---
abstract: 'A system of self-gravitating massive fermions is studied in the framework of the general-relativistic Thomas-Fermi model. We study the properties of the free energy functional and its relation to Einstein’s field equations. A self-gravitating fermion gas we then describe by a set of Thomas-Fermi type self-consistency equations.'
author:
- |
    Neven Bilić$^1$ and Raoul D. Viollier$^2$\
    $^1$Rudjer Bošković Institute, 10000 Zagreb, Croatia\
    E-mail: bilic@thphys.irb.hr\
    $^2$Department of Physics, University of Cape Town,\
    Rondebosch 7701, South Africa, E-mail: viollier@physci.uct.ac.za
title: ' General-Relativistic Thomas-Fermi model '
---

Thermodynamical properties of the self-gravitating fermion gas have been extensively studied in the framework of the Thomas-Fermi model \[1-6\]. The system was investigated in the nonrelativistic Newtonian limit. The canonical and grand-canonical ensembles for such a system have been shown to have a nontrivial thermodynamical limit [@thi; @her1]. Under certain conditions this system will undergo a phase transition that is accompanied by a gravitational collapse [@her1; @mes] which may have important astrophysical and cosmological implications [@bil1; @bil2].

In this paper we formulate the general-relativistic version of the model. The effects of general relativity become important if the total rest-mass of the system is close to the Oppenheimer-Volkoff limit [@opp]. There are three main features that distinguish the relativistic Thomas-Fermi theory from the Newtonian one: [*i*]{}) the equation of state is relativistic [*ii*]{}) the temperature and chemical potential are metric dependent local quantities [*iii*]{}) the gravitational potential satisfies Einstein’s field equations (instead of Poisson’s equation).

Let us first discuss the general properties of a canonical, self-gravitating relativistic fluid. Consider a nonrotating fluid consisting of $N$ particles in a spherical volume of radius $R$ in equilibrium at non-zero temperature. We denote by $u_{\mu}$ , $p$, $\rho$, $n$ and $\sigma$ the velocity, pressure, energy density, particle number density and entropy density of the fluid. A canonical ensemble is subject to the constraint that the number of particles $$\int_{\Sigma} n\, u^{\mu}d\Sigma_{\mu}
=N
\label{eq26}$$ should be fixed. The spacelike hypersurface $\Sigma$ that contains the fluid is orthogonal to the time-translation Killing vector field $k^{\mu}$ which is related to the velocity of the fluid $$k^{\mu}=\xi u^{\mu}\, ;  \;\;\;\;\;\;
\xi=(k^{\mu}k_{\mu})^{1/2}.
\label{eq50}$$ The metric generated by the mass distribution is static, spherically symmetric and asymptotically flat, i.e. $$ds^2=\xi^2 dt^2 -\lambda^2 dr^2 -
     r^2(d\theta^2+\sin \theta d\phi^2).
\label{eq00}$$ $\xi$ and $\lambda$ may be represented in terms of the gravitational potential and mass $$\xi=e^{\varphi (r)},
\label{eq01}$$ $$\lambda=\left(1-\frac{2{\cal{M}}(r)}{r}\right)^{-1/2}
\label{eq10}$$ with $${\cal{M}}(r)=\int^r_0 dr'\, 4\pi r'^2 \rho(r') \, .
\label{eq11}$$

The temperature $T$ and chemical potential $\mu$ are metric dependent local quantities. Their space-time dependence may be derived from the equation of hydrostatic equilibrium [@lan] $$\partial_{\nu}p=-(p+\rho)\xi^{-1}\partial_{\nu}\xi ,
\label{eq17}$$ and the thermodynamic identity (Gibbs-Duhem relation) $$d\frac{p}{T}=
 n d\frac{\mu}{T}-\rho d\frac{1}{T}.
\label{eq18}$$ The condition that the heat flow and diffusion vanish [@isr] $$\frac{\mu}{T}={\rm const}
\label{eq19}$$ together with (\[eq17\]) and (\[eq18\]) implies $$T \xi=T_0\, ; \;\;\;\;\;\;
\mu \xi=\mu_0  \, ,
\label{eq21}$$ where $T_0$ and $\mu_0$ are constants equal to the temperature and chemical potential at infinity. The temperature $T_0$ may be chosen arbitrarily as the temperature of the heat-bath. The quantity $\mu_0$ in a canonical ensemble is an implicit functional of $\xi$ owing to the constraint (\[eq26\]). First equation in (\[eq21\]) is the well known Tolman condition for thermal equilibrium in a gravitational field [@tol].

Following Gibbons and Hawking [@gib] we postulate the free energy of the canonical ensemble as $$F=M-\int_{\Sigma} T\sigma \, k^{\mu}d\Sigma_{\mu} \, ,
\label{eq30}$$ where $M$ is the total mass as measured from infinity. The entropy density of a relativistic fluid may be expressed as $$\sigma=\frac{1}{T}(p+\rho-\mu n).
\label{eq16}$$ Based on equation (\[eq21\]) the free energy may be written in the form analogous to ordinary thermodynamics $$F=M-T_0 S
\label{eq60}$$ with $M={\cal{M}}(R)$ and the total entropy $S$ defined as $$S = \int_0^R dr\,4\pi r^2 \lambda
\frac{1}{T}(p+\rho)-\frac{\mu_0}{T_0} N  ,
\label{eq70}$$ where we have employed the spherical symmetry to replace the proper volume integral as $$\int_{\Sigma} u^{\mu}d\Sigma_{\mu}
= \int_0^R dr 4\pi r^2 \lambda .
\label{eq80}$$

The following theorem demonstrates how the extrema of the free energy are related to the solutions of Einstein’s field equation.

Among all momentarily static, spherically symmetric configurations $\{\xi(r),{\cal{M}}(r)\}$ which for a given temperature $T_0$ at infinity contain a specified number of particles $$\int_0^R 4\pi r^2 dr \, \lambda(r)  n(r) = N
\label{eq25}$$ within a spherical volume of a given radius $R$, those and only those configurations that extremize the quantity F defined by [(\[eq60\])]{} will satisfy Einstein’s field equation $$\label{eq22}
\frac{d\xi}{dr}=\xi\frac{{\cal{M}}+4\pi r^3 p}{r(r-2{\cal{M}})} \, ,$$ with the boundary condition $$\xi(R)=\left(1-\frac{2 M}{R}\right)^{1/2}.
\label{eq23}$$

[**Proof.**]{} By making use of the identity (\[eq18\]), and the fact that $\delta(\mu/T)=\delta(\mu_0/T_0)$ and that $N$ is fixed by the constraint (\[eq25\]), from equations (\[eq60\]) and (\[eq70\]) we find $$\delta F= \delta M -
\int_0^R dr\, 4\pi r^2 \frac{T_0}{T}(p+\rho)
\delta \lambda
-  \int_0^R dr\, 4\pi r^2 \lambda \frac{T_0}{T} \delta\rho \, .
\label{eq90}$$ The variations $\delta\lambda$ and $\delta\rho$ can be expressed in terms of the variation $\delta {\cal{M}}(r)$ and its derivative $$\frac{d\delta {\cal{M}}}{dr} =4\pi r^2 \delta\rho.
\label{eq93}$$ This gives $$\delta F= \delta M -
\int_0^R dr\, 4\pi r^2
 \frac{T_0}{T}(p+\rho)
\frac{\partial\lambda}{\partial {\cal{M}}}
\delta {\cal{M}}
-\int_0^R dr\, \lambda\frac{T_0}{T}\frac{d\delta {\cal{M}}}{dr}.
\label{eq91}$$ By partial integration of the last term and replacing $T_0/T$ by $\xi$, we find $$\delta F =
\left[1-\lambda(R)\xi(R)\right]\delta M
- \int_0^R dr\, \left[4\pi r^2 \xi (p+\rho)
\frac{\partial\lambda}{\partial {\cal{M}}}
-\frac{d}{dr}(\lambda\xi)\right]\delta {\cal{M}} \, ,
\label{eq94}$$ where $\delta {\cal{M}}(r)$ is an arbitrary variation on the interval $[0,R]$, except for the constraint $\delta {\cal{M}}(0)=0$. Therefore $\delta F$ will vanish if and only if $$4\pi r^2 \xi (p+\rho)
\frac{\partial\lambda}{\partial {\cal{M}}}
-\frac{d}{dr}(\lambda\xi) =0
\label{eq95}$$ and $$1-\lambda(R)\xi(R) =0.
\label{eq96}$$ Using (\[eq10\]) and (\[eq11\]), we can write equation (\[eq95\]) in the form (\[eq22\]), and equation (\[eq96\]) gives the desired boundary condition (\[eq23\]). Thus, $\delta F=0$ if and only if a configuration $\{\xi,{\cal{M}}\}$ satisfies equation (\[eq22\]) with (\[eq23\]) as was to be shown.\
[*Remark 1.*]{} A solutions to equation (\[eq22\]) is dynamically stable if the free energy assumes a minimum.\
[*Remark 2.*]{} Our Theorem 1 is a finite temperature generalization of the result obtained for cold, catalyzed matter [@har].

We now proceed to the formulation of the general-relativistic Thomas-Fermi model. Consider the case of a self-gravitating gas consisting of $N$ fermions with the mass $m$ contained in a sphere of radius $R$. The equation of state may be represented in a parametric form using the well known momentum integrals over the Fermi distribution function [@ehl] $$n   = g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\frac{1}{1+e^{E/T-\mu/T}} \, ,
\label{eq13}$$ $$\rho = g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\frac{E}{1+e^{E/T-\mu/T}} \, ,
\label{eq14}$$ $$p = g T \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\ln (1+e^{-E/T+\mu/T}) \, ,
\label{eq15}$$ where $g$ denotes the spin degeneracy factor, $T$ and $\mu$ are local temperature and chemical potential, respectively, as defined in equation (\[eq21\]), and $E=\sqrt{m^2+q^2}$. Introducing a single parameter $$\alpha=
\frac{\mu}{T}
=\frac{\mu_0}{T_0} \, ,
\label{eq100}$$ and the substitution $$\xi=
\frac{\mu_0}{m}\psi \, ,
\label{eq102}$$ equations (\[eq13\])-(\[eq15\]) may be written in the form $$n   = g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\frac{1}{1+e^{(E\psi-m)\alpha}} \, ,
\label{eq104}$$ $$\rho= g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\frac{E}{1+e^{(E\psi-m)\alpha}} \, ,
\label{eq106}$$ $$p   = g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\frac{q^2}{3E}\frac{1}{1+e^{(E\psi-m)\alpha}} \, ,
\label{eq108}$$ Field equations are given by $$\frac{d\psi}{dr}=\psi\frac{{\cal{M}}+4\pi r^3 p}{r(r-2{\cal{M}})} \, ,
\label{eq42}$$ $$\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho,
\label{eq43}$$ with the boundary conditions $$\psi(R)=\frac{m}{\mu_0}\left(1-\frac{2 {\cal{M}}(R)}{R}\right)^{1/2}
\, ; \;\;\;\;\;
{\cal{M}}(0)=0.
\label{eq44}$$ Finally, the constraint (\[eq26\]) may be written as $$\int_0^Rdr\, 4\pi r^2 (1-2{\cal{M}}/r)^{-1/2}\, n(r)=N .
\label{eq45}$$ Given the ratio $\alpha$, the radius $R$, and the number of fermions $N$, the set of self-consistency equations (\[eq104\])-(\[eq45\]) defines the Thomas-Fermi equation. One additional important requirement is that a solution of the self-consistency equations (\[eq104\])-(\[eq45\]) should minimize the free energy defined by (\[eq60\]).

We now show that a solution of the Thomas-Fermi equation exists provided the number of fermions is smaller than a certain number $N_{\rm max}$ that depends on $\alpha$ and $R$. From (\[eq106\]) and (\[eq108\]) it follows that for any $\alpha>0$, the equation of state $\rho(p)$ is an infinitely smooth function and $d\rho /dp > 0$ for $p > 0$. Then, as shown by Rendall and Schmidt [@ren], there exist for any value of the central density $\rho_0$ a unique static, spherically symmetric solution of field equations with $\rho \rightarrow 0 $ as $r$ tends to infinity. In that limit ${\cal M}(r)\rightarrow\infty$, as may easily be seen by analysing the $r\rightarrow \infty$ limit of equations (\[eq42\]) and (\[eq43\]). However, the enclosed mass $M$ and the number of fermions $N$ within a given radius $R$ will be finite. We can then cut off the matter from $R$ to infinity and join on the empty space Schwarzschild solution by making use of equation (\[eq44\]). This equation together with (\[eq100\]) fixes the chemical potential and the temperature at infinity. Furthermore, it may be shown that our equation of state obeys a $\gamma$-low asymptotically at high densities, i.e., $\rho=$ const $n^{\gamma}$ and $p=(\gamma-1) \rho$, with $\gamma=4/3$. It is well known [@har] that in this case, there exist a limiting configuration $\{ \psi_{\infty}(r),{\cal{M}}(r)_{\infty}\}$ such that $M$ and $N$ approach non-zero values $M_{\infty}$ and $N_{\infty}$, respectively, as the central density $\rho_{0}$ tends to infinity. Thus, the quantity $N$ is a continuous function of $\rho_{0}$ on the interval $0 \leq \rho_0 < \infty$, with $N=0$ for $\rho_{0}=0$, and $N=N_{\infty}$ as $\rho_{0}\rightarrow\infty$. The range of $N$ depends on $\alpha$ and $R$ and its upper bound may be denoted by $N_{\rm max}(R,\alpha)$. Thus, for given $\alpha$, $R$ and $N<N_{\rm max}(R,\alpha)$ the set of self-consistency equations (\[eq104\])-(\[eq45\]) has at least one solution.

Next we show that, in the Newtonian limit, we recover the nonrelativistic Thomas-Fermi equation. Using the nonrelativistic chemical potential $\mu_{NR}=\mu_0-m$ and the approximation $\xi=e^{\varphi}\simeq 1+\varphi$, $E\simeq m+q^2/2m$ and ${\cal{M}}/r \ll 1$ , we find the usual Thomas-Fermi self-consistency equations [@mes; @bil1] $$n=\frac{\rho}{m}
 = g \int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\left(1+\exp(\frac{q^2}{2mT_0}+\frac{m}{T_0}\varphi
 -\frac{\mu_{NR}}{T_0}) \right)^{-1} \, ,
\label{eq49}$$ $$\frac{d\varphi}{dr}=\frac{{\cal{M}}}{r^2} \, ;
\;\;\;\;
\frac{d{\cal{M}}}{dr}=4\pi r^2 \rho \, ,
\label{eq41}$$ $$\varphi(R)=-\frac{m N}{R}
\, ; \;\;\;
{\cal{M}}(0)=0,
\label{eq47}$$ $$\int_0^R dr\,4\pi r^2 n(r)=N.
\label{eq46}$$ The free energy (\[eq60\]) in the Newtonian limit yields $$F=m N +\mu_{NR} N - \frac{1}{2}\int_0^R dr \, 4\pi r^2 n\varphi
-\int_0^R dr \, 4\pi r^2 p
\label{eq40}$$ with $$p= g T_0\int^{\infty}_{0} \frac{d^3q}{(2\pi)^3}\,
\ln\left(1+\exp(-\frac{q^2}{2mT_0}-\frac{m}{T_0}\varphi
 +\frac{\mu_{NR}}{T_0}) \right) \, ,
\label{eq48}$$ which, up to a constant, equals the Thomas-Fermi free energy [@her2].

A straightforward thermodynamic limit $N\rightarrow\infty$ as discussed by Hertel, Thirring and Narnhofer [@her1; @her2] is in our case not directly applicable. First, in contrast to the non-relativistic case, there exists, as we have demonstrated, a limiting configuration with maximal $M$ and $N$. Second, the scaling properties of the relativistic Thomas-Fermi equation are quite distinct from the nonrelativistic one. The following scaling property can be easily shown: If the configuration $\{\psi(r),{\cal{M}}(r)\}$ is a solution of the self consistency equations (\[eq104\])-(\[eq45\]), then the configuration $\{\tilde{\psi}=\psi(A^{-1}r),\tilde{{\cal{M}}}
 =A{\cal{M}}(A^{-1}r);A>0\}$ is also a solution with the rescaled fermion number $\tilde{N}=A^{3/2}N$, radius $\tilde{R}=AR$, asymptotic temperature $\tilde{T_0}=A^{-1/2}T_0$, and fermion mass $\tilde{m}=A^{-1/2}m$. The free energy is then rescaled as $\tilde{F}=AF$. Therefore, there exist a thermodynamic limit of $N^{-2/3}F$, with $N^{-2/3}R$, $N^{1/3}T_0$, $N^{1/3}m$ approaching constant values when $N\rightarrow\infty$. In that limit the Thomas-Fermi equation becomes exact.

It is obvious that application of this model to astrophysical systems should work very well if the interactions among individual particles are negligible. This applies, for example, to weakly interacting quasidegenerate heavy neutrino or neutralino matter \[6,7,16-19\]. or perhaps even to collisionless stellar systems [@shu; @chav].

Acknowledgment {#acknowledgment .unnumbered}
--------------

We acknowledge useful discussions with D. Tsiklauri. This work was supported by the Foundation for Fundamental Research (FFR) and the Ministry of Science and Technology of the Republic of Croatia under Contract No. 00980102.

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