Pathologies in Asymptotically Lifshitz Spacetimes

TL;DR

This paper investigates the stability and singularity issues of Lifshitz spacetimes, revealing that their ground states often contain naked singularities and that the initial value problem is generally ill-posed, challenging their viability as gravitational duals.

Contribution

It demonstrates that asymptotically Lifshitz spacetimes typically have naked singularities and that their initial value problem is ill-posed, questioning their stability and consistency as duals to field theories.

Findings

The ground state with flat spatial sections has a naked singularity.

The initial value problem for Lifshitz spacetimes is generally not well-posed.

Perturbations tend to violate Lifshitz asymptotics in finite time.

Abstract

There has been significant interest in the last several years in studying possible gravitational duals, known as Lifshitz spacetimes, to anisotropically scaling field theories by adding matter to distort the asymptotics of an AdS spacetime. We point out that putative ground state for the most heavily studied example of such a spacetime, that with a flat spatial section, suffers from a naked singularity and further point out this singularity is not resolvable by any known stringy effect. We review the reasons one might worry that asymptotically Lifshitz spacetimes are unstable and employ the initial data problem to study the stability of such systems. Rather surprisingly this question, and even the initial value problem itself, for these spacetimes turns out to generically not be well-posed. A generic normalizable state will evolve in such a way to violate Lifshitz asymptotics in finite time. Conversely, enforcing the desired asymptotics at all times puts strong restrictions not just on the metric and fields in the asymptotic region but in the deep interior as well. Generically, even perturbations of the matter field of compact support are not compatible with the desired asymptotics.