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Relativistic Gravothermal Instabilities

The thermodynamic instabilities of the self-gravitating, classical ideal gas are studied in the case of static, spherically symmetric configurations in General Relativity taking into account the Tolman-Ehrenfest effect. One type of instabilities is found at low energies, where thermal energy becomes too weak to halt gravity and another at high energies, where gravitational attraction of thermal pressure overcomes its stabilizing effect. These turning points of stability are found to depend on the total rest mass $\mathcal{M}$ over the radius $R$. The low energy instability is the relativistic generalization of Antonov instability, which is recovered in the limit $G\mathcal{M} \ll R c^2$ and low temperatures, while in the same limit and high temperatures, the high energy instability recovers the instability of the radiation equation of state. In the temperature versus energy diagram of series of equilibria, the two types of gravothermal instabilities make themselves evident as a double spiral! The two energy limits correspond also to radius limits. So that, stable static configurations exist only in between two marginal radii for any fixed energy with negative thermal plus gravitational energy. Ultimate limits of rest mass, as well as total mass-energy, are reported. Applications to neutron cores are discussed.