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

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 \(\nu =GNm/Rc^2\) (Roupas in Class Quantum Grav 32:135023, 2015; Alberti and Chavanis in arXiv:1908.10316). The double spiral shrinks as \(\nu \) increases and finally disappears when \(\nu _{{\mathrm{max}}}=0.1764\). 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 \(\varLambda =-ER/GN^2m^2\) and \(\eta =GNm^2/R k_\mathrm{B} T_{\infty }\) appropriate to the nonrelativistic limit, and considering \(\nu \rightarrow 0\), the hot spiral goes to infinity and the caloric curve tends to a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization \({{{\mathcal {M}}}}=GM/Rc^2\) and \({{{\mathcal {B}}}}={Rc^4}/{GNk_\mathrm{B} T_{\infty }}\) appropriate to the ultrarelativistic limit, and considering \(\nu \rightarrow 0\), the cold spiral goes to infinity and the caloric curve tends to 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.