Caustic avoidance in Horava-Lifshitz gravity

TL;DR

This work analyzes Horava-Lifshitz gravity with the projectability condition and without detailed balance to address whether caustics form on constant-time hypersurfaces. Using the ADM formulation, it shows that for plane-symmetric, codimension-one configurations with $\lambda \neq 1$ the would-be caustic is ruled out by gauge choices and the equations of motion, and more generally nonlinear higher-curvature terms provide repulsive gravity that induces a bounce for codimension $>1$, preventing caustics. It also discusses how the absence of a local Hamiltonian constraint yields a dust-like density $\rho_d$ (the dark matter as an integration constant) that enters the modified Einstein equations and can be positive in patches, while reducing to GR with dust in the IR as $\lambda \to 1$. The findings support a caustic-free IR limit and have implications for cosmology and the dark-matter-as-integration-constant scenario.

Abstract

There are at least four versions of Horava-Lishitz gravity in the literature. We consider the version without the detailed balance condition with the projectability condition and address one aspect of the theory: avoidance of caustics for constant time hypersurfaces. We show that there is no caustic with plane symmetry in the absence of matter source if λ\ne 1. If λ=1 is a stable IR fixed point of the renormalization group flow then λis expected to deviate from 1 near would-be caustics, where the extrinsic curvature increases and high-energy corrections become important. Therefore, the absence of caustics with λ\ne 1 implies that caustics cannot form with this symmetry in the absence of matter source. We argue that inclusion of matter source will not change the conclusion. We also argue that caustics with codimension higher than one will not form because of repulsive gravity generated by nonlinear higher curvature terms. These arguments support our conjecture that there is no caustic for constant time hypersurfaces. Finally, we discuss implications to the recently proposed scenario of ``dark matter as integration constant''.