Einstein–Maxwell Equivalence in the λ–R Hořava–Lifshitz Model

ABSTRACT In this work, we analyze the – model within the anisotropic electromagnetic–gravitational Hořava–Lifshitz theory. Starting from the Hamiltonian formulation of the electromagnetic–gravitational–dilaton system in dimensions obtained through a Kaluza–Klein reduction of the Hořava–Lifshitz theory, we consider the infrared regime with coupling constants , , and the dilaton fixed at its ground state . We show that the Hamiltonian can be written as the Einstein–Maxwell Hamiltonian plus a term proportional to the square of the trace of the canonical momentum . The preservation in time of the Hamiltonian constraint leads to an elliptic equation whose only asymptotically flat solution is , so the Hamiltonian reduces to the Einstein–Maxwell one on the constraint surface . We further analyze the constraint structure and show that the theory propagates four configuration–space physical degrees of freedom, corresponding to the two transverse–traceless gravitational modes and the two transverse electromagnetic modes of Einstein–Maxwell theory. We also show that the condition has a geometrical interpretation as the maximal slicing condition. Under asymptotically flat conditions and without additional boundary constraints, the anisotropic electromagnetic–gravitational Hořava–Lifshitz theory therefore becomes canonically equivalent to Einstein–Maxwell theory in maximal slicing. Consequently, gravitational and electromagnetic waves far from their sources propagate in exactly the same way in both theories.