Wave zone of the Ho\v{r}ava--Lifshitz gravity coupled to a gauge vector

TL;DR

This paper analyzes wave propagation in a specific Hořava-Lifshitz gravity model coupled with a gauge vector, showing that physical modes propagate independently at a common speed in the wave zone.

Contribution

It demonstrates the existence of a well-defined wave zone with decoupled gravitational and gauge vector modes in the non-projectable Hořava-Lifshitz theory at the kinetic conformal point.

Findings

Physical degrees of freedom satisfy independent wave equations.

Both gravitational and vector modes propagate at speed √β.

The Newtonian part does not affect wave propagation in the wave zone.

Abstract

We consider the anisotropic gravity-gauge vector coupling in the non-projectable Ho\v{r}ava-Lifshitz theory at the kinetic conformal point, in the low energy regime. We show that the canonical formulation of the theory, evaluated at its constraints, reduces to a canonical formulation solely in terms of the physical degrees of freedom. The corresponding reduced Hamilton defines the ADM energy of the system. We obtain its explicit expression and discuss its relation to the ADM energy of the Einstein-Maxwell theory. We then show that there exists, in this theory, a well--defined wave zone. In it, the physical degrees of freedom {\i}.e., the transverse--traceless tensorial modes associated to the gravitational sector and the transverse vectorial modes associated to the gauge vector interaction satisfy independent linear wave equations, without any coupling between them. The Newtonian part of the anisotropic theory, very relevant near the sources, does not affect the free propagation of the physical degrees of freedom in the wave zone. It turns out that both excitations, the gravitational and the vectorial one, propagate with the same speed $\sqrt{\beta}$, where $\beta$ is the coupling parameter of the scalar curvature of the three dimensional leaves of the foliation defining the Ho\v{r}ava--Lifshitz geometry.