Integral equations and Kerr-Schild fields I. Time-dependent, spherically-symmetric fields

Kerr-Schild solutions to the vacuum Einstein equations are considered from the viewpoint of a gauge-theoretic formulation of gravity. This formulation employs the spacetime algebra (the Clifford algebra of spacetime), which offers a number of novel insights into the nature of the solutions. Working with gauge fields defined over a background Minkowski spacetime allows the Einstein equations to be recast as integral equations, which can be used to explore the nature of the gravitational singularities. For the Schwarzschild and Vaidya solutions the fields are shown to result from a $\del$-function point source. For the Reissner-Nordstrom solution the integral equations reveal that inclusion of the gravitational fields removes the divergent self-energy familiar from classical electromagnetism.

C. J. L. Doran, Integral equations and Kerr-Schild fields I. Spherically-symmetric fields, Preprint (1998)