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Title: Unified first law and the thermodynamics of the apparent horizon in the FRW universe

Abstract

In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law first proposed by Hayward to treat the outertrapping horizon of a dynamical black hole can be used to the apparent horizon (a kind of inner trapping horizon in the context of the FRW cosmology) of the FRW universe. We discuss three kinds of gravity theorties: Einstein theory, Lovelock thoery, and scalar-tensor theory. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon. In Lovelock theory, treating the higher derivative terms as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. In scalar-tensor gravity, we find, by using the same procedure, the Clausius relation no longer holds. This indicates that the apparent horizon of the FRW universe in the scalar-tensor gravity corresponds to a system of nonequilibrium thermodynamics. We show this point by using the method developed recently by Eling et al. for dealing with the f(R) gravity.

Authors:
;  [1];  [2]
  1. Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080 (China)
  2. (China) and Graduate School of the Chinese Academy of Sciences, Beijing 100039 (China)
Publication Date:
OSTI Identifier:
21020154
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevD.75.064008; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BLACK HOLES; COSMOLOGY; ENERGY-MOMENTUM TENSOR; ENTROPY; FIELD EQUATIONS; GENERAL RELATIVITY THEORY; GRAVITATION; SCALARS; THERMODYNAMICS; TRAPPING; UNIVERSE

Citation Formats

Cai Ronggen, Cao Liming, and Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080. Unified first law and the thermodynamics of the apparent horizon in the FRW universe. United States: N. p., 2007. Web. doi:10.1103/PHYSREVD.75.064008.
Cai Ronggen, Cao Liming, & Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080. Unified first law and the thermodynamics of the apparent horizon in the FRW universe. United States. doi:10.1103/PHYSREVD.75.064008.
Cai Ronggen, Cao Liming, and Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080. Thu . "Unified first law and the thermodynamics of the apparent horizon in the FRW universe". United States. doi:10.1103/PHYSREVD.75.064008.
@article{osti_21020154,
title = {Unified first law and the thermodynamics of the apparent horizon in the FRW universe},
author = {Cai Ronggen and Cao Liming and Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080},
abstractNote = {In this paper we revisit the relation between the Friedmann equations and the first law of thermodynamics. We find that the unified first law first proposed by Hayward to treat the outertrapping horizon of a dynamical black hole can be used to the apparent horizon (a kind of inner trapping horizon in the context of the FRW cosmology) of the FRW universe. We discuss three kinds of gravity theorties: Einstein theory, Lovelock thoery, and scalar-tensor theory. In Einstein theory, the first law of thermodynamics is always satisfied on the apparent horizon. In Lovelock theory, treating the higher derivative terms as an effective energy-momentum tensor, we find that this method can give the same entropy formula for the apparent horizon as that of black hole horizon. This implies that the Clausius relation holds for the Lovelock theory. In scalar-tensor gravity, we find, by using the same procedure, the Clausius relation no longer holds. This indicates that the apparent horizon of the FRW universe in the scalar-tensor gravity corresponds to a system of nonequilibrium thermodynamics. We show this point by using the method developed recently by Eling et al. for dealing with the f(R) gravity.},
doi = {10.1103/PHYSREVD.75.064008},
journal = {Physical Review. D, Particles Fields},
number = 6,
volume = 75,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}
  • It is shown that the differential form of Friedmann equation of a FRW universe can be rewritten as the first law of thermodynamics dE=TdS+WdV at apparent horizon, where E={rho}V is the total energy of matter inside the apparent horizon, V is the volume inside the apparent horizon, W=({rho}-P)/2 is the work density, {rho} and P are energy density and pressure of matter in the universe, respectively. From the thermodynamic identity one can derive that the apparent horizon r-tilde{sub A} has associated entropy S=A/4G and temperature T={kappa}/2{pi} in Einstein general relativity, where A is the area of apparent horizon and {kappa}more » is the surface gravity at apparent horizon of FRW universe. We extend our procedure to the Gauss-Bonnet gravity and more general Lovelock gravity and show that the differential form of Friedmann equations in these gravities can also be written as dE=TdS+WdV at the apparent horizon of FRW universe with entropy S being given by expression previously known via black hole thermodynamics.« less
  • We propose the most general modified first-order Horava-Lifshitz gravity, whose action does not contain time derivatives higher than the second order. The Hamiltonian structure of this theory is studied in all the details in the case of the spatially-flat Friedmann-Robertson-Walker (FRW) space-time, demonstrating many of the features of the general theory. It is shown that, with some plausible assumptions, including the projectability of the lapse function, this model is consistent. As a large class of such theories, the modified Horava-Lifshitz F(R) gravity is introduced. The study of its ultraviolet properties shows that its z=3 version seems to be renormalizable inmore » the same way as the original Horava-Lifshitz proposal. The Hamiltonian analysis of the modified Horava-Lifshitz F(R) gravity shows that it is in general a consistent theory. The F(R) gravity action is also studied in the fixed-gauge form, where the appearance of a scalar field is particularly illustrative. Then the spatially-flat FRW cosmology for this F(R) gravity is investigated. It is shown that a special choice of parameters for this theory leads to the same equations of motion as in the case of traditional F(R) gravity. Nevertheless, the cosmological structure of the modified Horava-Lifshitz F(R) gravity turns out to be much richer than for its traditional counterpart. The emergence of multiple de Sitter solutions indicates the possibility of unification of early-time inflation with late-time acceleration within the same model. Power-law F(R) theories are also investigated in detail. It is analytically shown that they have a quite rich cosmological structure: early-/late-time cosmic acceleration of quintessence, as well as of phantom types. Also it is demonstrated that all the four known types of finite-time future singularities may occur in the power-law Horava-Lifshitz F(R) gravity. Finally, a covariant proposal for (renormalizable) F(R) gravity within the Horava-Lifshitz spirit is presented.« less
  • We present a kind of generalized Vaidya solutions in a generic Lovelock gravity. This solution generalizes the simple case in Gauss-Bonnet gravity reported recently by some authors. We study the thermodynamics of apparent horizon in this generalized Vaidya spacetime. Treating those terms except for the Einstein tensor as an effective energy-momentum tensor in the gravitational field equations, and using the unified first law in Einstein gravity theory, we obtain an entropy expression for the apparent horizon. We also obtain an energy expression of this spacetime, which coincides with the generalized Misner-Sharp energy proposed by Maeda and Nozawa in Lovelock gravity.
  • Author of ref. 1, M.R. Setare (JCAP 01 (2007) 023), by redefining the event horizon measured from the sphere of the horizon as the system's IR cut-off for an interacting holographic dark energy model in a non-flat universe, showed that the generalized second law of thermodynamics is satisfied for the special range of the deceleration parameter. His paper includes an erroneous calculation of the entropy of the cold dark matter. Also there are some missing terms and some misprints in the equations of his paper. Here we present that his conclusion is not true and the generalized second law ismore » violated for the present time independently of the deceleration parameter.« less
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