# 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:

- Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080 (China)
- (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}

}