Embedding nonrelativistic physics inside a gravitational wave

TL;DR

This paper explores how nonrelativistic physics can be embedded within certain gravitational wave spacetimes, revealing deep geometric and dynamical correspondences and extending previous results to more general classes.

Contribution

It introduces a new geometric definition for gravitational waves with a null Killing vector, enabling the generalization of nonrelativistic embeddings and related properties.

Findings

Null dimensional reduction leads to Schrödinger equations on curved space.

Geodesic completeness results are extended to a broader class of gravitational waves.

Classification of gravitational waves with constant scalar invariants is investigated.

Abstract

Gravitational waves with parallel rays are known to have remarkable properties: Their orbit space of null rays possesses the structure of a non-relativistic spacetime of codimension-one. Their geodesics are in one-to-one correspondence with dynamical trajectories of a non-relativistic system. Similarly, the null dimensional reduction of Klein-Gordon's equation on this class of gravitational waves leads to a Schroedinger equation on curved space. These properties are generalized to the class of gravitational waves with a null Killing vector field, of which we propose a new geometric definition, as conformally equivalent to the previous class and such that the Killing vector field is preserved. This definition is instrumental for performing this generalization, as well as various applications. In particular, results on geodesic completeness are extended in a similar way. Moreover, the classification of the subclass with constant scalar invariants is investigated.