Physics
Scientific paper
Nov 1969
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1969phrv..187.1767r&link_type=abstract
Physical Review, vol. 187, Issue 5, pp. 1767-1783
Physics
225
Scientific paper
A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin- 1/2 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, ∇α∇αφ=μ2φ, and in the fermion case the Dirac equation in general relativity γα∇αψ=μψ (where μ=mcℏ). The coefficients of the metric gαβ are determined by the Einstein equations with a source term given by the mean value <φ|Tμν|φ> of the energy-momentum 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 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 G2m5N43ℏ-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.
Bonazzola Silvano
Ruffini Remo
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