Regular Article

Caloric curves of self-gravitating fermions in general relativity

¹ Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France
² Living Systems Research, Roseggerstraße 27/2, A-9020 Klagenfurt am Wörthersee, Austria

^a e-mail: chavanis@irsamc.ups-tlse.fr

Received: 13 November 2019
Received in final form: 16 June 2020
Accepted: 10 September 2020
Published online: 11 November 2020

Abstract

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 < N_OV, where N_OV is the Oppenheimer–Volkoff limit, there exists an equilibrium state for any value of the temperature T and energy E as in the nonrelativistic case [P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)]. Gravitational collapse is prevented by quantum mechanics (Pauli’s exclusion principle). When N > N_OV, 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 ≪ N_OV and R ≫ R_OV with N^R3 fixed. The classical (non quantum) results are recovered for N ≫ N_OV and R ≫ R_OV 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 N_OV < N < 3.73 N_OV. We also relate the phase transitions from a gaseous state to a core-halo state in the microcanonical ensemble to the onset of red-giant structure and to the supernova phenomenon.