can be interpreted as the mass of the solution when m is a constant. Using ∂∂v F i = iF i − ˙ F = i − ˙ fr ! (cid:18) − fr (cid:19) i − , (2.11)in the equation (2.5), we have8 πGµ = p X i c i ( n − r n − i − ∂∂v F i . (2.12)Comparing equations (2.10) and (2.12), we obtain µ = ˙ m ( v )Ω n − r n − . (2.13)Thus, we obtain a kind of radiating Vaidya spacetime in a generic Lovelock gravity by solvingequation (2.10). When the dimension of spacetime is five, that is, n = 5, the solution reducesto the one reported by some authors in references [11, 12, 13, 14], while in n = 4, the solutionis just the familiar Vaidya solution of general relativity.Now we further generalize the Vaidya spacetime in Lovelock gravity to more general case.Note that for the metric (2.2) we have G rr = G vv , so the energy-momentum tensor of matterhas to satisfy T rr = T vv . Certainly, the matter of pure radiation discussed above satisfies theconstraint. In fact, they are T rr = T vv = 0. If we further assume the spherical part of theenergy-momentum tensor has the form T ii = σT rr = σT vv (where σ is a constant, and therepeat index i does not sum), then from the equation ∇ a T ab = 0 or the explicit expressionsof G ab in equations (2.4), (2.5) and (2.6), we can find ∂ v T vv + ∂ r T rv + n − r T rv = 0 , (2.14)and ∂ r T rr + ( n − − σ ) r T rr = 0 . (2.15)As a result, for the pure radiation matter with T rr = T vv = 0, one can find that T rv has to beproportional to 1 /r n − . This is consistent with the equation (2.13). Next, in the case with T rr = T vv = 0, the equation (2.15) tells us that T rr and T vv have the form T rr = T vv = C ( v ) r − ( n − − σ ) , (2.16)where C ( v ) is a function of v . The off-diagonal part of the energy-momentum tensor T ab ,i.e., the component T rv has to satisfy the equation (2.14). Now the equations of motion G vv = G rr = 8 πG C ( v ) r − ( n − − σ ) modify the equation (2.8) to p X i c i ( n − n − i − (cid:2) r n − F i (cid:3) ′ = 8 πG C ( v ) r ( n − σ . (2.17)Integrating this equation, we have p X i c i ( n − n − i − F i = 16 πG (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19) , (2.18)where m ( v ) is an arbitrary function, appearing as an integration constant again. Here,Θ( r ) = R drr ( n − σ , which is Θ( r ) = ln( r ) , (2.19)for σ = − / ( n −
2) and Θ( r ) = r ( n − σ +1 ( n − σ + 1 , (2.20)otherwise. Certainly, to ensure some energy condition for the energy-momentum tensor, theparameter σ and function m ( v ) and C ( v ) should satisfy certain consistency relations. Forexample, C ( v ) ≤ − ≤ σ ≤
0. These relations have been discussed in [13]. One cansee from the relation (2.11) that T rv is given by the partial derivative of the equation (2.18)with respect to v . Thus we have T rv = ˜ µ = ˙ m ( v )Ω n − r n − + ˙ C ( v )Θ( r ) r n − . (2.21)This is consistent with the equation (2.14). So the energy-momentum tensor of matter inthis case can be written as T ab = ˜ µl a l b − P ( l a n b + n a l b ) + σP q ab , (2.22)where n a is a null vector which satisfies l a n a = −
1. In coordinates { v, r, · · · } , we have l a = (1 , , , · · · ), and n a = ( f / , − , , · · · ). The tensor q ab is a projection operator which isgiven by q ab = g ab + l a n b + l b n a . So the metric (2.2) can be put into the form g ab = h ab + q ab ,where h ab = − l a n b − l b n a (2.23)is the metric of the two-dimensional spacetime transverse to the ( n − { v, r, · · · } , the line element of h ab can be expressed as − f ( v, r ) dv +2 dvdr . The quantity P is the radial pressure with the form P = −C ( v ) r − ( n − − σ ) .Generally, it is not easy to solve equation (2.18) when p is greater than one. However,for some special cases, one can explicitly get analytic solutions. Now, we give some simpleexamples.(1). Gauss-Bonnet theory: The simple radiating Vaidya solution in the Gauss-Bonnetgravity without a cosmological constant has been found in the references [11, 12, 13, 14].Here we can directly solve the second order algebraic equation (2.18), and give the solution ds GB = − f ( v, r ) dυ + 2 dvdr + r d Ω n − , (2.24)where f ( v, r ) satisfies the equation (2.18), which in this case becomes F + α ( n − n − F = 16 πGn − (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19) . (2.25)Here, we have set c = 0, c = 1 and c = α . This parameter α , called the Gauss-Bonnetcoefficient, has the dimension of length squared. Since it is an algebraic equation of ordertwo, the solutions have two branches in general. Only one branch has the general relativitylimit and is given by f ( v, r ) = 1 + r n − n − α " − s π ( n − n − αn − (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19) , (2.26)where, for simplicity, we have set G = 1. From f ( v, r ) = 0, we can obtain the trappinghorizon, which is also its apparent horizon in this case. The radius of apparent horizonsatisfies r n − A + α ( n − n − r n − A = 16 πn − (cid:18) m ( v )Ω n − + C ( v )Θ( r A ) (cid:19) . (2.27)Thus, in general, the radius of the apparent horizon is a function of v . This solution is thesame as the one in [13] if a cosmological constant is included.(2). Dimensionally continued Lovelock gravity: An interesting case is to choose somespecial values for the coefficients c i , i = 0 , · · · , p . In [15, 16, 17, 18] a set of special coefficientshas been chosen so that the equation (2.18) has a simple expression. In odd dimensions theaction is the Chern-Simons form for the AdS group, while in even dimensions it is calledBorn-Infeld theories constructed with the Lorentz part of the AdS curvature tensor. In theodd-dimensional case, say n = 2 p + 1, i.e., Chern-Simons theory, we can choose c i = ( n − i − (cid:18) pi (cid:19) ℓ − n +2 i , (2.28)where the parameter ℓ is a length scale. Note that c i differ from the coefficients α i in thereferences [15, 16, 17, 18] only by a factor of ( n − i )!. At the same time we choose 1 / πG as Ω n − πG = ℓ ( n − . (2.29)Then the equation (2.18) gives the solution f ( v, r ) = 1 − [ m ( v ) + Ω n − C ( v )Θ( r )] p + r ℓ , (2.30)For the even-dimensional case, say n = 2 p + 2, we set c i , i = 1 , · · · , p , to be c i = ( n − i − (cid:18) pi (cid:19) ℓ − n +2 i , (2.31)and the gravity coupling constant Ω n − πG = ℓ ( n − , (2.32)Then the equation (2.18) gives the solution f ( v, r ) = 1 − (cid:20) m ( v ) + Ω n − C ( v )Θ( r ) r (cid:21) p + r ℓ . (2.33)The solutions (2.30) and (2.33) reduce to the static cases if m ( v ) and C ( v ) do not depend oncoordinate v . In these cases, the solutions have a unique AdS vacuum [15, 16, 17, 18]. Thesesolutions (2.30) and (2.33) with vanishing C ( v ) have already been found by M. Nozawa andH. Maeda in [45].(3). Pure Lovelock gravity: In this theory [23, 24, 25], only two of coefficients c i arenon-vanishing: one is c and the other is c k with 1 ≤ k ≤ p . We can normalize them as c / ( n − n −
2) = − /ℓ and c k ( n − / ( n − k − α k − , where ℓ and α are twolength scales. Then the equation (2.18) becomes α k − F k = 1 ℓ + 16 πGn − (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19) . (2.34)Thus, in the case of even dimensions, if the right hand of (2.34) is non-negative, we have f ( v, r ) = 1 ± r α (cid:20) α ℓ + 16 πGα n − (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19)(cid:21) k , (2.35)while, in the other case of odd dimensions, f ( v, r ) = 1 + r α (cid:20) α ℓ + 16 πGα n − (cid:18) m ( v )Ω n − r n − + C ( v )Θ( r ) r n − (cid:19)(cid:21) k . (2.36)Thus, for k = 1, one gets a generalized Vaidya black hole with a cosmological constant.When m ( v ) and C ( v ) are two constants, these solutions reduce to the static solutions.For dynamical black holes, it is difficult to study their thermodynamical properties. Infour-dimensional Einstein gravity, Hayward has proposed a method to discuss this issue [30,31, 32, 33]. In the next section, we will discuss the thermodynamics of the apparent horizonof the new solutions in this section.0 III. ENTROPY AND ENERGY OF THE APPARENT HORIZON
In the early 1970s, it was found that four laws of black hole mechanics in general relativityare very analogous to four laws of thermodynamics. Due to Hawking’s discovery that blackhole radiates thermal radiation, it turns out that it is not just an analog, but they areindeed identical with each other. Then thermodynamics of black hole has been establishedsoundly (for a review, see [26]). In black hole thermodynamics, the temperature and entropyof a black hole are given by T EH = κ/ π and S EH = A/
4, where κ and A are the surfacegravity and area of event horizon of the black hole, respectively. In higher derivative theoryof gravity, such as Gauss-Bonnet gravity, it turns out that the area formula for black holeentropy no longer holds. In fact, black hole entropy gets corrections from these higherderivative terms. For a diffeomorphism invariant theory, Wald [27] showed that the entropyof black hole is a kind of Noether charge, and obtained a formula for black hole entropy,now called Wald formula, associated with the event horizon of black hole. For more details,see the review paper by Wald [26].In the literature, most of discussions on thermodynamics of black holes have been focusedon stationary black holes. Thermodynamics of a black hole is associated with the eventhorizon of the black hole, which is the boundary of the past of future infinity. Therefore theevent horizon is a null hypersurface and depends on some global structures of the spacetime,so it is difficult to study the event horizon of a dynamical (time-dependent) spacetime. Ingeneral relativity, there exists another kind of horizon, named apparent horizon, which is notheavily dependent of global structure of spacetime. In the standard definition of apparenthorizon, one first slices the spacetime, and then finds the boundary of trapped region ineach slice. This boundary (two-dimensional surface) is called the apparent horizon [28].Also the three-dimensional hypersurface of the union of all these two-dimensional surfaces isusually called the apparent horizon. Over the past years, several generalizations of horizonhave been proposed, such as the trapping horizon by Hayward, the isolated horizon and thedynamical horizon by Ashtekar et al . The relations and differences among those horizonshave been discussed in a recent review [29]. For a dynamical black hole, the outer trappinghorizon and the dynamical horizon are not null hypersurfaces but spacelike hypersurfaces ofthe spacetime [29]. Therefore, some well-known results associated with the event horizon,such as Wald entropy formula, may not be applicable to those horizons.1In this section, we will discuss the entropy and the energy associated with the apparenthorizon of the generalized Vaidya solution in Lovelock gravity theory (2.18). Before goingon, let us first give a brief review on the work of Hayward [30, 31, 32, 33]. Although his workfocuses on four-dimensional Einstein gravity, it can be straightforwardly generalized to higherdimensional Einstein gravity [34, 35]. These discussions reveal the deep relation betweenequations of motion and thermodynamics of the spacetimes. For relevant discussions, seealso [36, 37, 38, 39, 40, 41]. For an n -dimensional spherically symmetric spacetime ( M , g ab ),we can write the metric in the double null form ds = h ab dx a dx b + r ( x ) d Ω n − , (3.1)where { x a } are coordinates of the two-dimensional spacetime ( M, h ab ) which is transverse tothe ( n − r ( x ) is the radius of the sphere. Similarly, one can also dividethe energy-momentum tensor of matter into two parts. One part denoted by T ab (do notbe confused with the total energy-momentum tensor) corresponds to the two-dimensionalspacetime h ab . From this energy-momentum tensor, one can define two important physicalquantities: the work density W = − / h ab T ab , which corresponds to the work term in thefirst law, and the energy supply Ψ a = T ba ∂ b r + W ∂ a r . By using these two quantities and theMisner-Sharp energy inside the sphere with radius r , E = ( n − π Ω n − r n − (cid:0) − h ab ∂ a r∂ b r (cid:1) , (3.2)one can put some components of the Einstein equations into the so-called unified first law dE = A Ψ +
W dV , (3.3)where A = Ω n − r n − and V = Ω n − r n − / ( n −
1) are the area and the volume of an ( n − r . For a vector ξ tangent to the trapping horizon of the spacetime,Hayward showed that on trapping horizon, one has [32] A Ψ a ξ a = κ π L ξ A = κ π L ξ S = T δS , (3.4)where S = A/ T = κ/ π , and κ is the surface gravity defined by κ = D a D a r/
2. Here, D a is the covariant derivative associated with metric h ab . The δ operator in the right hand sideof the equation (3.4) should be understood as follows: take a Lie derivative with respectto ξ , and then evaluate it on the apparent horizon. Note that the left hand side of the2equation (3.4) represents the amount of energy crossing the trapping horizon. Therefore theequation (3.4) may be understood as the Clausius relation for mechanics of dynamic blackholes: δQ = T δS with δQ = A Ψ a ξ a . By projecting the unified first law onto the trappinghorizon, the first law of thermodynamics is given by δE = T δS + W δV . (3.5)In the generalized Vaidya spacetime found in the previous section, the apparent horizonis just a kind of trapping horizon, so we do not emphasize the difference between these twoconcepts in the present paper. For the generalized Vaidya spacetime (2.2), the apparenthorizon is given by f ( v, r ) = 0. From the definition of surface gravity, it is easy to find κ = f ′ ( v, r A ) /
2. We now discuss thermodynamics on apparent horizon/trapping horizonfor the dynamical solutions in Lovelock gravity theory. Following [35], we may move allterms of G ab except for the Einstein tensor into the right hand side of the field equations G ab = − πT ab and rewrite them in the standard form for Einstein gravity G ab = 8 πT ab = 8 π (cid:16) T ( m ) ab + T ( e ) ab (cid:17) . (3.6)Here, T ( m ) ab is the energy-momentum tensor of matter (2.22). The effective energy-momentumtensor T ( e ) ab has the form 8 πT ( e ) ab = H ab = G ab − G ab , (3.7)where G ab is the Einstein tensor of the spacetime. In this way, we can go along the lineof Einstein gravity, although we are discussing a gravity theory beyond the Einstein grav-ity. Thus, the work term and the energy supply defined on the two-dimensional spacetime( M , h ab ) are, respectively, W = − h ab T ab = W ( m ) + W ( e ) , Ψ a = T ba ∂ b r + W ∂ a r = Ψ ( m ) a + Ψ ( e ) a , (3.8)where W ( m ) and Ψ ( m ) a are defined by using T ( m ) ab , while W ( e ) and Ψ ( e ) a by T ( e ) ab , and thecontraction is taken over the two-dimensional spacetime h ab . It is easy to find in our case, W ( m ) = − P = C ( v ) r − ( n − − σ ) , (3.9)so the energy supply for the matter is given byΨ ( m ) a = ˜ µl a . (3.10)3Now, because we are treating an effective Einstein gravity (i.e. view the non-Einstein term H ab as an energy-momentum tensor of higher curvature terms), the Misner-Sharp energy(3.2) is applicable. For the generalized Vaidya spacetime, we can replace the term h ab ∂ a r∂ b r in equation (3.2) by f ( v, r ). It is easy to see that equations (3.8) and (3.2) satisfy the unifiedfirst law (3.3).On the apparent horizon, we have the Clausius-like equation A Ψ a ξ a = A Ψ ( m ) a ξ a + A Ψ ( e ) a ξ a = κ π δA . (3.11)where ξ is a vector field tangent to the apparent horizon. This vector can be determinedas follows. Since f = 0 on the ( n − f with respect to ξ should vanish, so L ξ f = 0 on this surface. This means that on theapparent horizon we have ξ v ∂ v f + ξ r ∂ r f = 0 . (3.12)Noting that κ = f ′ ( v, r A ) /
2, we have ξ r /ξ v = − ˙ f / κ . In the case of ˙ f = ∂ v f = 0, we have ξ r = 0. Therefore on the apparent horizon, it is reasonable to set ξ a to be ξ a = ˙ f κ ! l a + n a . Here, the normalization of ξ is not important and will not play a key role in the followingdiscussion. It is easy to find ξ a ξ a = − ˙ f /κ . So the generator or the tangent vector of theapparent horizon is not a null vector unless ˙ f = 0, i.e., the static case. On the other hand,for the static cases, one has ξ a = n a = ( ∂/∂v ) a on the apparent horizon, as expected.Note that the heat flow δQ is determined by the pure matter energy-momentum tensor orpure matter energy-supply. This point has been emphasized in [35]. Thus, on the apparenthorizon we can rewrite the equation (3.11) as δQ ≡ Aξ a Ψ ( m ) a = κ π δA − Aξ a Ψ ( e ) a . (3.13)An interesting question is whether the right hand side of the equation (3.13 ) can be cast intoa form T δS of the Clausius relation as in the Einstein gravity. The answer is affirmative,since the right hand side of the (3.13) indeed can be written in a form of
T δS , where S will be given below. One can get this entropy expression by directly substituting theexpression of H ab in equation (3.7) into the equation (3.13). In fact, the authors in [35]4obtain an entropy expression of apparent horizon in Lovelock gravity, but in a setting ofFRW universe. Here let us stress that it is not always possible to rewrite the right hand sideof the equation (3.13) in a form T δS , scalar-tensor theory and f ( R ) gravitational theorybeing the counterexamples [35].In order to find the entropy expression on the apparent horizon, here we use a littledifferent method from the one in [35]. By using the explicit form of Ψ ( m ) a in equation (3.10),we have δQ ≡ A Ψ ( m ) a ξ a = A ˜ µl a ξ a = − A ˜ µ . (3.14)¿From equations (2.11), (2.18) and (2.21), it is not hard to find − A ˜ µ ˙ f = A π X i c i i ( n − n − i − r − i . (3.15)Hence we have on the apparent horizon, δQ = − A ˜ µ = κ π ˙ f κ ! A X i c i i ( n − n − i − r − i ! = κ π L ξ S = κ π δS , (3.16)where all the calculations are done on the apparent horizon. This is very similar to theClausius relation δQ = T δS if we define the temperature by T = κ/ π and the entropy ofapparent horizon S by S = A X i c i i ( n − n − i )! r − iA . (3.17)The entropy (3.17) is the same as that in the static cases if one replaces r A with the eventhorizon radius r + of static black holes in Lovelock gravity [19].Now, we have shown that A Ψ ( m ) term can be written as T δS term in the first law. Canthe equations of motion in Lovelock gravity be rewritten as the unified first law (3.3) inEinstein gravity theory, where the left-hand side is completely determined by spacetimegeometry, while the right hand side is determined by matter in spacetime? We can rewrite(3.3) as dE − A Ψ ( e ) − W ( e ) dV = A Ψ ( m ) + W ( m ) dV . (3.18)Note that the left hand side of the equation is totally determined by geometry becauseΨ ( e ) and W ( e ) are defined by geometric quantities as well as E . This implies that the lefthand side of the equation (3.18) can give us an energy form like the Misner-Sharp energy inEinstein gravity. Indeed it is not hard to show that the left hand side of the equation can5be cast into a form of dE L , where the function E L is given by E L = Ω n − π p X i ( n − c i ( n − i − r n − i − (1 − f ( v, r )) i . (3.19)Thus in the Vaidya-like spacetime for Lovelock gravity theory we arrive at a generalizedunified first law dE L = A Ψ ( m ) + W ( m ) dV . (3.20)After projecting on the apparent horizon, we obtain the first law of apparent horizon δE L = T δS − P δV . (3.21)where we have used W ( m ) = − P . If we write the function f ( v, r ) back into the form h ab ∂ a r∂ b r , the energy (3.19) is nothing but the generalized Minser-Sharp energy suggestedby H. Maeda and M. Nozawa in recent papers [43, 44], where the generalized Misner-Sharpenergy is shown to be a quasilocal conserved charge associated with a locally conservedcurrent constructed from the generalized Kodama vector. Here let us stress that our proce-dure to find the entropy and the energy expressions is a direct generalization of the methodproposed by Hayward to Lovelock gravity theory. The two methods, though different, leadto the same result. It would be interesting to use our method to study the unified first lawin other gravity theories and to get the corresponding Miner-Sharp energy.In some sense, Lovelock gravity theory is special since this theory is ghost-free, and theequations of motion are in fact second order for metric, as in the case of Einstein gravity.These properties may help us separating matter from geometry effectively. For other theorieswithout such properties, such as f ( R ) gravity, the relation between equations of motion andthermodynamics is complicated. Recently, Eling, Guedens and Jacobson have pointed outthat equations of motion for f ( R ) gravity correspond to a non-equilibrium thermodynamicsof spacetime. To get correct equations of motion, an entropy production term has to beadded to the Clausiu relation [37]. However, to what extent can the spacetime be describedby equilibrium thermodynamics is still an open question.In summary, the definition of heat variation δQ is important in our procedure. Theabove discussion shows that δQ should be defined by pure matter energy supply or purematter energy-momentum tensor, through which we can find a reasonable entropy associatedwith the apparent horizon. With this entropy and the unified first law in Einstein gravity6theory, we can obtain a proper energy form, i.e. generalized Misner-Sharp energy, completelydetermined by spacetime geometry. Thus we can establish the generalized unified first lawfor Lovelock gravity theory.The generalized unified first law (3.21) of the apparent horizon is a version of physicalprocess for the first law of black hole thermodynamics, i.e., an active version of the first law.In this version of the first law, one considers a spacetime and matter energy-momentumtensor, and calculates the variation of thermodynamical quantities which are induced by thematter (it enters into horizon). The first law then establishes a relation among the variationsof those thermodynamic quantities. In the passive version or phase space version of first law,on the other hand, one compares the thermodynamical quantities of two spacetimes differingfrom each other by a small amount of variables, and then gives variations of these quantities,which lead to the first law. This physical process of the first law can shed some light onthe passive version of the first law. For the solution (2.18), here we give some suggestionon the passive version of the first law. By using equations (2.18) and the definitions of theapparent horizon and the surface gravity, the surface gravity can be written as a functionof the radius of the apparent horizon κ = 12 r A P i i ˜ c i r − i +2 A "X i ( n − i − c i r − i +2 A + 16 π C ( v ) r ( n − σ − n +4 A , (3.22)where r A = r A ( v ) and the coefficients ˜ c i are defined by˜ c i = c i ( n − n − i − . (3.23)The energy inside the apparent horizon is given by M L = Ω n − π p X i ˜ c i r n − i − A , (3.24)and the pressure on the apparent horizon by P = −C ( v ) r − ( n − − σ ) A . (3.25)Then it is easy to check that the following relation holdsd M L = T d S − P d V . (3.26)where the entropy is given by (3.17), while d r A denotes a variation of the apparent horizon,i.e., from r A to r A + d r A . This is a variation of the state parameter r A of the dynamical7black hole. It should be noted that the variation operator “d” is not an exterior derivative“ d ” of the spacetime manifold, but a difference of two nearby points in the solution spaceof the theory. Equation (3.26) is just the passive version of the first law. Another point wewish to stress is that the physical meaning of energy M L is ambiguous. It is not an ADMmass of the spacetime (although it is an ADM mass in the static limit), but a generalizedMisner-Sharp energy inside the apparent horizon. By using equation (2.18), the energy M L can be expressed as M L = m ( v ) + Ω n − C ( v )Θ( r A ) . (3.27)Thus, we see that the generalized Misner-sharp energy inside the apparent horizon for thesolution (2.18) gets the physical meaning as a generalized Bondi mass if the matter is pureradiation. However, for other type of matters, such as the term leading to the nonvanishing C ( v ), the Bondi mass should be modified. Further, it turns out difficult to define the Bondimass at null infinity in odd dimensions. This has been pointed out in references [46, 47].For the static case, r A = r + , equations (3.22) and (3.26) reduce to the first law of black holethermodynamics in Lovelock gravity theory [19]. IV. CONCLUSION AND DISCUSSION
In this paper, we have obtained a kind of generalized Vaidya solutions (2.18) in a genericLovelock gravity. Explicit forms of these solutions are given for some special cases, such asGauss-Bonnet gravity, dimensional continued gravity and pure Lovelock gravity. We havealso investigated the unified first law on the apparent horizon of this spacetime, and giventhe entropy (3.17) associated with the apparent horizon and the energy (3.19) for ( n − r . In our procedure, we treated all terms of Lovelock gravityexcept for the Einstein tensor in equations of motion as an effective energy-momentum tensorin Einstein gravity theory, to which we could apply the unified first law proposed by Haywardfor Einstein gravity theory. Defining the heat flux by pure matter energy-momentum tensor,we obtained an entropy expression (3.17) associated with the apparent horizon. Substitutingthis entropy into the unified first law leads to an energy form (3.19) for Lovelock gravity,which is the same as the one suggested by H. Maeda and M. Nozawa [43, 44]. This methodcan be further extended to other theories of gravity, for example, brane world theory.Here a comment not discussed in section III is in order. We did not discuss whether the8apparent horizon (or trapping horizon) is “outer” and “inner” [32]. In general, for higherdimensional cases, the parameter p in (1.1) is more than one, so the equation (2.18) is ahigher order algebraic equation of f ( v, r ) and may have multi-horizons. We have assumedin section III that the apparent horizon is an outer apparent horizon (trapping horizon) ofblack hole spacetime with positive surface gravity 1 / f ′ ( v, r ) | r = r A >
0. In fact, the resultof this paper also holds for the cosmological event horizon (inner apparent horizon) withnegative surface gravity 1 / f ′ ( v, r ) | r = r A <
0. In this case the Hawking temperature of theapparent horizon is T = | κ/ π | .Note that, while investigating the thermodynamical properties of the new solution, we justfocussed on the mechanics of the dynamical black hole. To establish the thermodynamics ofapparent horizon, one has to show that there exists Hawking radiation with the temperature T = κ/ π as in the case of static black holes. In fact, this may be proved by following somerecent works on Hawking radiation and entropy of dynamical spacetime [48, 49, 50, 51, 52](for a recent review, see [53]).In a dynamical spacetime the apparent horizon is not necessarily the same as the eventhorizon. For the solution (2.18), the apparent horizon is simply given by an algebraicequation f ( v, r A ( v )) = 0, from which the event horizon is very different. The event horizonrequires the global information of the spacetime. However, if we assume the event horizonis a null hypersurface of the spacetime, it is not hard to find that the event horizon ofsolution (2.18) is given by f ( v, r + ( v )) = 2 dr + ( v ) /dv . Therefore, in general, it is not easyto give an explicit expression for the location of event horizon because we have to solvea differential equation instead of an algebraic equation. For some simple cases, such asthe four-dimensional Vaidya spacetime and the five-dimensional Vaidya spacetime with theGauss-Bonnet term, we may give the explicit formula of the location of the event horizonfor some simple mass functions m ( v ). Taking the four-dimensional Vaidya spacetime in theEinstein gravity as an example, the event horizon is given by r + = 2 m ( v ) / (1 − r + ). Someauthors have calculated the associated temperature and entropy by using brick wall modeland gravity anomalies method [54, 55]: T = 1 − · r + πm ( v ) = 14 πr + , S = A πr . (4.1)At this stage a serious problem may arise. For a dynamical spacetime such as the Vaidyasolution, which horizon, the apparent horizon or the event horizon or both of them, is indeed9associated with Hawking radiation and the entropy? This issue has been commented in [56](references therein). But it is fair to say that this is still an open question. Obviously, thisis a quite interesting issue worth further studying. Acknowledgments