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

TL;DR

This paper explores the geometric properties of Schroedinger space-times with various dynamical exponents, highlighting their singularities, coordinate completeness, and the existence of global timelike vectors, with implications for non-relativistic conformal field theories.

Contribution

It provides a detailed analysis of the global structure of Schroedinger space-times for different z values, including the construction of global coordinates at z=2 and the absence of global timelike Killing vectors for z>2.

Findings

Schroedinger space-times are singular for 1<z<2.

Poincare coordinates are incomplete for z ≥ 2.

Global coordinates exist at z=2, linking geodesic completeness and harmonic trapping.

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.