Holographic Renormalization for Asymptotically Lifshitz Spacetimes

TL;DR

This paper develops a holographic renormalization framework for asymptotically Lifshitz spacetimes with critical exponent z=2, constructing boundary actions and stress tensors to ensure a well-defined variational principle.

Contribution

It introduces two new boundary actions with local counterterms for Lifshitz spacetimes and analyzes their implications for boundary stress tensors and asymptotic symmetries.

Findings

Constructed boundary actions with local counterterms for z=2 Lifshitz spacetimes

Identified the appropriate boundary stress tensor definitions under different conditions

Analyzed asymptotic symmetries and boundary data constraints

Abstract

A variational formulation is given for a theory of gravity coupled to a massive vector in four dimensions, with Asymptotically Lifshitz boundary conditions on the fields. For theories with critical exponent z=2 we obtain a well-defined variational principle by explicitly constructing two actions with local boundary counterterms. As part of our analysis we obtain solutions of these theories on a neighborhood of spatial infinity, study the asymptotic symmetries, and consider different definitions of the boundary stress tensor and associated charges. A constraint on the boundary data for the fields figures prominently in one of our formulations, and in that case the only suitable definition of the boundary stress tensor is due to Hollands, Ishibashi, and Marolf. Their definition naturally emerges from our requirement of finiteness of the action under Hamilton-Jacobi variations of the fields. A second, more general variational principle also allows the Brown-York definition of a boundary stress tensor.