Thermodynamics of scalar-tensor theory with non-minimally derivative coupling

With the usual definitions for the entropy and the temperature associated with the apparent horizon, we show that the unified first law on the apparent horizon is equivalent to the Friedmann equation for the scalar--tensor theory with non-minimally derivative coupling. The second law of thermodynamics on the apparent horizon is also satisfied. The results support a deep and fundamental connection between gravitation, thermodynamics, and quantum theory.


I. INTRODUCTION
The discovery of black hole thermodynamics [1] shows a deep connection between gravitation and thermodynamics. In particular, the black hole temperature which is proportional to the surface gravity at the event horizon and Hawking radiation [2] tell us that this relation may be linked to quantum gravity [3]. Instead of proportional to the volume, the Bekenstein-Hawking entropy is equal to one quarter of the area of the event horizon of the black hole measured in Planck units [2,4]. Based on this area law of entropy, Bekenstein then argued a universal entropy bound for a weakly self-gravitating physical system in an asymptotically flat space-time [5]. This leads to the proposal of the holographic principle [6][7][8]. The holographic principle was supported by the AdS/CFT correspondence which states that the type IIB superstring theory on AdS 5 × S 5 is equivalent to the N = 4 super-Yang-Mills theory with gauge group U(N) in four dimensions [9]. The AdS/CFT correspondence relates a gravitational theory in d-dimensional anti-de Sitter space with a conformal field theory living in a (d − 1)-dimensional boundary space. The Hawking radiation and the holography show that the thermodynamic property of gravitation is unique. These special properties may provide some physical insights into the nature of quantum gravity. By applying the area law of entropy for all local acceleration horizons, it was found that Einstein equation could be derived from the first law of thermodynamics [10]. The relation was then discussed in cosmology, and the equivalence between the first law of thermodynamics and Friedmann equation was derived [11][12][13][14][15]. The relation between thermodynamics and gravitation was discussed extensively in the literature, and the relation holds also in more general theories of gravity [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].
The simplest generalization of Einstein's general relativity is Brans-Dicke theroy [29]. In Brans-Dicke theory, gravitation is propagated by massless spin zero scalar field in addition to the massless spin 2 graviton. The scalar degree of freedom can also arise upon compactification of higher dimensions. In general, the scalar field φ is coupled to the curvature scalar R as f (φ)R. More general couplings for the scalar field are also possible [30][31][32][33].
In this paper, we discuss the thermodynamics of the scalar-tensor theory with nonminimally derivative coupling ω 2 G µν φ ,µ φ ,ν . The paper is organized as follows. In section II, we review the scalar-tensor theory with non-minimally derivative coupling. The relation between the first law of thermodynamics and Friedmann equation is presented in section III. We discuss the second law of thermodynamics in section IV and conclusions are drawn in Section V.

COUPLING
The action for the general scalar-tensor theory with non-minimally derivative coupling is where the Planck mass M 2 pl = (8πG) −1 = κ −2 , ξ and η are coupling constants whose dimensions depend on the function F (φ) and N(φ), respectively, V (φ) corresponds to the scalar field potential, and S b is the action for the background matter which includes the dust and the radiation. Varying the action (1) with respect to the metric g µν , we get: where T b µν is the energy momentum tensor for the background matter, and T c µν is the effective energy momentum tensor for the scalar field. For convenience, according to the coupling terms with coefficients Z(φ), ξ and η, we write T c µν in the following form: The equation of the motion for the scalar field φ is To discuss the cosmological evolution, we take the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric, where k = 0, −1, +1 represents a flat, open and closed universe respectively. Substituting the FRW metric (8) into Eqs. (2)- (6) and assuming that the scalar field φ is spatially homogeneous, we obtain Friedmann equations as The equation of motion for the scalar field becomes: In the above equations, we have third-order time derivative ... φ . The presence of higher than two time derivatives usually introduces the Boulware-Deser ghost [55] and more degrees of freedom. In order to overcome these problems, we need to keep the equations to contain no more than second time derivatives. This can be achieved by choosing η + 2ξ = 0 and N(φ) = F (φ) [34]. Without loss of generality, we can choose Z(φ) = 1 by re-defining the scalar field φ so that we have the canonical kinetic term. With the choice η = −2ξ = −ω 2 and N(φ) = F (φ) = 1, the general scalar-tensor theory with non-minimally derivative coupling which contains only up to second derivatives is Now the energy-momentum tensor T c µν for the scalar field reduces to: Using the FRW metric (8), we obtain T c 00 and T c 11 as: Therefore, the effective energy density and pressure for the scalar field are given by: The Friedman equations are: If the non-minimally derivative coupling is absent, ω 2 = 0, we recover the standard result of Einstein gravity with canonically scalar field.

III. THE RELATION BETWEEN THE FIRST LAW OF THERMODYNAMICS AND FRIEDMANN EQUATION
In this section, we discuss the equivalence between the first law of thermodynamics on the apparent horizon and Friedmann equation. For a spherically symmetric space-time with the metric ds 2 = g ab dx a dx b +r 2 dΩ 2 , where the unit spherical metric dΩ 2 = dθ 2 + sin 2 θdφ 2 , the apparent horizon is defined as f = g abr ,ar,b = 0, the dynamical surface gravity at the apparent horizon is κ = ∇ a ∇ ar /2 [16], and the Hawking temperature associated with the apparent horizon is T = |κ|/2π. For the FRW metric (8), the apparent horizon is: The surface gravity at apparent horizon is and the associated temperature is T A = κ/2π. The entropy enclosed by the apparent horizon is S A = πr 2 A /G. Therefore, we have For the scalar-tensor theory with non-minimally derivative coupling, the effective total energy density is ρ tot = ρ b + ρ c . The total energy of the system inside the apparent horizon is E = ρ tot V , where the volume V = 4πr 3 A /3. So the energy change is where dr A =ṙ A dt, and dρ tot =ρ tot dt. By using the energy conservation for the total energy, we have Substituting the above result into Eq. (23), we get The work term W dV with W = (ρ tot − p tot )/2 is Applying the unified first law, we get Taking the time derivative of the apparent horizonr A defined in Eq. (20), we geṫ Combining Eqs. (29) and (30), we geṫ Using the energy conservation equation (24), and integrating the above equation (31), we obtain the Friedman equation Therefore, we derive the Friedmann equation from the unified first law on the apparent horizon for the scalar-tensor theory with non-minimally derivative coupling.
Now we would like to derive the unified first law starting from the Friedmann equation.
Substituting Eq. (19) into Eq. (30), we obtain We multiply −[1 −ṙ A /(2Hr A )]/G to both sides of Eq. (33), then Eq. (33) becomes Combining the above Eq. (34) with Eq. (27), we get So the unified first law is derived from the Friedmann equation together with the energy conservation equation. Therefore, with the usual definitions for the entropy and the Hawking temperature associated with the apparent horizon, we show that the unified first law on the apparent horizon is equivalent to the Friedmann equation for the scalar-tensor theory with non-minimally derivative coupling.

IV. THE SECOND LAW OF THERMODYNAMICS ON THE APPARENT HORI-ZON
As discussed in the previous section, the entropy of the apparent horizon S A = A/(4G) = By using the Friedmann Eqs. (18) and (19), we geṫ As long as ρ tot + p tot ≥ 0, we haveṠ A ≥ 0, and the second law of thermodynamics on the apparent horizon is satisfied.

V. CONCLUSIONS
With the usual definition of the area law of entropy S A = πr 2 A /(4G) of the apparent horizon, and the temperature T A = −[1 −ṙ A /(2Hr A )]/(2πr A ), as well as the energy con-servation for the effective total energy densityρ tot + 3H(ρ tot + p tot ) = 0, we show that the unified first law of thermodynamics dE = T A dS A + W dV is equivalent to the Friedmann equation for the scalar-tensor theory with non-minimally derivative coupling. The result further supports the argument that the apparent horizon is a physical boundary and the relation between the first law of thermodynamics and Friedmann equation holds for more general theory of gravity, and suggests a deep and fundamental connection between gravitation, thermodynamics and quantum theory. Furthermore, we show that the second law of thermodynamics on the apparent horizon is also satisfied for the scalar-tensor theory with non-minimally derivative coupling as long as ρ tot + p tot ≥ 0.