Geodesic incompleteness and partially covariant gravity

Abstract

We study the issue of length renormalization in the context of fully covariant gravity theories as well as non-relativistic ones such as Ho\v{r}ava-Lifshitz gravity. The difference of their symmetry groups implies a relation between the lengths of paths in spacetime in the two types of theory. Provided that certain asymptotic conditions hold, this relation allows us to transfer analytic criteria for the standard spacetime length to be finite to the Perelman length to be likewise finite, and therefore formulate conditions for geodesic incompleteness in partially covariant theories. We also discuss implications of this result for the issue of singularities in the context of such theories.