Kerr–Schild gravitational fields in matter and the Kerr-interior problem in general relativity

Stationary Kerr–Schild gravitational fields in the interior of elastic–solid bodies are considered. It turns out that the Einstein field equations for such systems split into two parts: a ‘‘differential’’ part which may be written as a linear partial differential equation of the Maxwell type for a suitably defined, three-dimensional Euclidean vector and an ‘‘algebraic’’ (generally nonlinear) part which allows the calculation of the stresses supporting the bodies. These results are applied to the problem of finding the sources of the Kerr space–time, and a new class of solutions of the Einstein field equations is found. Such solutions describe sources of the Kerr metric composed by an elastic body and a shell distribution of elastic stress energy located on the matching surface. Some explicit examples of ‘‘half local’’ (singular) Kerr–Schild sources of the Kerr space–time are also given.