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- 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, _^=^2, and in the fermion case the Dirac equation in general relativity ^_= (where =mc). The coefficients of the metric g_ are determined by the Einstein equations with a source term given by the mean value 〈|T_|〉 of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector 〈| 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 to 10^54 g cm^-3, and number of particles from 10 to 10^40) as well as in the relativistic region (density 10^54 g cm^-3, 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 (Planckmass) {m}^210^40 (for m10^-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^2m^5N^4{3}^-2 and reaches a value mc^2 for NN₂ₑ₈ₓ (Planckmass) {m}^310^57 (for m10^-24 g, implying mass 10^33 g, radius 10^6 cm, density 10^15 g/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/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.