An Alternative Approach for General Covariant Horava-Lifshitz Gravity and Matter Coupling

TL;DR

The paper addresses the spin-0 graviton issue and the uncertain role of $\lambda$ in Horava-Lifshitz gravity by proposing a manifestly gauge-invariant, general covariant framework with an extended $U(1)$ symmetry. It introduces a minimal substitution recipe to couple matter while preserving $U(1)\times$Diff$(\mathcal{M},\mathcal{F})$, reproduces the Horava–Melby-Thompson action for $\lambda=1$, and shows that $\lambda$ remains a running parameter while the extended symmetry eliminates the scalar graviton even for $\lambda\neq1$. The work provides explicit matter couplings (nonrelativistic scalar and vector fields) and clarifies the role of the Newton prepotential $\nu$ and the gauge field $A$, along with the constraint $R-2\Omega=0$, which has implications for cosmology and IR phenomenology. Overall, the framework offers a practical, gauge-invariant path to study HL gravity with matter and motivates further investigation into the IR behavior and cosmological viability of these theories.

Abstract

Recently, Horava and Melby-Thompson proposed in arXiv:1007.2410 a nonrelativistic gravity theory with extended gauge symmetry that is free of the spin-0 graviton. We propose a minimal substitution recipe to implement this extended gauge symmetry which reproduce the results obtained by them. Our prescription has the advantage of being manifestly gauge invariant and immediately generalizable to other fields, like matter. We briefly discuss the coupling of gravity with scalar and vector fields found by our method. We show also that the extended gauge invariance in gravity does not force the value of $λ$ to be $λ=1$ as originally claimed. However, the spin-0 graviton is eliminated even for general $λ$.