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 1054 g cm-3, and number of particles from 10 to 1040) as well as in the relativistic region (density 1054 g cm-3, number of particles 1040), 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[(Planck mass) m]2 1040 (for m 10-25 g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For N fermions, the binding energy of typical particles is G2m5N43-2 and reaches a value mc2 for N Ncrit[(Planck mass) m]3 1057 (for m 10-24 g, implying mass 1033 g, radius 106 cm, density 1015 g/cm3). 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 1042 g/cm3, 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. © 1969 The American Physical Society.
