Statistical mechanics of self-gravitating systems in general relativity: I. The quantum Fermi gas

Abstract

We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all the necessary equations that are needed to construct the caloric curves of selfgravitating fermions in general relativity as done in recent works. We consider the nonrelativistic limit c → +∞ and recover the equations obtained within the framework of Newtonian gravity. We also discuss the inequivalence of statistical ensembles as well as the relation between the dynamical and thermodynamical stability of self-gravitating systems in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONSelf-gravitating fermions play an important role in different areas of astrophysics. They appeared in the context of white dwarfs, neutron stars and dark matter halos where the fermions are electrons, neutrons and massive neutrinos respectively. We start by a brief history of the subject giving an exhaustive list of references. 1 Soon after the discovery of the quantum statistics by Fermi [3,4] and Dirac [5] in 1926, Fowler [6] used this "new thermodynamics" to solve the puzzle of the extreme high density of white dwarfs, which could not be explained by classical physics [7]. He understood that white dwarfs owe their stability to the quantum pressure of the degenerate electron gas resulting from the Pauli exclusion principle [8]. He considered a completely degenerate electron gas at T = 0 based on the fact that the temperature in white dwarfs is much smaller than the Fermi temperature (T ≪ T F ). He also used Newtonian gravity which is a very good approximation to describe white dwarfs in general. The first models [9-11] of white dwarfs were based on the nonrelativistic equation of state of a Fermi gas and provided the corresponding mass-radius relation. Stoner [9] developed an analytical approach based on a uniform density approximation for the star. Chandrasekhar [11] derived the exact mass-radius relation of nonrelativistic white dwarfs by applying the theory of polytropes of index n = 3/2 [12]. It was then realized that special relativity must be taken into account at high densities. When the relativistic equation of state is employed it was found that white dwarfs can exist only below a maximum mass M max = 1.42 M ⊙ [13-25], now known as the Chandrasekhar limiting mass. Frenkel [13] was the first to mention that relativistic effects become important when the mass of white dwarfs becomes larger than the solar mass but he did not envision the existence of an upper mass limit. The maxi...

Predictive model of BEC dark matter halos with a solitonic core and an isothermal atmosphere

“…These core-halo structures have been computed in [88] in relation to the Lynden-Bell theory of violent relaxation but they also appear in numerous works on fermionic DM halos in which the DM particle is a fermion like a massive neutrino (see the Introduction of Ref. [42] and Sec. V.A of [79] for an exhaustive list of references).…”

“…On the other hand, the Lynden-Bell DF (D5), and more generally any DF of the form f = f (ǫ), implies the condition of hydrostatic equilibrium (see, e.g., [42] and the Remark of Appendix F)…”

“…By using the Madelung [50] transforma- 3 Some authors have considered the case where the DM particle is a fermion like a massive neutrino (see the Introduction of Ref. [42] for a short review and an exhaustive list of references). In this model, gravitational collapse is prevented by the quantum pressure arising from the Pauli exclusion principle.…”

New Bounds for the Mass of Warm Dark Matter Particles Using Results from Fermionic King Model

“…It requires a relativistic approach analogous the one developed by Chavanis and Alberti in Refs. [41,[74][75][76], but including the incidence of evaporation effects. In general, one could expected many qualitative behaviors similar to the ones reported in Refs.…”

“…In general, one could expected many qualitative behaviors similar to the ones reported in Refs. [41,74,76] replacing the box radius by the tidal radius. Accordingly, the relativistic approach of fermionic King model should be hallmarked by three characteristic lengths: the tidal radius R, the Fermi radius R F given by Equation (24), and the Schwarzschild radius R S = GM/c 2 .…”