A functional renormalization group equation for foliated spacetimes

TL;DR

This paper derives an exact functional renormalization group equation for Hořava-Lifshitz gravity, revealing how gravitational couplings evolve and demonstrating the persistence of a non-Gaussian fixed point across different formalisms and signatures.

Contribution

It introduces a novel RG flow equation for foliated spacetimes in Hořava-Lifshitz gravity, connecting it to metric formalism and analyzing fixed points in Lorentzian signature.

Findings

The non-Gaussian fixed point persists in the ADM formalism.

The flow equation is invariant under foliation-preserving diffeomorphisms.

The fixed point structure is robust across different metric signatures.

Abstract

We derive an exact functional renormalization group equation for the projectable version of Ho\v{r}ava-Lifshitz gravity. The flow equation encodes the gravitational degrees of freedom in terms of the lapse function, shift vector and spatial metric and is manifestly invariant under background foliation-preserving diffeomorphisms. Its relation to similar flow equations for gravity in the metric formalism is discussed in detail, and we argue that the space of action functionals, invariant under the full diffeomorphism group, forms a subspace of the latter invariant under renormalization group transformations. As a first application we study the RG flow of the Newton constant and the cosmological constant in the ADM formalism. In particular we show that the non-Gaussian fixed point found in the metric formulation is qualitatively unaffected by the change of variables and persists also for Lorentzian signature metrics.