Nonlocal Properties of Fermionic Matter in Strong Gravitational Fields
A self-consistent solution of the coupled two-particle-Dirac and Einstein field equations in the mean field Hartree approximation is presented. The purpose of this work is to answer the question whether the nonlocal properties introduced by describing the matter field quantum mechanically by Dirac wave functions give rise to major modifications of the equation of state of the fermionic matter, compared to the classical Thomas-Fermi-model for a gravitating system of spin-1/2-particles, where the influence of all gravitational forces (i.e. of all derivatives of the metric) is neglected and a liquid drop model for the energy-momentum-tensor results. Our numerical solution indeed leads to an energy-momentum-tensor for the total system with an anisotropic pressure, the radial pressure being up to 10 percent smaller than the pressure in the angular directions. This anisotropy reflects the nonlocal structure of the Dirac wave function and thus is of principal interest, but it is found to be too small to play a role in the problem of gravitational collapse of macroscopic matter distributions.