The anisotropic coupling of gravity and electromagnetism in Hořava-Lifshitz theory

Abstract

We analyze the electromagnetic-gravity interaction in a pure Hořava-Lifshitz framework. To do so we formulate the Hořava-Lifshitz gravity 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 coupling constant in the kinetic term of the action is $\lambda =1/4$. It is the kinetic conformal point for the non-relativistic electromagnetic-gravity interaction. In distinction, the corresponding kinetic conformal value for pure Hořava-Lifshitz gravity in $3+1$ dimensions is $\lambda =1/3$. We analyze the geometrical structure of the critical and noncritical cases, they correspond to different theories. The physical degrees of freedom propagated by the noncritical theory are the transverse traceless graviton, the transverse gauge vector and two scalar fields. In the critical theory one of the scalars is absent, only the dilaton scalar field is present. The gravity and vector excitations propagate with the same speed, which at low energy can be taken to be the speed of light. The field equations for the gauge vector in the non-relativistic theory have exactly the same form as the relativistic electromagnetic field equations arising from the Kaluza-Klein reduction of General Relativity, and are equal to them for a particular value of one of the coupling constants. The potential in the Hamiltonian is a polynomial of finite degree in the gauge vector and its covariant derivatives.

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• Binary X-ray sources in massive Brans-Dicke gravity
• Wave zone in the Hořava-Lifshitz theory at the kinetic-conformal point in the low energy regime
• Gravitational waves in Hořava-Lifshitz anisotropic gravity
• Propagation of anisotropic gravitational and electromagnetic waves at very high energies
• Global dynamics of the Horava-Lifshitz cosmological model in a non-flat universe with non-zero cosmological constant