# Pure electromagnetic-gravitational interaction in Ho\v{r}ava-Lifshitz theory at the kinetic conformal point

## TL;DR

This paper develops a Hořava-Lifshitz gravity model coupled with electromagnetism via Kaluza-Klein reduction, showing it reproduces Einstein-Maxwell theory at low energies and maintains quantum stability at the kinetic conformal point.

## Contribution

It introduces a consistent electromagnetic-gravitational coupling in Hořava-Lifshitz theory at the kinetic conformal point, including higher order derivatives, and demonstrates its equivalence to Einstein-Maxwell theory in the IR limit.

## Key findings

- Propagating degrees of freedom match Einstein-Maxwell theory.
- The theory is stable under quantum corrections at the kinetic conformal point.
- Maxwell equations emerge naturally for a specific coupling constant value.

## Abstract

We introduce the electromagnetic-gravitational coupling in the Ho\v{r}ava-Lifshitz framework, in $3+1$ dimensions, by considering the Ho\v{r}ava-Lifshitz gravity theory in $4+1$ dimensions at the kinetic conformal point and then performing a Kaluza-Klein reduction to $3+1$ dimensions. The action of the theory is second order in time derivatives and the potential contains only higher order spacelike derivatives up to $z=4$, $z$ being the critical exponent. These terms include also higher order derivative terms of the electromagnetic field. The propagating degrees of freedom of the theory are exactly the same as in the Einstein-Maxwell theory. We obtain the Hamiltonian, the field equations and show consistency of the constraint system. The kinetic conformal point is protected from quantum corrections by a second class constraint. At low energies the theory depends on two coupling constants, $\beta$ and $\alpha$. We show that the anisotropic field equations for the gauge vector is a deviation of the covariant Maxwell equations by a term depending on $\beta-1$. Consequently, for $\beta=1$, Maxwell equations arise from the anisotropic theory at low energies. We also prove that the anisotropic electromagnetic-gravitational theory at the IR point $\beta=1$, $\alpha=0$, is exactly the Einstein-Maxwell theory in a gravitational gauge used in the ADM formulation of General Relativity.