General proof of the entropy principle for self-gravitating fluid in f(R) Gravity
aa r X i v : . [ g r- q c ] M a y Preprint typeset in JHEP style - HYPER VERSION
General proof of the entropy principle forself-gravitating fluid in f ( R ) Gravity
Xiongjun Fang a , Minyong Guo b and Jiliang Jing a ∗ a Department of Physics, and Key Laboratory of Low Dimensional QuantumStructures and Quantum Control of Ministry of Education, Hunan NormalUniversity, Changsha, Hunan 410081, P. R. China b Department of Physics, Beijing Normal University, Beijing 100875, P. R.China
Abstract:
The discussions on the connection between gravity and thermodynamics attract muchattention recently. We consider a static self-gravitating perfect fluid system in f ( R )gravity, which is an important theory could explain the accelerated expansion of theuniverse. We first show that the Tolman-Oppenheimer-Volkoff equation of f ( R ) the-ories can be obtained by thermodynamical method in spherical symmetric spacetime.Then we prove that the maximum entropy principle is also valid for f ( R ) gravity ingeneral static spacetimes beyond spherical symmetry. The result shows that if theconstraint equation is satisfied and the temperature of fluid obeys Tolmans law, theextrema of total entropy implies other components of gravitational equations. Con-versely, if f ( R ) gravitational equation hold, the total entropy of the fluid should beextremum. Our work suggests a general and solid connection between f ( R ) gravityand thermodynamics. Keywords: maximum entropy principle, Tolman-Oppenheimer-Volkoff equation, f ( R ) gravity. ∗ Corresponding author. Email: [email protected] ontents
1. Introduction 12. TOV equations obtained by two different methods 3
3. Maximum entropy principle for general static spacetime in f ( R ) gravity 84. Summary and discussions 115. Appendix 12 δρ P ab and P ab
1. Introduction
Black holes are mysterious and important objects which have been studied for along time. The mathematical analogy between laws of black hole mechanics and theordinary laws of thermodynamics leads to the discovery of black hole thermodynamics[1]. Hawking proved that the black hole behaves like a black body with a temperaturewhich is proportional to its surface gravity, and one quarter of the horizon area isthe entropy of the black hole [2, 3]. Since then, black hole thermodynamics hasbeen widely studied, and people believe that it could help us to catch sight of someimportant fundamental theories.It is generally believed that the gravity equation is the basic equation of nature.However, Jacobson put forward a new point of view that the Einstein equation isan equation of state [4]. Over the years this point has been accepted by more andmore people. R. Cai proved that applying the first law of thermodynamics to theapparent horizon of a Friedmann-Robertson-Walker universe, one could derive theFriedmann equations describing the dynamics of the universe [5]. And Verlindeconsidered gravity could be explained as an entropic force [6]. Bravetti presentedan example of spacetimes effectively emerging from the thermodynamic limit overan unspecified microscopic theory without any further assumptions [7]. All these– 1 –iscussion showed that the gravity equation may not be the basic assumption butthe thermodynamics relation is.In fact, before the establish of black hole thermodynamics, Cocke [8] proposedmaximum entropy principle for self-gravitating fluid. In contrast to black hole, thelocal thermodynamic quantities of self-gravitating fluid, such as entropy density s ,energy density ρ and local temperature T , are well defined. The presence of gravityonly affects the distribution of these local quantities. Roughly speaking, there are twodifferent ways to approach the distribution of the self-gravitating fluid system. Oneway is called the dynamical method, and the other way is called the thermodynamicalmethod. In dynamical method, using Einstein’s equation and applying conservationof matter, people could obtain the distribution of the fluid. But in thermodynamicalmethod, using only constrain Einstein equation and the ordinary thermodynamicalrelation, then the fluid could be configured such that its total entropy attains a max-imum value. For a spherical radiation system, Sorkin, Wald and Zhang [9] showedthat if the total entropy be an extremum and the Einstein constraint equation holds,then the Tolman-Oppenheimer-Volkoff (TOV) equation of hydrostatic equilibriumcan be derived. Gao [10] generalized SWZ’s work to arbitrary perfect fluid in staticspherical spacetime and got the TOV equation for this fluid. This issue has beenfurther explored in the past years [11-14]. Recently, series of general proofs of themaximum entropy principle in the case of static spacetime without the spherical sym-metry has been completed, included Einstein-Maxwell theory and Lovelock theory[15-18].However, whether the principle is valid for f ( R ) gravity is not clear. f ( R ) gravityis a natural generalization of Einstein gravity. The motivation of modifying gener-alization of GR comes from high-energy physics for adding higher order invariantsto the gravitational action, as well as comes from cosmology and astrophysics forseeking generalizations of GR. As an important modified theories, f ( R ) theories canexplain the accelerated expansion of the universe because it contains the αR terms[19]. f ( R ) gravity comes about by a straightforward generalization of the Lagrangianin the Einstein action to become a general function of Ricci scalar R , i.e., S = 12 κ d x √− gf ( R ) , (1.1)where f as a series expansion of Rf ( R ) = n = ∞ X n = −∞ α n R n . (1.2)There are two advantages in f ( R ) gravity [20, 21]. First, f ( R ) actions are simpleenough to be deal with, but sufficiently general to encapsulate some of the basiccharacteristics of higher-order gravity. Second, there are some reasons to believe– 2 –hat f ( R ) gravity is unique among higher-order gravity theories which can avoidOstrogradski instability. Thus it is interesting to discuss the maximum entropyprinciple in this generalized gravity theories.In Sec. II, we discuss the maximum entropy in spherical static spacetime. TOVequation constrains the structure of a spherically symmetric body of isotropic mate-rial which is in static gravitational equilibrium. We first briefly review how to obtainthe modified TOV equation of f ( R ) gravity by traditional dynamical method. Andthen in addition to the constraint equation, we only make use of the ordinary ther-modynamical relations to derive the same TOV equation in f ( R ) gravity. These twodifferent methods obtained the same results shows that the maximum entropy inspherical static spacetime in f ( R ) theories is valid.In Sec. III, with the help of Tolman’s law, which was also used in [13,15,18], weextend the entropy principle to general static spacetimes without spherical symmetrycondition. It should be noticed that, the total entropy S appears to depend on δh ab and δR (See Appendix A). Fixing the induced metric h ab and scalar curvature R and their first derivatives on the boundary, in order to allow us to use integrationby parts and drop the boundary terms. And it also shows that the components of δR vanishes automatically. So the vanishing of the components of δh ab just givesthe spatial components of gravitational equation of f ( R ) theories. The derivation ofthis section shows that the maximum entropy is also valid in f ( R ) theories withoutspherical symmetry.Throughout our discussion, symbol a, b, c represent the abstract index notation, ∇ a and D a are the covariant derivative associated with the metric of spacetime andthe covariant derivative associated with the induced metric h ab , respectively, and (cid:3) ≡ ∇ a ∇ a .
2. TOV equations obtained by two different methods
As for the issue of the distribution of fluid, dynamical method is a classical one, andis widely used. First we briefly review how to get TOV equation in f ( R ) gravity bydynamical method [22]. In 4-dimensional spacetime the action of the f(R) gravitycan be expressed as S = 12 κ Z d x √− gf ( R ) + S M , (2.1)where κ ≡ π , and S M is the action term of perfect fluid matter. Variation withrespect to the metric gives f R ( R ) R ab − f ( R ) g ab − [ ∇ a ∇ b − g ab (cid:3) ] f R ( R ) = 8 πT ab , (2.2)– 3 –here T ab = − √− g δS M δg ab , (2.3)and f R ( R ) ≡ ∂f /∂R .For a static perfect fluid system with spherical symmetries in the f(R) gravity,the line element of the spacetime can be written as ds = − e φ ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdφ ) . (2.4)The energy-momentum tensor of the perfect fluid is T ab = ρu a u b + ph ab , (2.5)where ρ is the matter density and p is the pressure, and in our chosen coordinate wehave u a = − e φ ( r ) ( dt ) a . The components of the field equations become8 πT tt = 8 πρ = f R ( R ) r + e − λ (2 rλ ′ − r f R ( R ) + 12 f ( R ) − R f R ( R ) − e − λ f ′′ R ( R ) + e − λ ( λ ′ − r ) f ′ R ( R ) , (2.6)and 8 πT rr = 8 πp = − f R ( R ) r + e − λ (1 + 2 rφ ′ ) r f R ( R ) − f ( R )+ R f R ( R ) + e − λ rφ ′ r f ′ R , (2.7)where ′ ≡ d/dr .To get the TOV equation, we consider Bianchi identity, which leads to energy-momentum conservation equation ∇ a T ab = 0. The radial component of the conser-vation equation gives p ′ = − ( ρ + p ) φ ′ . (2.8)If we assume e − λ = 1 − mr . (2.9)– 4 –hen the modified TOV equations take the following convenient form8 πp + 2 mr f R ( R ) + f ( R )2 − R f R ( R ) − r − m ) r f ′ R ( R )= − r − mr (cid:18) f R ( R ) r + f ′ R ( R ) (cid:19) p ′ p + ρ . (2.10)Notice that if the Lagrangian degenerate to Einstein theory, i.e. f ( R ) = R , Eq.(2.10) will reduce to the well-known TOV equation in Ref. [23]. In previous subsection, we have shown how to get the TOV equation in f ( R ) gravityby using dynamical method. Generally, to get the TOV equation, one has to useradial-radial component (Eq. (2.7)) of the gravitational equation. Now, we shallderive TOV equation in f ( R ) gravity by using thermodynamical method. We onlyuse conservation equation (Eq. (2.6)) and the extrema of total entropy, admitsthe condition of the variation of total particle number vanish in static spacetime, toderive TOV equation of f ( R ) gravity. It implies that the maximum entropy principlecontains part of information of gravitational equations in f ( R ) gravity.The perfect fluid system satisfies the familiar first law dS = 1 T dE + PV dV − µT dN , (2.11)where S , E , N represent the total entropy, energy, and particle number within thevolume V . Their density variables are denoted by s , ρ and n , respectively. ApplyingEq. (2.11) to a unit volume, it is easy to find ds = 1 T dρ − µT dn , (2.12)and the integrated form which be called the Gibbs-Duhem relation s = p + ρ − µnT . (2.13)Now we apply the maximum entropy principle to this self-gravitating perfectfluid in f ( R ) gravity. In spherical symmetry case, the total entropy and the totalparticle number from r = 0 to r = r can be written as [10] S = 4 π Z r s ( r ) (cid:20) − m ( r ) r (cid:21) − / r dr , (2.14) N = 4 π Z r n ( r ) (cid:20) − m ( r ) r (cid:21) − / r dr . (2.15)– 5 –he variation of total particle number vanishes, which means δN = 0, is a desiredcondition in maximum entropy principle. Following the standard method of Lagrangemultipliers, the equation of variation becomes δS + λδN = 0 . (2.16)Define the ”total Lagrangian” by L ( m, m ′ , n ) = s ( ρ ( m, m ′ ) , n ) (cid:18) − m ( r ) r (cid:19) − r + λn ( r ) (cid:18) − m ( r ) r (cid:19) − r . (2.17)So the constrained Euler-Lagrange equation is given by ∂L∂n = 0 , (2.18) ddr ∂L∂m ′ = ∂L∂m . (2.19)Eq. (2.18) yields, ∂s∂n = λ . (2.20)By using thermodynamic relation we have µ = λT . (2.21)Now consider Eq. (2.19), after some calculation, we have ∂L∂m = 18 πT (cid:18) r f ′′ R + 3 r f ′ R (cid:19) r (cid:18) − mr (cid:19) − / + ( nλ + s ) r (cid:18) − mr (cid:19) − / , (2.22)and ∂L∂m ′ = 18 πT (cid:18) r f R ( R ) + 1 r f ′ R ( R ) (cid:19) r (cid:18) − mr (cid:19) − / . (2.23)By substituting these results into Eq. (2.19), together with Eqs. (2.13) and (2.21),– 6 –e could obtain the Euler-Lagrangian equation of m in an explicit form T ′ πT (cid:18) r f R ( R ) + 1 r f ′ R ( R ) (cid:19) r = 18 πT (cid:18) − r f R ( R ) − r f ′ R ( R ) + 2 r f ′ R ( R ) + 1 r f ′′ R ( R ) (cid:19) r + 18 πT (cid:18) r f R ( R ) + 1 r f ′ R ( R ) (cid:19) (2 r + rm ′ − m ) (cid:18) − mr (cid:19) − − πT (cid:18) r f ′′ R + 3 r f ′ R (cid:19) r − p + ρT r (cid:18) − mr (cid:19) − . (2.24)The constraint Eq. (2.21) yield µ ′ = λT ′ . (2.25)Considering the fundamental thermodynamic relation dp = sdT + ndµ , (2.26)it follows that p ′ = sT ′ + nµ ′ = T ′ T ( ρ + p ) . (2.27)Thus we obtain the equation p ′ ρ + p (cid:18) r f R ( R ) + 1 r f ′ ( R ) (cid:19) r (1 − mr )= (cid:18) − r f R ( R ) + 1 r f ′ R + 1 r f ′′ R (cid:19) r (cid:18) − mr (cid:19) + (cid:18) r f R + 1 r f ′ R (cid:19) (2 r + rm ′ − m ) − (cid:18) r f ′′ R ( R ) + 3 r f ′ R ( R ) (cid:19) r (1 − mr ) − πpr − πρr . (2.28)By substituting Eq. (2.6) into Eq. (2.28), finally we get − p ′ ρ + p (cid:18) r f R ( R ) + 1 r f ′ ( R ) (cid:19) r (1 − mr )= 8 πp + f ( R )2 + (cid:18) mr − R (cid:19) f R ( R ) − r − m ) r f ′ R ( R ) . (2.29)This TOV equation in f ( R ) gravity is exactly the same as Eq. (2.10).It is necessary to point out that we only use the maximum entropy principle andthe time-time component gravitational equation. No other assumptions are needed.Which tells us that in spherical symmetry case, the extrema of entropy contain– 7 –art of information of gravitational equation. And this is a direct evidence for thefundamental relationship between gravitation and thermodynamics.
3. Maximum entropy principle for general static spacetime in f ( R ) gravity In previous sections, we got TOV equation by using dynamical and thermodynamicmethods, respectively. But all these discussions based on static spherical symmetrycondition. What we want to know is that whether the maximum entropy principleis valid for general static spacetime in f ( R ) gravity.Consider a general static spacetime ( M , g ab ) and Σ as a three dimensional hy-persurface denoting a moment of the static observers. Let C be a region on Σ witha boundary ¯ C and h ab be the induced metric on Σ, and R be the scalar curvatureof spacetime, respectively. We assume that the Tolman’s law holds in C , like[15],which means that the local temperature T of the fluid satisfies T χ = T , (3.1)where χ is the red-shift factor for static observers and T is a constant which canbe viewed as the red-shift temperature of the fluid. Without loss of generality, wetake T = 1 in Tolman’s law. Assume that the constraint equation is satisfied in C .We also need h ab , R and their first derivatives are fixed on ¯ C , which allow us to useintegration by parts and drop the boundary term. Then, it can be proved that theother components of gravitational equation are implied by the extrema of the totalfluid entropy for all variations of data in C .We consider a general perfect fluid as discussed in previous section. The inducedmetric on Σ is given by h ab = g ab + u a u b . (3.2)The stress-energy tensor T ab takes the form of Eq. (2.5). And from conservation law ∇ a T ab = 0, we can show that ∇ a p = − ( ρ + p ) A a , (3.3)where A a is the four-acceleration of the observer. Since u a = ξ a χ , (3.4)where ξ a is the Killing vector. One can show that A a = ∇ a χ/χ , (3.5)– 8 –nd thus ∇ a p = − ( ρ + p ) ∇ a χ/χ . (3.6)Meanwhile, the local first law can also be expressed in the form dp = sdT + µdn . (3.7)Comparing with Eqs. (3.6) and (3.7), and applying the Tolman’s law Eq. (3.1) andthe Gibbs-Duhem relation Eq. (2.13), we finally have µχ = µ/T = constant . (3.8)If we choose energy density ρ and particle number density n as two independentthermodynamical variables. Then in static spacetime, the total entropy S is anintegral of the entropy density s over the region C on Σ, S = Z C √ hs ( ρ, n ) . (3.9)where h is the determinant of h ab . Thus, the variation of the total entropy can bewritten as δS = Z C sδ √ h + √ hδs . (3.10)Using the local first law of thermodynamics, T ds = dρ − µdn , (3.11)we find δS = Z C sδ √ h + √ h (cid:18) ∂s∂ρ δρ + ∂s∂n δn (cid:19) = Z C sδ √ h + √ h (cid:18) T δρ − µT δn (cid:19) . (3.12)Notice that µ/T is constant, which can be taken out of the integral. It is natural torequire the total number of particles N = Z C √ hn , (3.13)– 9 –o be invariant, which leads to Z C √ hδn = − Z C nδ √ h . (3.14)So we have δS = Z C (cid:16) s + nµT (cid:17) δ √ h + √ h T δρ = Z C ρ + pT δ √ h + √ h T δρ . (3.15)Using [24] δ √ h = 12 √ hh ab δh ab , (3.16)and denoting δS = Z C δL , (3.17)we have δL = 12 ρ + pT √ hh ab δh ab + √ h T δρ . (3.18)After some calculation, we get the exactly express of the term √ hδρ/T in AppendixA. By substituting Eq. (5.25) into Eq. (3.18), we have δL = ρ + p T √ hh ab δh ab + δL ρ = ρ + p T √ hh ab δh ab + √ h π D b D a ( χf R ( R )) δh ab − √ h π D c [ h ab D c ( χf R ( R ))] δh ab − √ h π f R ( R ) χR (3) ab δh ab − √ h π D a χ ( D b f R ( R )) δh ab + √ h π h ab D c ( χD c f R ( R )) δh ab . (3.19)So far, we only applied the constraint equation G = T . This shows explicitlythat δS is determined only by the variation of h ab . Since δS = 0 in our assumption,it is clear shows that δL = 0. Together with Eq. (5.9) and Tolman’s law Eq. (3.1),it is easily find8 πph ab T δh ab = h ab T δh ab (cid:20) h ac h bd R cd f R ( R ) − f ( R ) h ab − h ac h bd ∇ c ∇ d f R ( R ) + h ab (cid:3) f R ( R ) (cid:21) −
12 ( P ab + P ab ) δh ab , (3.20)– 10 –here P denotes the terms parallel to h ab , P ab = χh ab (cid:20) f R ( R ) R cd u c u d + 12 f ( R ) − u c u d ∇ c ∇ d f R ( R ) − (cid:3) f R ( R ) (cid:21) − χ f ( R ) h ab + h ab χ (cid:3) f R ( R ) − h ab D c D c ( χf R ( R )) + h ab D c ( χD c f R ( R )) , (3.21)and P denotes the other terms, P ab = χ [ f R ( R ) R cd h ac h bd − h ac h bd ∇ c ∇ d f R ( R )] + D b D a ( χf ′ ( R )) − f ′ ( R ) χR (3) ab − D a χ ( D b f R ( R )) . (3.22)We show that P ab and P ab are vanish in Appendix B. So Eq. (3.20) gives theprojection of gravitational equation8 πp = h ac h bd R cd f R ( R ) − f ( R ) h ab − h ac h bd ∇ c ∇ d f R ( R ) + h ab (cid:3) f R ( R ) . (3.23)In the above proof, under the fixed boundary condition, we only used the con-straint Eq. (5.9) to derive Eq. (3.19). By applying the extrema of entropy δS = 0,we obtained the spatial components of gravitational equation. It is easy to checkthat the proof is reversible. From the spatial components of gravitational equation,one can show δL = 0 in Eq. (3.19).
4. Summary and discussions
The TOV equation is an important equation for self-gravitating system. In thispaper, we first briefly review how to get modified TOV equation in f ( R ) gravityby dynamical method. By applying both time-time and radial-radial gravitationalequations, and the radial component constraint equation, one can show that thegeneral TOV equation in f ( R ) gravity is given by Eq. (2.10). Then, by applyingthe maximum entropy principle to a general self-gravitating fluid, we have obtainedthe same TOV equation of hydrostatic equilibrium by thermodynamical method.We should point out that, to derive the TOV equation by the extrema of the totalentropy S , we only need the constraint equation and fix the total particle numberof the system. Comparing with these two different processes, it tells us that themaximum entropy principle is valid in spherical symmetry case in f ( R ) theories.We also prove that the maximum entropy principle of perfect fluid is valid to casewithout any symmetry on spacelike hypersurface. Arbitrary select a region, considerthe fluid obeys Tolman’s law and the constraint equation is satisfied in this region.Then fix the induced metric h ab and scalar tensor R and their first derivative on theboundary of selected region. Our calculation shows that δS appears to depend on– 11 – h ab and δR , and the component of δR always vanish. It shows that the variation oftotal entropy in this region is exactly linear to δh ab . The assumption of δS = 0 meansthe coefficient of δh ab vanish, which imply that the other components of gravitationalequation satisfied.Till now, the maximum entropy principle has been set up in series of theories.Although it is unclear whether this principle could be used for a general Lagrangianof gravity theories which needs further study, we consider that there may exist somedeeper connection between gravity theories and thermodynamics. Acknowledgements
We thank S. Gao and X. He for many helpful discussions. This work is supportedby the National Natural Science Foundation of China under Grant No. 11475061;the SRFDP under Grant No. 20114306110003; the Open Project Program of StateKey Laboratory of Theoretical Physics, Institute of Theoretical Physics, ChineseAcademy of Sciences, China Grant No. Y5KF161CJ1. Xiongjun Fang is also sup-ported by China Postdoctoral Science Foundation funded project.
5. Appendix δρ In this Appendix, we will show the detailed calculation of δρ in Eq. (3.18). First,note that the extrinsic curvature of Σ, is defined byˆ B ab = h ac h bd ∇ d u c . (5.1)It is straightforward to show ˆ B ab = ∇ b u a + A a u b , (5.2)where A a is the four-acceleration of the observer . And in static spacetime we haveˆ B ab = 0 , (5.3)so we get a very helpful formula in our calculation, ∇ b u a = − A a u b . (5.4)Then, we show [24] that the curvature R (3) abcd of hypersurface Σ is related to thespacetime curvature R abcd by R (3) abcd = h af h bg h ck h j d R fgk j , (5.5)– 12 –rom this definition, it is easy to find R (3) ab = R ab + R aebl u e u l + R fb u f u a + R ak u k u b + u a u b R fk u f u k , (5.6)and R (3) = R + 2 R ab u a u b . (5.7)Meanwhile, from Eq. (5.6) we also have R (3) ab = R (3) cd h ac h bd = R ab + R aebl u e u l h ac h bd . (5.8)From Eq. (2.5), one can easily get the Hamiltonian constraint of the theory,8 πρ = u a u b R ab f R ( R ) + 12 f ( R ) − u a u b ∇ a ∇ b f R ( R ) − (cid:3) f R ( R ) . (5.9)Using Eq. (5.7), the variation of ρ is8 πδρ = δ (cid:20) u a u b R ab f R ( R ) + 12 f ( R ) − u a u b ∇ a ∇ b f R ( R ) − (cid:3) f R ( R ) (cid:21) = 12 δ [( R (3) − R ) f R ( R )] + 12 δf ( R ) − δ [ u c u d ∇ c ∇ d f R ( R )] − δ [ (cid:3) f R ( R )]= 12 f R ( R ) δR (3) + 12 ( R (3) − R ) f RR ( R ) δR − δ [ u c u d ∇ c ∇ d f R ( R )] − δ [ (cid:3) f R ( R )] , (5.10)where we denote f RR ( R ) ≡ ∂ f∂R , and we use δf ( R ) = f R ( R ) δR . So δρ term in Eq.(3.18) becomes δL ρ = √ h δρT = √ h πT f R ( R ) δR (3) + √ h πT ( R (3) − R ) f RR ( R ) δR − √ h πT δ [ u a u b ∇ a ∇ b f R ( R )] − √ h πT δ [ (cid:3) f R ( R )] . (5.11)Now, we deal with terms of δL ρ , respectively. Notice that the Tolman’s law Eq.(3.1) is used in our calculation. The first term of Eq. (5.11) can be calculate as8 πδL ρ = √ h T f R ( R ) δR (3) = 12 χ √ hf R ( R )( h ab δR (3) ab + R (3) ab δh ab ) . (5.12)– 13 –he standard calculation of δR (3) ab yields h ab δR (3) ab = √ hD a ( D b δh ab − h bc D a δh bc ) , (5.13)Using integration by parts and dropping boundary terms, we have8 πδL ρ = 12 √ hD b D a ( χf R ( R )) δh ab − √ hD c [ h ab D c ( χf R ( R ))] δh ab − f R ( R ) χ √ hR (3) ab δh ab . (5.14)The third term of Eq. (5.11) can be calculate as8 πδL ρ = − √ hT δ [ u a u b ∇ a ∇ b f R ( R )] , (5.15)and the fourth term of Eq. (5.11) can be written as8 πδL ρ = − √ hT δ [ (cid:3) f ′ ( R )] = − √ hT δ ( ∇ a ∇ a f ′ ( R )) . (5.16)Note that D a D a f R ( R ) = h ab ∇ a ∇ b f R ( R )= ∇ a ∇ a f R ( R ) + u a u b ∇ a ∇ b f R ( R ) , (5.17)and δ ( D a D a f R ( R )) = D a δ ( D a f R ( R )) + δC cac D a f R ( R ) , (5.18)where δC cac = 12 h cb D a δh cb . (5.19)So we have 8 πδL ρ + 8 πδL ρ = − √ hT [ D a δ ( D a f R ( R )) + δC cac D a f R ( R )]= −√ hD a χ ( D b f R ( R )) δh ab − √ h ( D a D a χ ) f RR ( R ) δR + 12 √ hh ab D c ( χD c f R ( R )) δh ab . (5.20)Considering the second term of Eq. (5.11), together with Eqs. (5.20) and (5.7), we– 14 –enote all the terms contained f RR ( R ) δR by δL ρδR , then8 πδL ρδR = −√ h ( D a D a χ ) f RR ( R ) δR + √ h χ ( R (3) − R ) f RR δR = −√ h ( D a D a χ ) f RR ( R ) δR + √ h χu a u b R ab f RR δR . (5.21)We can calculate that χR ab u a u b = R acbc u a u b = − χu a ∇ a ∇ c u c + χu a ∇ c ∇ a u c = χu a ∇ c ( − u a A c ) = χ ∇ a A a . (5.22)And we know ∇ a A a = A a A a + D a A a . (5.23)So the f RR ( R ) δR terms vanish, δL ρδR = 0 . (5.24)At last, the calculation of δρ term of Eq. (3.18) shows δL ρ = √ h π D b D a ( χf R ( R )) δh ab − √ h π D c [ h ab D c ( χf R ( R ))] δh ab − √ h π f R ( R ) χR (3) ab δh ab − √ h π χA a D b f R ( R ) δh ab − √ h π D a χ ( D b f R ( R )) δh ab + √ h π h ab D c ( χD c f R ( R )) δh ab + √ h π χA a ( D b f R ( R )) δh ab = √ h π D b D a ( χf R ( R )) δh ab − √ h π D c [ h ab D c ( χf R ( R ))] δh ab − √ h π f R ( R ) χR (3) ab δh ab − √ h π D a χ ( D b f R ( R )) δh ab + √ h π h ab D c ( χD c f R ( R )) δh ab . (5.25)This result tells us that the variation of ρ term is exactly determined only by thevariation of induced metric δh ab . – 15 – .2 Appendix B: Calculation of P ab and P ab In this Appendix, we give the detailed calculation of P ab and P ab . We first calculate P ab = χh ab (cid:20) f R ( R ) R cd u c u d + 12 f ( R ) − u c u d ∇ c ∇ d f R ( R ) − (cid:3) f R ( R ) (cid:21) − χf ( R ) h ab + h ab χ (cid:3) f R ( R ) − h ab D c D c ( χf R ( R ))+ h ab D c ( χD c f R ( R ))= h ab [ χf R ( R ) R cd u c u d − χu c u d ∇ c ∇ d f R ( R ) − D c D c ( χf R ( R )) + D c ( χD c f R ( R ))] . (5.26)In static spacetime, we have − χu c u d ∇ c ∇ d f R ( R )= − χu c ∇ c [ u d ∇ d f R ( R )] + χu c ( ∇ c u d ) ∇ d f R ( R )= χA d D d f R ( R ) . (5.27)And notice Eqs. (5.22) and (5.23), χR cd u c u d = χ ∇ c A c = χ ( A c A c + D c A c ) . (5.28)By substituting Eqs. (5.27) and (5.28) in Eq. (5.26), P ab = h ab [ χ ( A c A c + D c A c ) f R ( R ) + χA d D d f R ( R ) − D c D c ( χf R ( R )) + D c ( χD c f R ( R ))]= h ab [ A c ( D c χ ) f R ( R ) + χ ( D c A c ) f R ( R ) − ( D c D c χ ) f R ( R )]= 0 . 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