# Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe

## Abstract

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} 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.

- Authors:

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

- Publication Date:

- OSTI Identifier:
- 21020372

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.75.084003; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BLACK HOLES; COSMOLOGY; ENERGY DENSITY; ENTROPY; FIELD EQUATIONS; GENERAL RELATIVITY THEORY; GRAVITATION; NONLUMINOUS MATTER; THERMODYNAMICS; UNIVERSE

### Citation Formats

```
Akbar, M., and Cai Ronggen.
```*Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.084003.

```
Akbar, M., & Cai Ronggen.
```*Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe*. United States. doi:10.1103/PHYSREVD.75.084003.

```
Akbar, M., and Cai Ronggen. Sun .
"Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe". United States.
doi:10.1103/PHYSREVD.75.084003.
```

```
@article{osti_21020372,
```

title = {Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe},

author = {Akbar, M. and Cai Ronggen},

abstractNote = {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} 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.},

doi = {10.1103/PHYSREVD.75.084003},

journal = {Physical Review. D, Particles Fields},

number = 8,

volume = 75,

place = {United States},

year = {Sun Apr 15 00:00:00 EDT 2007},

month = {Sun Apr 15 00:00:00 EDT 2007}

}