Boundary conditions for metric fluctuations in Lifshitz

TL;DR

This paper investigates boundary conditions for metric fluctuations in Lifshitz spacetimes, revealing that for certain dynamical exponents, alternative boundary conditions are possible without instability, with implications for holographic models.

Contribution

It analyzes the normalizability and stability of linearized metric and matter fluctuations in Lifshitz backgrounds, proposing new boundary conditions for specific cases.

Findings

Slow fall-off modes are normalizable for z > 2.

Alternative boundary conditions for momentum density are stable.

Alternative boundary conditions for energy density cause instabilities.

Abstract

We consider the quantisation of linearised fluctuations of the metric and matter fields about a Lifshitz background, exploring the possibility of alternative boundary conditions, allowing the slow fall-off modes to fluctuate. We find that for $z >2$, slow fall-off modes for some of the linearised fluctuations are normalizable, which opens up the possibility of considering alternative boundary conditions. Analysing stability, we find that alternative boundary conditions for the momentum density are allowed, but alternative boundary conditions for the energy density lead to an instability of the type we recently discovered in a similar analysis for scalar fields on a fixed Lifshitz background. Our investigation is in the context of the simple massive vector model, but we would expect the conclusions to be more general.