Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

TL;DR

This paper analyzes the thermodynamic behavior of classical self-gravitating gases in general relativity, revealing how relativistic effects influence stability and the structure of caloric curves, with new insights into the ultrarelativistic limit.

Contribution

It introduces the asymptotic behavior of caloric curves in the ultrarelativistic limit, connecting relativistic and nonrelativistic regimes, and compares these models with black-body radiation and truncated distributions.

Findings

Caloric curve forms a double spiral that shrinks with increasing compactness.

In the nonrelativistic limit, the caloric curve approaches a single cold spiral.

In the ultrarelativistic limit, the caloric curve approaches a single hot spiral.

Abstract

We study the statistical mechanics of classical self-gravitating systems confined within a box of radius $R$ in general relativity. It has been found that the caloric curve $T_{\infty}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $\nu=GNm/Rc^2$. The double spiral shrinks as $\nu$ increases and finally disappears when $\nu_{\rm max}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $\nu$ decreases. Using a normalization $\Lambda=-ER/GN^2m^2$ and $\eta=GNm^2/R k_B T_{\infty}$ appropriate to the nonrelativistic limit, and considering $\nu\rightarrow 0$, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization ${\cal M}=GM/Rc^2$ and ${\cal B}={Rc^4}/{GNk_B T_{\infty}}$ appropriate to the ultrarelativistic limit, and considering $\nu\rightarrow 0$, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the self-gravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.