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

Xavier Bekaert, Kevin Morand

TL;DR

The paper develops an ambient Leibnizian framework that unifies nonrelativistic (Galilean/Newton–Cartan) and ultrarelativistic (Carrollian) geometries by treating them as projections or embeddings of a single ambient spacetime. It analyzes compatible connections in both torsionfree and torsional regimes, showing the space of ambient Galilean connections is affine and classified by ambient gravitational fieldstrengths, with explicit maps linking ambient and base geometries. The results extend the Eisenhart lift to a broader ambient setting and demonstrate that any Galilean or Carrollian manifold can be obtained from or embedded into an ambient Galilean manifold. These developments lay groundwork for incorporating Lorentzian ambient Bargmann structures in future work and deepen the geometric relationship between nonrelativistic and ultrarelativistic gravities.

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.

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

TL;DR

The paper develops an ambient Leibnizian framework that unifies nonrelativistic (Galilean/Newton–Cartan) and ultrarelativistic (Carrollian) geometries by treating them as projections or embeddings of a single ambient spacetime. It analyzes compatible connections in both torsionfree and torsional regimes, showing the space of ambient Galilean connections is affine and classified by ambient gravitational fieldstrengths, with explicit maps linking ambient and base geometries. The results extend the Eisenhart lift to a broader ambient setting and demonstrate that any Galilean or Carrollian manifold can be obtained from or embedded into an ambient Galilean manifold. These developments lay groundwork for incorporating Lorentzian ambient Bargmann structures in future work and deepen the geometric relationship between nonrelativistic and ultrarelativistic gravities.

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.

Paper Structure

This paper contains 21 sections, 38 theorems, 111 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Newtonian structures embedded inside Bargmannian structures
  3. Nonrelativistic theories of gravitation and kinematical algebras
  4. Outline of the paper
  5. Notations
  6. Ambient spacetimes as principal bundles
  7. Ambient geometry
  8. Projection and lift of a function
  9. Projection and lifts of a vector field
  10. Projection of a 1-form
  11. Projection of a Koszul connection
  12. Ambient Leibnizian structures
  13. Ambient Galilean connections
  14. Torsionfree connections
  15. Torsional connections
  16. ...and 6 more sections

Key Result

Proposition 2.2

Let $\mathscr{M}$ be a manifold. The following structures on $\mathscr{M}$ are in bijective correspondence: The relation between these geometrical structures is as follows: the congruence $I$ is the family of integral curves of the vector field $\xi$ as well as the vertical foliation of the manifold $\mathscr{M}$.

Figures (2)

  • Figure 1: Kinematical content of Newton-Cartan geometry
  • Figure 2: Newtonian structures embedded inside Bargmannian structures

Theorems & Definitions (62)

  • Definition 2.1: Platonic screen Bekaert2013c
  • Proposition 2.2: Ambient structure
  • Definition 2.3: Screen worldvolume Bekaert2013c
  • Definition 2.4: Ehresmann connection
  • Definition 2.5: Lift of a function
  • Definition 2.6: Projectable and invariant vector fields
  • Proposition 2.7
  • Definition 2.8: Lift of a vector field
  • Proposition 2.9
  • Proposition 2.10: Projectable 1-form
  • ...and 52 more