  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.
