Imbedding a Schwarzschild mass into cosmology

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 ds^2 (R, T) =A^-1 (R, T) dR^2+R^2d^2-B (R, T) dT^2, and coordinates (R, ), where 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 R₁. Inside R₁ 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.