The existence of singularities in general relativity despite isolated failures of geodesic focusing

Abstract

A key ingredient of the singularity theorems of general relativity is that geodesics will focus under certain conditions. While the conditions that are generally imposed are fairly weak and can be expected to hold for most geodesics in any reasonable space-time, the question arises of what the consequences might be if these conditions happen to fail to hold for isolated geodesics. The standard singularity theorems in the literature do not cover such cases. A result is presented here that demonstrates that a singularity will exist even in cases where geodesic focusing does not occur for some geodesics.