Equilibrium of a Large Assembly of Particles in General Relativity

A new approach to the problem of the equilibrium of a relativistic configuration is proposed, taking into account quantum-gravitational effects. This approach is based upon the quantization of the motion of the particles in the gravitational field. The case of an assembly of spinless particles is investigated by using the formalism of second quantization and starting from a Klein-Gordon equation which includes in the operator ${\nabla }_i{\nabla }^i$ the gravitational field. The mean values $〈T_{\mathrm{ik}}〉$ of the operator $T_{\mathrm{ik}}$ are computed. It is then possible to derive from the Einstein equations a system of differential equations which represents the equilibrium state for an assembly of a large number of particles. The corresponding problem for Fermi particles can be investigated in a similar way.