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Gravitational phase transitions and instabilities of self-gravitating fermions in general relativity

We discuss the occurrence of gravitational phase transitions and instabilities in a gas of self-gravitating fermions within the framework of general relativity. In the classical (nondegenerate) limit, the system undergoes a gravitational collapse at low energies $E<E_c$ and low temperatures $T<T_c$. This is called &#34;gravothermal catastrophe&#34; in the microcanonical ensemble and &#34;isothermal collapse&#34; in the canonical ensemble. When quantum mechanics is taken into account and when the particle number is below the Oppenheimer-Volkoff limit ($N<N_{\rm OV}$), complete gravitational collapse is prevented by the Pauli exclusion principle. In that case, the Fermi gas undergoes a gravitational phase transition from a gaseous phase to a condensed phase. The condensed phase represents a compact object like a white dwarf, a neutron star, or a dark matter fermion ball. When $N>N_{\rm OV}$, there can be a subsequent gravitational collapse below a lower critical energy $E<E&#39;&#39;_c$ or a lower critical temperature $T<T&#39;_c$ leading presumably to the formation of a black hole. The evolution of the system is different in the microcanonical and canonical ensembles. In the microcanonical ensemble, the system takes a &#34;core-halo&#34; structure. The core consists in a compact quantum object or a black hole while the hot halo is expelled at large distances. This is reminiscent of the red giant structure of low-mass stars or the implosion-explosion of massive stars (supernova). In the canonical ensemble, the system collapses as a whole towards a compact object or a black hole. This is reminiscent of the implosion of supermassive stars (hypernova).