Perturbations in the Schwarzschild Metric

Exact solutions of the general-relativistic field equations are known only for very simple physical systems. In order to obtain solutions of the field equations for more realistic systems, we consider an expansion of the field equations about some known exact solution, keeping only terms linear in the perturbation. We then choose the unperturbed metric to be the Schwarzschild metric, corresponding to the exterior of any spherically symmetric mass distribution, and consider the case where the perturbing matter is not located close to the Schwarzschild radius. A discussion of the solution for a scalar Green's function in curved space leads directly to an explicit expression for the perturbed metric in terms of the perturbing matter.