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

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.

Recovering Recovery: On the Relationship between Gauge Symmetry and Trautman Recovery

“…Proposition 3.3.1: This is proved as Prop. 3.20 of Bekaert and Morand (2016). Proposition 4.2.1: This is proved in Appendix C of Bekaert and Morand (2016).…”

“…3.20 of Bekaert and Morand (2016). Proposition 4.2.1: This is proved in Appendix C of Bekaert and Morand (2016). Proposition 4.2.2: The proof is contained in the discussion leading up to Prop.…”

“…But while this fact is useful for computations, it is hardly illuminating; and moreover, it is difficult to see what its relationship might be (if any) to what philosophers refer to as the 'gauge symmetry' of Neo-Newtonian models! 4 We must thus seek out a more incisive analysis of this gauge symmetry if we are to achieve our end; we claim that such an analysis will stem from the recent work of Bekaert and Morand (2016) (henceforth BM) on 'representing' a Newton-Cartan connection.…”

“…Proposition 4.2.2: The proof is contained in the discussion leading up to Prop. 3.25 of Bekaert and Morand (2016). Note that the relevant model space is the vector space of principal connections, and that the relevant affine isomorphism is modeled on the identity map.…”

Torsional Newton–Cartan geometry from Galilean gauge theory

“…Newton Cartan (NC) geometry [1,2,3,4,5,6,7,8,9] is a resurgent area of interest in the literature [10,11,12,13,14,15], mainly because of its applications in various crucial topics such as fractional quantum Hall effect (FQHE) [14,16,17,18,19,20,21,22,23], Horava -Lifshitz gravity [14,24,25], Galilean anomalies and hydrodynamics [26] etc. Inspite of its ubiquity there exists diverse and in some cases conflicting opinions about the details of NC geometry in the current literature.…”