We show that in the Hořava–Lifshitz theory at the kinetic-conformal point, in the low energy regime, a wave zone for asymptotically flat fields can be consistently defined. In it, the physical degrees of freedom, the transverse traceless tensorial modes, satisfy a linear wave equation. The Newtonian contributions, among which there are terms which manifestly break the relativistic invariance, are non-trivial but do not obstruct the free propagation (radiation) of the physical degrees of freedom. For an appropriate value of the couplings of the theory, the wave equation becomes the relativistic one in agreement with the propagation of the gravitational radiation in the wave zone of General Relativity. Previously to the wave zone analysis, and in general grounds, we obtain the physical Hamiltonian of the Hořava–Lifshitz theory at the kinetic-conformal point in the constrained submanifold. We determine the canonical physical degrees of freedom in a particular coordinate system. They are well defined functions of the transverse-traceless modes of the metric and coincide with them in the wave zone and also at linearized level.

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We consider the anisotropic gravity-gauge vector coupling in the non-projectable Hořava-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 ı.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 √ β, where β is the coupling parameter of the scalar curvature of the three dimensional leaves of the foliation defining the Hořava-Lifshitz geometry.

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