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

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 $T_{\mathrm{\infty }}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $\nu =GNm\mathrm{/}R{c}^2$. The double spiral shrinks as $\nu$ increases and finally disappears when ${\nu }_{\mathrm{m}\mathrm{a}\mathrm{x}}=0.1764$. 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 $\mathrm{\Lambda }=-ER\mathrm{/}G{N}^2{m}^2$ and $\eta=GNm^2/R k_B T_{\infty}$appropriatetothenonrelativisticlimit,andconsidering$\nu\rightarrow 0$, 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 $\nu \to 0$, 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.