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We analyze the electromagnetic-gravity interaction in a pureHo\v{r}ava-Lifshitz framework. To do so we formulate the Ho\v{r}ava-Lifshitzgravity in $4+1$ dimensions and perform a Kaluza-Klein reduction to $3+1$dimensions. We use this reduction as a mathematical procedure to obtain the$3+1$ coupled theory, which at the end is considered as a fundamental,self-consistent, theory. The critical value of the dimensionless couplingconstant in the kinetic term of the action is $\lambda=1/4$. It is the kineticconformal point for the non-relativistic electromagnetic-gravity interaction.In distinction, the corresponding kinetic conformal value for pureHo\v{r}ava-Lifshitz gravity in $3+1$ dimensions is $\lambda=1/3$. We analyzethe geometrical structure of the critical and noncritical cases, theycorrespond to different theories. The physical degrees of freedom propagated bythe noncritical theory are the transverse traceless graviton, the transversegauge vector and two scalar fields. In the critical theory one of the scalarsis absent, only the dilaton scalar field is present. The gravity and vectorexcitations propagate with the same speed, which at low energy can be taken tobe the speed of light. The field equations for the gauge vector in thenon-relativistic theory have exactly the same form as the relativisticelectromagnetic field equations arising from the Kaluza-Klein reduction ofGeneral Relativity, and are equal to them for a particular value of one of thecoupling constants. The potential in the Hamiltonian is a polynomial of finitedegree in the gauge vector and its covariant derivatives.

The anisotropic coupling of gravity and electromagnetism in Ho\v{r}ava-Lifshitz theory