Nonstatic Analogs of Schwarzschild's Interior Solution in General Relativity

Abstract

A class of nonstatic solutions of Einstein's field equations representing the gravitational field within a spherically symmetric distribution of matter, possessing the property that the four-dimensional stream lines are normal to the hypersurfaces $\ensuremath{\rho}=\mathrm{const}$, has been developed containing an undetermined function $\ensuremath{\phi}$ of the density $\ensuremath{\rho}$ of the distribution. This class contains solutions describing contracting as well as expanding distributions with a pressure gradient. In particular, Schwarzschild's interior solution for a homogeneous perfect fluid sphere belongs to this class, when the function $\ensuremath{\phi}$ is chosen as the density $\ensuremath{\rho}$ itself. If, however, $\ensuremath{\phi}$ is chosen to vary as ${\ensuremath{\rho}}^{\frac{2}{3}}$, one gets the Oppenheimer-Snyder-solution for continued gravitational contraction. At the instant when the boundary radius of this distribution takes a stationary value, if the derivative of the function $\ensuremath{\phi}$ is greater than 1 in relativistic units, then the pressure gradient dominates over gravitational attraction and the fluid sphere expands. The situation is reversed if the derivative is less than 1. When the derivative is equal to 1, the stationary boundary becomes static and one gets Schwarzschild's interior solution.