Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective

TL;DR

This paper explores a new ambient metric framework unifying Newton-Cartan and Carrollian geometries, analyzing compatible connections and their geometric properties in nonrelativistic gravity theories.

Contribution

It introduces a novel non-Lorentzian ambient metric structure that generalizes existing frameworks and characterizes the space of compatible torsional connections.

Findings

The new ambient structure projects to general Galilean connections.

The ambient metric embeds Carrollian connections.

Characterization of torsional connections preserving the new structure.

Abstract

Connections compatible with degenerate metric structures are known to possess peculiar features: on the one hand, the compatibility conditions involve restrictions on the torsion; on the other hand, torsionfree compatible connections are not unique, the arbitrariness being encoded in a tensor field whose type depends on the metric structure. Nonrelativistic structures typically fall under this scheme, the paradigmatic example being a contravariant degenerate metric whose kernel is spanned by a one-form. Torsionfree compatible (i.e. Galilean) connections are characterised by the gift of a two-form (the force field). Whenever the two-form is closed, the connection is said Newtonian. Such a nonrelativistic spacetime is known to admit an ambient description as the orbit space of a gravitational wave with parallel rays. The leaves of the null foliation are endowed with a nonrelativistic structure dual to the Newtonian one, dubbed Carrollian spacetime. We propose a generalisation of this unifying framework by introducing a new non-Lorentzian ambient metric structure of which we study the geometry. We characterise the space of (torsional) connections preserving such a metric structure which is shown to project to (resp. embed) the most general class of (torsional) Galilean (resp. Carrollian) connections.