The contraction of gravitating spheres

Abstract

Earlier ideas associating an invariant integral of the energy invariant with the number of nucleons in a gravitating body are shown to be fallacious, and thus do not provide a means of following through the contraction of such a body. It is shown how the full field equations of general relativity give a feasible and rigorous method of examining contracting models. Schwarzschild-type co-ordinates are introduced and are used to examine the slow adiabatic contraction of a sphere of constant density. The particle paths are found and the pressure-density relation permitting such slow adiabatic contraction is examined. It is shown that the simple 4/3 power law of Newtonian theory has to be replaced by a steeper dependence of pressure on density for high gravitational potentials. Radiation co-ordinates are introduced to examine radiating contracting systems, and equations fully specifying such a system are obtained. A simple example is given in outline to illustrate the method.