Integral equations and Kerr-Schild fields II. The Kerr solution

Stationary vacuum solutions of Kerr-Schild type are analysed within the framework of gauge-theory gravity. The complex structure at the heart of these fields is shown to have a clear geometric origin, with the role of the unit imaginary fulfilled by the spacetime pseudoscalar. A number of general results for stationary Kerr-Schild fields are obtained, and it is shown that the stress-energy tensor is a total divergence. The Kerr solution is analysed from this gauge-theory viewpoint, leading to a novel picture of its global properties. In particular, application of Gauss’ theorem to the stress-energy tensor reveals the presence of a disk of tension surrounded by the matter ring singularity. Remarkably, the tension profile over this disk has a simple classical interpretation. It is also shown that the matter in the ring follows a light-like path, as one expects for the endpoint of rotating, collapsing matter. Some implications of these results for physically-realistic black holes are discussed.

C. J. L. Doran, A. N. Lasenby and S. F. Gull, Integral equations and Kerr-Schild fields II. The Kerr solution, Preprint (1998)