Caloric curves of classical self-gravitating systems in general relativity

TL;DR

This paper analyzes the caloric curves of classical self-gravitating systems in general relativity, revealing how relativistic effects influence stability and collapse behavior through the dependence on the compactness parameter.

Contribution

It introduces a detailed analysis of caloric curves in relativistic self-gravitating systems, highlighting the impact of the compactness parameter on stability and the merging of cold and hot spirals.

Findings

Caloric curves form double spirals depending on the compactness parameter.

Relativistic effects cause the spirals to shrink and eventually disappear.

System stability decreases as the compactness parameter increases.

Abstract

We determine the caloric curves of classical self-gravitating systems at statistical equilibrium in general relativity. In the classical limit, the caloric curves of a self-gravitating gas depend on a unique parameter $\nu=GNm/Rc^2$, called the compactness parameter, where $N$ is the particle number and $R$ the system's size. Typically, the caloric curves have the form of a double spiral. The "cold spiral", corresponding to weakly relativistic configurations, is a generalization of the caloric curve of nonrelativistic classical self-gravitating systems. The "hot spiral'", corresponding to strongly relativistic configurations, is similar (but not identical) to the caloric curve of the ultrarelativistic self-gravitating black-body radiation. We introduce two types of normalization of energy and temperature in order to obtain asymptotic caloric curves describing respectively the cold and the hot spirals in the limit $\nu\rightarrow 0$. As the number of particles increases, the cold and the hot spirals approach each other, merge at $\nu'_S=0.128$, form a loop above $\nu_S=0.1415$, reduce to a point at $\nu_{\rm max}=0.1764$, and finally disappear. Therefore, the double spiral shrinks when the compactness parameter $\nu$ increases, implying that general relativistic effects render the system more unstable. We discuss the nature of the gravitational collapse at low and high energies with respect to a dynamical (fast) or a thermodynamical (slow) instability.