On the motion of particles in covariant Horava-Lifshitz gravity and the meaning of the A-field

TL;DR

This paper investigates particle motion in covariant Horava-Lifshitz gravity, revealing that the equivalence principle is generally violated at low energies and that the effective metric does not reproduce Newtonian gravity in weak fields.

Contribution

It introduces an effective relativistic metric for covariant Horava-Lifshitz gravity and analyzes particle trajectories, highlighting deviations from classical gravity and the conditions for equivalence principle recovery.

Findings

The equivalence principle is not generally recovered at low energies.

The effective metric cannot reproduce Newtonian gravity in weak fields.

The spatial Laplacian of A must vanish for equivalence principle recovery.

Abstract

We studied the low energy motion of particles in the general covariant version of Horava-Lifshitz gravity proposed by Horava and Melby-Thompson. Using a scalar field coupled to gravity according to the minimal substitution recipe proposed by da Silva and taking the geometrical optics limit, we could write an effective relativistic metric for a general solution. As a result, we discovered that the equivalence principle is not in general recovered at low energies, unless the spatial Laplacian of A vanishes. Finally, we analyzed the motion on the spherical symmetric solution proposed by Horava and Melby-Thompson, where we could find its effective line element and compute spin-0 geodesics. Using standard methods we have shown that such an effective metric cannot reproduce Newton's gravity law even in the weak gravitational field approximation.