Postulational approach to schwarzschild’s exterior solution with application to a class of interior solutions

Abstract

In order to clarify the physical ideas underlying Schwarzschild’s exterior solution, a postulational derivation is given that does not make use of the field equations. Basically this amounts to replacing two field equations by two postulates, one of which is a strong version of the principle of equivalence and the other, Newton’s inverse square law. These postulates are more general than the approach would indicate and there is actually a class of solutions for static systems with spherical symmetry which satisfy them. The energy-stress tensors producing these solutions have the important property that energy density equals radial stress. Two examples of interior solutions that fulfill these postulates are given: a solid sphere and a hollow shell. The basic properties of these solutions are described and compared with those of Schwarzschild’s interior solution. The solid sphere solution is used to complete a previous discussion of the clock paradox. In an Appendix, the field equations for static systems with spherical symmetry are written in a form that indicates the limitations of the postulates.