Systems of SelfGravitating Particles in General Relativity and the Concept of an Equation of State
Abstract
A method of selfconsistent fields is used to study the equilibrium configurations of a system of selfgravitating scalar bosons or spin 1/2 fermions in the ground state without using the traditional perfectfluid approximation or equation of state. The manyparticle system is described by a secondquantized free field, which in the boson case satisfies the KleinGordon equation in general relativity, ∇_{α}∇^{α}φ=μ^{2}φ, and in the fermion case the Dirac equation in general relativity γ^{α}∇_{α}ψ=μψ (where μ=mcℏ). The coefficients of the metric g_{αβ} are determined by the Einstein equations with a source term given by the mean value <φT_{μν}φ> of the energymomentum tensor operator constructed from the scalar or the spinor field. The state vector <φ 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 energymomentum tensor is anisotropic. A critical mass is found for a system of N~[(Planck mass)m]^{2}~10^{40} (for m~10^{25} g) selfgravitating 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^{43}ℏ^{2} and reaches a value ~mc^{2} for N~N_{crit}~[(Planck mass)m]^{3}~10^{57} (for m~10^{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 selfconsistent 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 OppenheimerVolkoff treatment in extremely good approximation. There exists a gravitational spinorbit 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.
 Publication:

Physical Review
 Pub Date:
 November 1969
 DOI:
 10.1103/PhysRev.187.1767
 Bibcode:
 1969PhRv..187.1767R