General Relativistic Fluid Spheres

Abstract

In Part I of this paper certain well known results concerning the Schwarzschild interior solution are generalized to more general static fluid spheres in the form of inequalities comparing the boundary value of ${g}_{44}$ with certain expressions involving only the mass concentration and the ratio of the central energy density to the central pressure. A minimal theorem appropriate to the relativistic domain is derived for the central pressure, corresponding to a well-known classical result. Inequalities involving the proper energy and the potential energy are also considered, as is the introduction of the physical radius in place of the coordinate radius. A singularity-free elementary algebraic solution of the field equations is presented and exact values obtained from it compared with the limits prescribed by some of the inequalities. In Part II an answer is given to the question whether the total amount of radiation emitted during the symmetrical gravitational contraction of an amount of matter whose initial energy, at complete dispersion, is ${W}_{0}$ can ever exceed ${W}_{0}$.