The geometry of weak gravitational singularities

Abstract

We introduce a new class of weak curvature singularity with the critical feature that, although the curvature may be unbounded on approach to the singularity, the curvature remains square Lebesgue integrable.

Previous work looking at analogous singularities in the context of Yang-Mills gauge theory, suggests that a two-dimensional singularity, with square Lebesgue integrable curvature, will have a connection approaching that of a flat connection on approach to the singularity.  By considering a 2-dimensional, timelike and static, weak singularity and by treating General Relativity as a gauge theory, we are able to apply these methods to gravitational singularities.  We show that 1. A limit holonomy exists and is independent of position on the singularity; 2. The connection tends to the conical connection in an L²₁ Sobolev norm; and 3. The metric tends to the conical metric in an L²₂ Sobolev norm.

In the final chapter we review previous work on the use of Colombeau’s theory of generalized functions in describing the curvature of conical spacetimes as distributions.  Using the results stated above, we are able to extend this work to a class of weak curvature singularities.  We show that the distributional part of the curvature of a weak, two-dimensional, timelike and static, curvature singularity is associated (in the sense of Colombeau algebras) to the distributed curvature of a 4-dimesnonal cone, which may be described in terms of 2-dimensoinal delta functions.

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