Solving Einstein's field equations with Mathematica

We report an example using Mathematica to solve the semi-classical Einstein field equations in spherical coordinates. Metric variations resulting from the Casimir effect are calculated for an ideal massless superconducting sphere. Expressions for the change in scalar curvature inside the superconducting boundary are developed. We first consider the static case when the sphere is superconducting. Metric equations are then developed for the evolution of a scalar quantum field after the sphere transitions to the normal state.