Shear-free Dust Solution in General Covariant Ho\v{r}ava-Lifshitz Gravity

TL;DR

This paper investigates dust solutions in covariant Hořava-Lifshitz gravity, finding that minimal coupling does not replicate general relativity solutions, but non-minimal coupling allows for gravitational collapse and bounce phenomena.

Contribution

It demonstrates the existence of dust solutions with non-minimal coupling in Hořava-Lifshitz gravity, including collapse and bounce behaviors not present in general relativity.

Findings

Minimal coupling does not produce GR-like dust solutions in HL gravity.

Non-minimal coupling admits gravitational collapse with singularity formation.

A bounce solution is possible, unlike in GR.

Abstract

In this paper, we have studied non stationary dust spherically symmetric spacetime, in general covariant theory ($U(1)$ extension) of the Ho\v{r}ava-Lifshitz gravity with the minimally coupling and non-minimum coupling with matter, in the post-newtonian approximation (PPN) in the infrared limit. The Newtonian prepotential $\varphi$ was assumed null. The aim of this work is to know if we can have the same spacetime, as we know in the General Relativity Theory (GRT), in Ho\v{r}ava-Lifshitz Theory (HLT) in this limit. We have shown that there is not an analogy of the dust solution in HLT with the minimally coupling, as in GRT. Using non-minimum coupling with matter, we have shown that the solution admits a process of gravitational collapse, leaving a singularity at the end. This solution has, qualitatively, the same temporal behaviour as the dust collapse in GRT. However, we have also found a second possible solution, representing a bounce behavior that is not found in GRT.