Imbedding a Schwarzschild mass into cosmology

Abstract

We develop a method for imbedding a Schwarzschild mass into a zero-curvature universe. We work with curvature coordinates ($R$,$T$), in terms of which the metric has the form $d{s}^{2}(R,T)={A}^{\ensuremath{-}1}(R,T)d{R}^{2}+{R}^{2}d{\ensuremath{\Omega}}^{2}\ensuremath{-}B(R,T)d{T}^{2}$, and coordinates ($R$,$\ensuremath{\tau}$), where $\ensuremath{\tau}$ is measured by radially moving geodesic clocks. We solve the field equations for a stress-energy tensor that corresponds to a radially moving perfect geodesic fluid outside some boundary ${R}_{b}$. Inside ${R}_{b}$ we take the stress-energy tensor to be composed of a perfect-fluid part and a Schwarzschild matter part. Specific examples of imbedding a mass into a de Sitter universe and a pressure-free Einstein---de Sitter universe are given, and we show how to extend our methods to general zero-curvature universes. A consequence of our results is that there will be spiralling of planetary orbits when a mass such as our Sun is imbedded in a universe. We relate our work to recent work done by Dirac with regard to his Large Numbers hypothesis.