On a Non-Static Fluid Sphere of Uniform Density in General Relativity

Abstract

Exact singularity free models for a spherically symmetric, non-static fluid of uniform density, but non-uniform pressure are investigated. It is found that the model is conformally flat and the pressure gradient is non-positive. The rate of change of (1/2π) × (circumference) as measured by an observer riding in a shell of matter is increasing throughout the sphere. If this "velocity" is smaller than the velocity of light, the surface of an oscillating or a bouncing model does not penetrate the Schwarzschild radius. For a model which continually contracts this "velocity" is equal to or greater than the velocity of light before or when the outer boundary of the material penetrates the Schwarzschild radius. The necessary and sufficient criterium for the interior solution to be matched to the Schwarzschild exterior solution is given. Apparent horizons exist if and only if the surface is inside the Schwarzschild surface, then there exist in fact two horizons: The absolute Schwarzschild surface and an apparent horizon in the interior of the fluid matter. Some models reminding us of phenomena of astrophysical interest are given (Quasars, supernova explosion, contraction of a white dwarf forming a neutron star, vibrating neutron stars).