Symmetries and Conservation Laws in Horava Gravity

Abstract: Horava gravity has been proposed as a renormalizable quantum gravity without the ghost problem through anisotropic scaling dimensions which break Lorentz symmetry in UV. In the Hamiltonian formalism, due to the Lorentz-violating terms, the constraint structure looks quite different from that of general relativity (GR) but we have recently found that "there exists the case where we can recover the same number of degrees of freedom as in GR", in a rather general set-up. In this paper, we study its Lagrangian perspectives and examine the full diffeomorphism (Diff) symmetry and its associated conservation laws in Horava gravity. Surprisingly, we find that the full Diff symmetry in the action can also be recovered when a certain condition, called "super-condition", which super-selects the Lorentz-symmetric sector in Horava gravity, is satisfied. This indicates that the broken Lorentz symmetry, known as "foliation-preserving" Diff, is just an "apparent" symmetry of the Horava gravity action and rather its "full action symmetry can be as large as the Diff in GR ". The super-condition exactly corresponds to the tertiary constraint in Hamiltonian formalism which is the second-class constraint and provides a non-trivial realization of the Lorentz symmetry otherwise being absent apparently. From the recovered Lorentz symmetry in the action, we obtain the conservation laws with the Noether currents as in covariant theories. The general formula for the conserved Noether charges reproduces the mass of four-dimensional static black holes with an "arbitrary" cosmological constant in Horava gravity, and is independent of ambiguities associated with the choice of asymptotic boundaries. We also discuss several challenging problems, including its implications to Hamiltonian formalism, black hole thermodynamics, radiations from colliding black holes.