Caloric curves of self-gravitating fermions in general relativity

We study the nature of phase transitions between gaseous and condensed states in the self-gravitating Fermi gas at finite temperature in general relativity. The condensed states can represent compact objects such as white dwarfs, neutron stars, or dark matter fermion balls. The caloric curves depend on two parameters: the system size R and the particle number N. When N NOV, there is no equilibrium state below a critical energy and below a critical temperature. In that case, the system is expected to collapse toward a black hole. We plot the caloric curves of the general relativistic Fermi gas, study the different types of phase transitions that occur in the system, and determine the phase diagram in the (R, N) plane. The nonrelativistic results are recovered for N ≪ NOV and R ≫ ROV with NR3 fixed. The classical (non quantum) results are recovered for N ≫ NOV and R ≫ ROV with N∕R fixed. We discuss the commutation of the limits c → +∞ and ℏ → 0. We study the relativistic corrections to the nonrelativistic caloric curves and the quantum corrections to the classical caloric curves. We highlight a situation of physical interest where a self-gravitating Fermi gas, by cooling, first undergoes a phase transition toward a compact object (white dwarf, neutron star, dark matter fermion ball), then collapses into a black hole. This situation occurs in the microcanonical ensemble when NOV