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Pure electromagnetic-gravitational interaction in Hořava-Lifshitz theory at the kinetic conformal point

We introduce the electromagnetic-gravitational coupling in the Hořava-Lifshitz framework, in $3+1$ dimensions, by considering the Hořava-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, $β$ and $α$. We show that the anisotropic field equations for the gauge vector is a deviation of the covariant Maxwell equations by a term depending on $β-1$. Consequently, for $β=1$, Maxwell equations arise from the anisotropic theory at low energies. We also prove that the anisotropic electromagnetic-gravitational theory at the IR point $β=1$, $α=0$, is exactly the Einstein-Maxwell theory in a gravitational gauge used in the ADM formulation of General Relativity.