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

A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin-½ 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, ${\nabla }_{\alpha }{\nabla }^{\alpha }\phi ={\mu }^2\phi$, and in the fermion case the Dirac equation in general relativity ${\gamma }^{\alpha }{\nabla }_{\alpha }\psi =\mu \psi$ (where $\mu =\frac{\mathrm{mc}}{\hslash }$). The coefficients of the metric $g_{\alpha \beta }$ are determined by the Einstein equations with a source term given by the mean value $〈\phi \left|T_{\mu \nu }\right|\phi 〉$ of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector $〈\phi |$ 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 ${10}^{-80\mathrm{}}$ to ${10}^{54}$ g ${\mathrm{cm}}^{-3\mathrm{}}$, and number of particles from 10 to ${10}^{40}$) as well as in the relativistic region (density ∼${10}^{54}$ g ${\mathrm{cm}}^{-3\mathrm{}}$, number of particles ∼${10}^{40}$), 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 $N\sim {\left[\frac{\left(\mathrm{Planck}\mathrm{mass}\right)}{m}\right]}^2\sim {10}^{40}$ (for $m\sim {10}^{-25\mathrm{}}$ g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For $N$ fermions, the binding energy of typical particles is $G^2{m}^5{N}^{\frac{4}{3}}{\hslash }^{-2\mathrm{}}$ and reaches a value $\sim {\mathrm{mc}}^2$ for $N\sim N_{\mathrm{crit}}\sim {\left[\frac{\left(\mathrm{Planck}\mathrm{mass}\right)}{m}\right]}^3\sim {10}^{57}$ (for $m\sim {10}^{-24\mathrm{}}$ g, implying mass ∼${10}^{33}$ g, radius ∼${10}^6$ cm, density ∼${10}^{15}$ 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.