Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view

TL;DR

This paper investigates the structure of compatible connections in generalized Newton-Cartan gravity, focusing on the classification and properties of torsionfree and torsional cases, and highlighting differences from relativistic geometries.

Contribution

It characterizes the affine structure of compatible Newton-Cartan connections and explores the equivalence problem in both torsionfree and torsional scenarios.

Findings

Characterization of the affine space of compatible connections.

Identification of model vector spaces for these connections.

Extension of Newtonian connections to include torsional cases.

Abstract

The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connections along the two corresponding alternative roads in a series of papers. In the present one, we focus on compatible connections and investigate the equivalence problem (i.e. the search for the necessary data allowing to uniquely determine connections) in the torsionfree and torsional cases. More precisely, we characterise the affine structure of the spaces of such connections and display the associated model vector spaces. In contrast with the relativistic case, the metric structure does not single out a privileged origin for the space of metric-compatible connections. In our construction, the role of the Levi-Civita connection is played by a whole class of privileged origins, the so-called torsional Newton-Cartan (TNC) geometries recently investigated in the literature. Finally, we discuss a generalisation of Newtonian connections to the torsional case.