Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State

Abstract

A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin-\textonehalf{} fermions in the ground state without using the traditional perfect-fluid approximation or equation of state. The many-particle system is described by a second-quantized free field, which in the boson case satisfies the Klein-Gordon equation in general relativity, ${\ensuremath{\nabla}}_{\ensuremath{\alpha}}{\ensuremath{\nabla}}^{\ensuremath{\alpha}}\ensuremath{\varphi}={\ensuremath{\mu}}^{2}\ensuremath{\varphi}$, and in the fermion case the Dirac equation in general relativity ${\ensuremath{\gamma}}^{\ensuremath{\alpha}}{\ensuremath{\nabla}}_{\ensuremath{\alpha}}\ensuremath{\psi}=\ensuremath{\mu}\ensuremath{\psi}$ (where $\ensuremath{\mu}=\frac{\mathrm{mc}}{\ensuremath{\hbar}}$). The coefficients of the metric ${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$ are determined by the Einstein equations with a source term given by the mean value $〈\ensuremath{\varphi}|{T}_{\ensuremath{\mu}\ensuremath{\nu}}|\ensuremath{\varphi}〉$ of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector $〈\ensuremath{\varphi}|$ corresponds to the ground state of the system of many particles. In both cases, for completeness, a nonrelativistic Newtonian approximation is developed, and the corrections due to special and general relativity explicitly are pointed out. For $N$ bosons, both in the region of validity of the Newtonian treatment (density from ${10}^{\ensuremath{-}80}$ to ${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, and number of particles from 10 to ${10}^{40}$) as well as in the relativistic region (density \ensuremath{\sim}${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, number of particles \ensuremath{\sim}${10}^{40}$), we obtain results completely different from those of a traditional fluid analysis. The energy-momentum tensor is anisotropic. A critical mass is found for a system of $N\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{2}\ensuremath{\sim}{10}^{40}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}25}$ g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For $N$ fermions, the binding energy of typical particles is ${G}^{2}{m}^{5}{N}^{\frac{4}{3}}{\ensuremath{\hbar}}^{\ensuremath{-}2}$ and reaches a value $\ensuremath{\sim}{\mathrm{mc}}^{2}$ for $N\ensuremath{\sim}{N}_{\mathrm{crit}}\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{3}\ensuremath{\sim}{10}^{57}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}24}$ g, implying mass \ensuremath{\sim}${10}^{33}$ g, radius \ensuremath{\sim}${10}^{6}$ cm, density \ensuremath{\sim}${10}^{15}$ g/${\mathrm{cm}}^{3}$). For densities of this order of magnitude and greater, we have given the full self-consistent relativistic treatment. It shows that the concept of an equation of state makes sense only up to ${10}^{42}$ g/${\mathrm{cm}}^{3}$, and it confirms the Oppenheimer-Volkoff treatment in extremely good approximation. There exists a gravitational spin-orbit coupling, but its magnitude is generally negligible. The problem of an elementary scalar particle held together only by its gravitational field is meaningless in this context.