  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.
