Geodesic incompleteness and partially covariant gravity

TL;DR

This paper investigates how length measurements and geodesic incompleteness relate in covariant and non-relativistic gravity theories, providing criteria for singularities based on length finiteness transfer.

Contribution

It establishes a relation between spacetime length in covariant and non-covariant theories, enabling criteria transfer for geodesic incompleteness and singularity analysis.

Findings

Length relations depend on symmetry group differences.

Criteria for geodesic incompleteness can be transferred between theories.

Implications for understanding singularities in covariant and non-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.