Relativistic gravitational phase transitions and instabilities of the Fermi gas

We describe microcanonical phase transitions and instabilities of the ideal Fermi gas in general relativity at nonzero temperature confined in the interior of a spherical shell. The thermodynamic behaviour is governed by the compactness of rest mass, namely of the total rest mass over radius of the system. For a fixed value of rest mass compactness, we study the caloric curves as a function of the size of the spherical box. At low compactness values, low energies and for sufficiently big systems the system is subject to a gravothermal catastrophe, which cannot be halted by quantum degeneracy pressure, and the system collapses. For small systems, there appears no instability at low energies. For intermediate sizes, between two marginal values, gravothermal catastrophe is halted and a microcanonical phase transition occurs from a gaseous phase to a condensed phase with a nearly degenerate core. The system is subject to a relativistic instability at low energy, when the core gets sufficiently condensed above the Oppenheimer-Volkoff limit. For sufficiently high values of rest mass compactness the microcanonical phase transitions are suppressed. They are replaced either by an Antonov type gravothermal catastrophe for sufficiently big systems or by stable equilibria for small systems. At high energies the system is subject to the 'relativistic gravothermal instability', identified by Roupas (2015 Class. Quantum Grav. 32 135023), for all values of compactness and any size.