# Entropy and Gravity

## Abstract

**:**

## 1. Introduction

^{3}the universe has infinitely large spatial extension. Then the evolution of “the entropy within a surface co-moving with the Hubble flow” is representative of the evolution of “the entropy of the universe”.

## 2. Entropy Change during Gravitational Contraction

^{3}and temperature ${T}_{1}\sim {10}^{4}$ K. There are about $N\sim {10}^{57}$ baryons in the Sun. The average density and temperature of the Sun are, respectively ${\rho}_{2}\sim {10}^{30}$ and ${T}_{2}\sim {10}^{7}$. The entropy change $\text{\Delta}{S}_{M}$ when the matter of the Sun changed from its state as a nearly homogeneous cosmic plasma 300,000 years after the Big Bang to a star, can be calculated from Equation (2). Using that $M={\rho}_{1}{V}_{1}={\rho}_{2}{V}_{2}$ leads to:

**Figure 1.**Thermal entropy (modulus an undetermined constant and in units of (3/2)Nk) as a function of R for a gas cloud contracting or expanding under the action of its own gravity. (a) Contraction. Here R < R

_{0}. Then the entropy first increases towards a maximum at R = R

_{0}/2, and then decreases. (b) Expansion. Here R > R

_{0}. The entropy of the gas increases.

**Figure 2.**Evolution of a system (a) in which gravity may be neglected, and (b) in a self gravitating system where the box is much larger than the Jean’s length of the gas it contains.

## 3. Entropy of Black Holes and Cosmic Horizons

## 4. The Weyl Curvature Hypothesis

- Expanding universe with large $\text{\Lambda}$: In the initial epoch the dust dominates and ${S}_{\text{G}1}$ is increasing linearly with $t$. The universe is becoming more and more inhomogeneous. At late times $\text{\Lambda}$ dominates, and ${S}_{\text{G}1}$ stops growing, evolving asymptotically towards a constant value. The universe inflates.
- Expanding universe with small $\text{\Lambda}$: Again the dust dominates initially. In this case ${S}_{\text{G}1}$ increases forever, but with a decreasing rate and again approaches a constant value asymptotically.
- Expanding universe with vanishing $\text{\Lambda}$: The quantity ${S}_{\text{G}1}$ again increases forever, this time approaching asymptotically a function $f\left(t\right)=c+b{t}^{p}$ where c and b are constants and $p=3$ if ${F}^{2}>1$ and $p=1$ if ${F}^{2}=1$. (The function $F\left(r\right)$ is defined in refs. [19] and [20].)
- Recollapsing universe: Due to the dust term the final singularity will be more inhomogeneous than the initial singularity.

## 5. The Weyl Curvature Hypothesis and Black Hole Entropy

^{2}is a quantity proportional to the Weyl curvature invariant. However, it cannot be given by eq.(32) since the Ricci curvature invariant vanishes in the Schwarzschild spacetime. So Rudjord et al. [21] replaced the Wainwright, Anderson expression (32) by:

## 6. Is There a Maximal Entropy for the Universe?

## 7. Entropy Change During the Inflationary Era

**Figure 3.**Time-variation of the candidate gravitational entropies ${S}_{G1}$ (upper curve) and ${S}_{G2}$ in a comoving volume during the beginning of the inflationary era.

^{-5}.

## 8. The Entropy of the Universe

## 9. Conclusions

## Acknowledgements

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Grøn, Ø.
Entropy and Gravity. *Entropy* **2012**, *14*, 2456-2477.
https://doi.org/10.3390/e14122456

**AMA Style**

Grøn Ø.
Entropy and Gravity. *Entropy*. 2012; 14(12):2456-2477.
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**Chicago/Turabian Style**

Grøn, Øyvind.
2012. "Entropy and Gravity" *Entropy* 14, no. 12: 2456-2477.
https://doi.org/10.3390/e14122456