Lorentz covariant treatment of the Kerr–Schild geometry

It is shown that a Lorentz covariant coordinate system can be chosen in the case of the Kerr–Schild geometry which leads to the vanishing of the pseudo energy–momentum tensor and hence to the linearity of the Einstein equations. The retarded time and the retarded distance are introduced and the Liénard–Wiechert potentials are generalized to gravitation in the case of world-line singularities to derive solutions of the type of Bonnor and Vaidya. An accelerated version of the de Sitter metric is also obtained. Because of the linearity, complex translations can be performed on these solutions, resulting in a special relativistic version of the Trautman–Newman technique and Lorentz covariant solutions for spinning systems can be derived, including a new anisotropic interior metric that matches to the Kerr metric on an oblate spheroid.