Lifshitz Singularities

TL;DR

Horowitz and Way analyze Lifshitz spacetimes, which realize anisotropic scaling with $t \to \lambda^z t$ and $x \to \lambda x$, and identify a null curvature singularity at $r \to 0$ for $z \neq 1$. They show that higher-dimensional embeddings and perturbative $\alpha'$ corrections do not resolve this singularity, and derive a plane-wave limit near the singularity with $W(U)=\frac{1-z}{z^2 U^2}$ to study string dynamics. In this plane-wave background, first-quantized strings experience infinite excitation when crossing the Lifshitz singularity for $z\neq1$, indicating a genuine string-theoretic instability and backreaction that likely destroys the Lifshitz fixed point in the infrared. The results suggest that Lifshitz spacetimes cannot reliably describe deep infrared Lifshitz critical points without new physics, such as temperature effects or symmetry-breaking mechanisms, to regularize the singular region.

Abstract

Lifshitz spacetimes are possible gravitational duals to strongly coupled field theories with an anisotropic scaling symmetry. These spacetimes however, have a null curvature singularity. We find that higher dimensional embeddings of Lifshitz also have a similar singularity. We study the propagation of test strings in this background and find that they become infinitely excited if they try to propagate through the singularity. This means that the Lifshitz geometry is unstable and will receive large corrections in string theory.