Asymptotically Schr\"odinger Space-Times

TL;DR

This paper reviews the emergence of asymptotically Schr"odinger space-times via TsT transformations, defines Schr"odinger space-times and boundaries in a coordinate-independent way, and constructs their Fefferman-Graham expansions.

Contribution

It introduces a coordinate-independent framework for Schr"odinger space-times and their boundaries, extending the geometric understanding from AdS to Schr"odinger geometries.

Findings

Derived a coordinate-independent definition of Schr"odinger space-times.

Established a method to define Schr"odinger boundaries using defining functions.

Constructed Fefferman-Graham expansions for locally Schr"odinger space-times.

Abstract

We first review how asymptotically Schr\"odinger space-times arise in a natural way by performing a TsT transformation on asymptotically AdS space-times and some of its consequences. We then give a coordinate independent definition of a pure Schr\"odinger space-time in terms of an AdS metric and an AdS null Killing vector. Then, by analogy with the AdS case, we provide a local and coordinate independent definition of a Schr\"odinger boundary in terms of a defining function. We use this to construct the Fefferman-Graham expansions of locally Schr\"odinger space-times.