Geometry of Schroedinger Space-Times, Global Coordinates, and Harmonic Trapping

TL;DR

This paper analyzes the geometry of Schrödinger space-times with dynamical exponent $z>1$ and compares them to AdS with $z=1$, focusing on global structure and holographic implications. It shows that Schrödinger metrics are singular for $1<z<2$ and that Poincaré coordinates are incomplete for $z\geq 2$, then constructs a global coordinate system for the special case $z=2$ by diagonalizing the global-time generator $H+C$ (with $P_-$ fixed), revealing a plane-wave deformation that acts as harmonic trapping. This global metric is geodesically complete due to the trapping term, and its holographic interpretation provides an IR regulator for the dual non-relativistic CFT, with scalar fields exhibiting harmonic-oscillator type behavior in global coordinates. For $z>2$ there is no global timelike Killing vector, and time-dependence must appear in any global description; the analysis also connects to AdS global coordinates via trapping coordinates and discusses scalar field dynamics in the $z=2$ background. Overall, the work clarifies the global geometry of non-relativistic holographic backgrounds and establishes a concrete global framework for $z=2$ Schrödinger space-times, including scalar field dynamics.

Abstract

We study various geometrical aspects of Schroedinger space-times with dynamical exponent z>1 and compare them with the properties of AdS (z=1). The Schroedinger metrics are singular for 1<z<2 while the usual Poincare coordinates are incomplete for z \geq 2. For z=2 we obtain a global coordinate system and we explain the relations among its geodesic completeness, the choice of global time, and the harmonic trapping of non-relativistic CFTs. For z>2, we show that the Schroedinger space-times admit no global timelike Killing vectors.