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 Hořava–Lifshitz gravity. The difference in their symmetry groups implies a relation among 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 and 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.

Global Dynamics of the Hořava–Lifshitz Cosmological Model in a Non-Flat Universe with Non-Zero Cosmological Constant

“…Here the function a(t) is the dimensionless rescaling factor of the expanding universe, and γ ij is a constant curvature metric of being maximally symmetric. Without loss of generality, we normalize κ 2 and N to the number one, and then we can describe the cosmological model as…”

“…It is named Hořava-Lifshitz gravity together with the scalar field theory of Lifshitz. This theory has inspired many studies and applications in length renormalization [2], entropy argument [3], cosmology , dark energy [31][32][33][34][35], black holes [36][37][38][39][40], gravitational waves [41] and electromagnetics [42][43][44][45]. More information can also be found in the review articles [46][47][48] and the references therein.…”