Asymptotically Schr\"odinger Space-Times

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.