Horizon thermodynamics in f(R) theory

Abstract

We investigate whether the new horizon first law proposed recently still work in f (R) theory. We identify the entropy and the energy of black hole as quantities proportional to the corresponding value of integration, supported by the fact that the new horizon first law holds true as a consequence of equations of motion in f (R) theories. The formulas for the entropy and energy of black hole found here are in agreement with the results obtained in literatures. For applications, some nontrivial black hole solutions in f (R) theories have been considered, the entropies and the energies of black holes in these models are firstly computed, which may be useful for future researches.It is well-known that there is a profound connection between gravity and thermodynamics: the spacetime with horizons can be described by thermodynamic laws. Bekenstein found that the area of a black hole can be seen as its entropy [1]. Four laws of black hole mechanics were proposed in [2]. It has been shown that the entropy of a black hole can be taken as the Noether charge associated with the diffeomorphism invariance of the theory of gravity [3,4]. From the first law of thermodynamics Einstein equation has been derived in [5]. Non-equilibrium thermodynamics of spacetime has been investigated in [6]. By using a more general definition of the Noether charge entropy, the equations of motion of generalized theories of gravity are equivalent to the thermodynamic relation δQ = T δS [7]. This attempt also has been considered in modified gravity theories: such as f (R) theory [8], Lancos-Lovelock gravity [9], and the scalar-Gauss-Bonnet gravity [10]. In [11], a general formalism for understanding the thermodynamics of horizons in spherically symmetric spacetimes was developed. For stationary axis-symmetric horizons and time dependent evolving horizons, it has been shown that the near horizon structure of Einstein equations can be expressed as a thermodynamic identity under the virtual displacement of the horizon [12]. It also has been shown that the gravitational field equations of n + 1 -dimensional topological black holes with constant horizon curvature, in cubic and quartic quasi-topological gravity, can be recast in the form of the first law of thermodynamics [13]. All these studies were based on some assumptions, such as horizon, null surfaces, Unruh temperature, and so on. In [14], without assuming a temperature or a horizon the thermal entropy density has been obtained for any arbitrary spacetime, implying that gravity possesses thermal effects, or, thermal entropy density possesses effects of gravity. These results has been generalized to the case of nonzero chemical potential [15].When assuming a horizon equation of state, one can get a horizon first law by considering a virtual displacement, from which the entropy can be obtained [11]. Recently a new horizon first law was suggested in [16], in this approach both the entropy and the free energy are derived concepts, and from which the standard horizon first law is recovered by a Legendr...

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