Integral equations, Kerr-Schild fields and gravitational sources

TL;DR

This paper explores Kerr-Schild solutions to Einstein's equations using integral equations, revealing the nature of sources, the role of complex structures, and global properties of black holes, including tension distributions and matter trajectories.

Contribution

It provides a new integral equation perspective on Kerr-Schild solutions, identifying source structures and global properties of black holes, with novel geometric and physical insights.

Findings

Schwarzschild and Vaidya solutions originate from point sources

Inclusion of gravity removes classical electromagnetic self-energy divergence

Kerr solution features a tension disk and matter ring with light-like matter paths

Abstract

Kerr-Schild solutions to the vacuum Einstein equations are considered from the viewpoint of integral equations. We show that, for a class of Kerr-Schild fields, the stress-energy tensor can be regarded as a total divergence in Minkowski spacetime. If one assumes that Minkowski coordinates cover the entire manifold (no maximal extension), then Gauss' theorem can be used to reveal the nature of any sources present. For the Schwarzschild and Vaidya solutions the fields are shown to result from a delta-function point source. For the Reissner-Nordstrom solution we find that inclusion of the gravitational fields removes the divergent self-energy familiar from classical electromagnetism. For more general solutions a complex structure is seen to arise in a natural, geometric manner with the role of the unit imaginary fulfilled by the spacetime pseudoscalar. The Kerr solution is analysed leading to a novel picture of its global properties. Gauss' theorem 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.