Embedding nonrelativistic physics inside a gravitational wave

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.

56 pages, 9 figures, v3:Minor corrections

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• Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view
• Embedding Galilean and Carrollian geometries I. Gravitational waves
• Scaling and conformal symmetries for plane gravitational waves
• Conformal and projective symmetries in Newtonian cosmology
• Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields
• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Null lifts and projective dynamics
• Eisenhart lift of Koopman-von Neumann mechanics
• New metric-affine generalizations of gravitational wave geometries
• Equivalence of a harmonic oscillator to a free particle and Eisenhart lift
• More on Jacobi metric: Randers-Finsler metrics, frame dragging and geometrisation techniques
• On Twisted Spacetimes: a new class of Galilean cosmological models