13.2 Thermodynamics and horizon entropy

It is known that in Einstein gravity the gravitational entropy S of stationary black holes is proportional to the horizon area A, such that S = A ∕(4G), where G is gravitational constant [75Jump To The Next Citation Point]. A black hole with mass M obeys the first law of thermodynamics, T dS = dM [59Jump To The Next Citation Point], where T = κs∕(2π) is a Hawking temperature determined by the surface gravity κs [293Jump To The Next Citation Point]. This shows a deep physical connection between gravity and thermodynamics. In fact, Jacobson [324Jump To The Next Citation Point] showed that Einstein equations can be derived by using the Clausius relation TdS = dQ on local horizons in the Rindler spacetime together with the relation S ∝ A, where dQ and T are the energy flux across the horizon and the Unruh temperature seen by an accelerating observer just inside the horizon respectively.

Unlike stationary black holes the expanding universe with a cosmic curvature K has a dynamically changing apparent horizon with the radius r¯ = (H2 + K ∕a2)− 1∕2 A, where K is a cosmic curvature [108Jump To The Next Citation Point] (see also [296]). Even in the FLRW spacetime, however, the Friedmann equation can be written in the thermodynamical form T dS = − dE + W dV, where W is the work density present in the dynamical background [8]. For matter contents of the universe with energy density ρ and pressure P, the work density is given by W = (ρ − P )∕2 [297Jump To The Next Citation Point298Jump To The Next Citation Point]. This method is identical to the one established by Jacobson [324], that is, dQ = − dE + W dV.

In metric f (R) gravity Eling et al. [228Jump To The Next Citation Point] showed that a non-equilibrium treatment is required such that the Clausius relation is modified to dS = dQ ∕T + diS, where S = F A∕ (4G ) is the Wald horizon entropy [610Jump To The Next Citation Point] and diS is the bulk viscosity entropy production term. Note that S corresponds to a Noether charge entropy. Motivated by this work, the connections between thermodynamics and modified gravity have been extensively discussed – including metric f (R) gravity [6Jump To The Next Citation Point7Jump To The Next Citation Point281Jump To The Next Citation Point431619Jump To The Next Citation Point620Jump To The Next Citation Point2301035150Jump To The Next Citation Point157] and scalar-tensor theory [281Jump To The Next Citation Point619Jump To The Next Citation Point620Jump To The Next Citation Point108Jump To The Next Citation Point].

Let us discuss the relation between thermodynamics and modified gravity for the following general action [53Jump To The Next Citation Point]

∫ [ ] 4 √ --- f(R, ϕ,X ) I = d x − g --16-πG----+ ℒM , (13.16 )
where X ≡ − (1∕2)g μν∇μ ϕ∇ νϕ is a kinetic term of a scalar field ϕ. For the matter Lagrangian ℒM we take into account perfect fluids (radiation and non-relativistic matter) with energy density ρM and pressure P M. In the FLRW background with the metric ds2 = h dx αdxβ + ¯r2dΩ2 αβ, where ¯r = a(t)r and 0 x = t, 1 x = r with the two dimensional metric 2 2 hαβ = diag(− 1,a (t)∕[1 − Kr ]), the Friedmann equations are given by
2 K 8πG H + -2-= -----(ρd + ρM ), (13.17 ) a 3F H˙ − K--= − 4πG--(ρd + Pd + ρM + PM ), (13.18 ) a2 F ρ˙M + 3H (ρM + PM ) = 0, (13.19 )
where F ≡ ∂f∕∂R and
[ ] -1--- 1- ˙ ρd ≡ 8πG f,X X + 2(F R − f) − 3H F , (13.20 ) [ ] Pd ≡ -1--- ¨F + 2H ˙F − 1-(FR − f) . (13.21 ) 8πG 2
Note that ρd and Pd originate from the energy-momentum tensor (d) T μν defined by
1 [ 1 1 ] T(μdν)≡ ----- -gμν(f − RF ) + ∇ μ∇ νF − gμν□F + --f,X ∇ μϕ∇ νϕ , (13.22 ) 8πG 2 2
where the Einstein equation is given by
8πG ( (d) (M)) G μν = ----- Tμν + T μν . (13.23 ) F
Defining the density ρd and the pressure Pd of “dark” components in this way, they obey the following equation
--3-- 2 2 ˙ ρ˙d + 3H (ρd + Pd) = 8 πG (H + K ∕a )F . (13.24 )
For the theories with ˙ F ⁄= 0 (including f (R) gravity and scalar-tensor theory) the standard continuity equation does not hold because of the presence of the last term in Eq. (13.24View Equation).

In the following we discuss the thermodynamical property of the theories given above. The apparent horizon is determined by the condition hαβ∂αr¯∂ β¯r = 0, which gives ¯rA = (H2 + K ∕a2)−1∕2 in the FLRW spacetime. Taking the differentiation of this relation with respect to t and using Eq. (13.18View Equation), we obtain

Fd ¯rA 3 ------= ¯rAH (ρd + Pd + ρM + PM )dt. (13.25 ) 4πG

In Einstein gravity the horizon entropy is given by the Bekenstein–Hawking entropy S = A ∕(4G ), where A = 4π ¯r2 A is the area of the apparent horizon [5975293]. In modified gravity theories one can introduce the Wald entropy associated with the Noether charge [610]:

AF S = ----. (13.26 ) 4G
Then, from Eqs. (13.25View Equation) and (13.26View Equation), it follows that
--1-- 3 ¯rA- 2π¯rA dS = 4π¯rAH (ρd + Pd + ρM + PM )dt + 2G dF . (13.27 )
The apparent horizon has the Hawking temperature T = |κs|∕(2π), where κs is the surface gravity given by
1 ( ¯r˙A ) ¯rA ( K ) 2 πG κs = − --- 1 − ------ = − --- ˙H + 2H2 + --2 = − -----¯rA (ρT − 3PT) . (13.28 ) ¯rA 2H ¯rA 2 a 3F
Here we have defined ρT ≡ ρd + ρM and PT ≡ Pd + PM. For the total equation of state wT = PT ∕ρT less than 1/3, as is the case for standard cosmology, one has κ ≤ 0 s so that the horizon temperature is given by
1 ( r˙¯ ) T = ----- 1 − --A--- . (13.29 ) 2π¯rA 2H ¯rA
Multiplying the term 1 − ˙¯r ∕(2H ¯r ) A A for Eq. (13.27View Equation), we obtain
3 2 T 2 TdS = 4πr¯AH (ρd + Pd + ρM + PM )dt − 2π ¯rA(ρd + Pd + ρM + PM )d¯rA + G-π ¯rAdF. (13.30 )

In Einstein gravity the Misner-Sharp energy [428] is defined by E = ¯r ∕(2G ) A. In f (R) gravity and scalar-tensor theory this can be extended to E = ¯rAF ∕(2G) [281Jump To The Next Citation Point]. Using this expression for f (R,ϕ, X ) theory, we have

2 2 E = ¯rAF--= V 3F-(H---+-K-∕a-)-= V (ρd + ρM ), (13.31 ) 2G 8πG
where V = 4π¯r3∕3 A is the volume inside the apparent horizon. Using Eqs. (13.19View Equation) and (13.24View Equation), we find
3 2 r¯A- dE = − 4πr¯AH (ρd + Pd + ρM + PM )dt + 4π ¯rA(ρd + ρM )d¯rA + 2G dF . (13.32 )
From Eqs. (13.30View Equation) and (13.32View Equation) it follows that
2 ¯rA TdS = − dE + 2π¯rA(ρd + ρM − Pd − PM )dr¯A + 2G-(1 + 2πr¯AT ) dF . (13.33 )
Following [297298108Jump To The Next Citation Point] we introduce the work density W = (ρd + ρM − Pd − PM )∕2. Then Eq. (13.33View Equation) reduces to
¯rA- T dS = − dE + W dV + 2G (1 + 2πr¯AT ) dF , (13.34 )
which can be written in the form [53Jump To The Next Citation Point]
T dS + TdiS = − dE + W dV , (13.35 )
( ) 1 ¯rA E dF diS = − ----- (1 + 2π ¯rAT )dF = − -- + S ---. (13.36 ) T 2G T F
The modified first-law of thermodynamics (13.35View Equation) suggests a deep connection between the horizon thermodynamics and Friedmann equations in modified gravity. The term diS can be interpreted as a term of entropy production in the non-equilibrium thermodynamics [228]. The theories with F = constant lead to diˆS = 0, which means that the first-law of equilibrium thermodynamics holds. The theories with dF ⁄= 0, including f (R) gravity and scalar-tensor theory, give rise to the additional non-equilibrium term (13.36View Equation) [672816196201085053Jump To The Next Citation Point].

The main reason why the non-equilibrium term diS appears is that the energy density ρd and the pressure Pd defined in Eqs. (13.20View Equation) and (13.21View Equation) do not satisfy the standard continuity equation for F˙ ⁄= 0. On the other hand, if we define the effective energy-momentum tensor T (D ) μν as Eq. (2.9View Equation) in Section 2, it satisfies the continuity equation (2.10View Equation). This correspond to rewriting the Einstein equation in the form (2.8View Equation) instead of (13.23View Equation). Using this property, [53] showed that equilibrium description of thermodynamics can be possible by introducing the Bekenstein–Hawking entropy Sˆ= A∕ (4G ). In this case the horizon entropy ˆS takes into account the contribution of both the Wald entropy S in the non-equilibrium thermodynamics and the entropy production term.

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