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On the intrinsic torsion of spacetime structures

José Figueroa-O'Farrill

TL;DR

<3-5 sentence high-level summary>The paper develops a unified framework to study spacetime geometries via $G$-structures and their intrinsic torsion, using the Spencer differential to identify torsion as the obstruction living in a cokernel. It then explicitly classifies four families of non-Lorentzian geometries—Galilean, Carrollian, Aristotelian and Bargmannian—into 3, 4, 16 and 27 intrinsic-torsion classes respectively, with geometric characterisations and interrelations (notably Bargmannian structures relating to both Galilean and Carrollian via null reductions and null hypersurfaces). The results illuminate how intrinsic torsion encodes physically meaningful data such as the derivatives of clock forms and Carrollian/ Bargmannian second fundamental forms, and they lay groundwork for realizing these structures in homogeneous and null-reduction settings. The work deepens understanding of non-Lorentzian spacetime geometries and provides a structured taxonomy that connects nonrelativistic limits, null reductions, and embedded Carrollian geometries.

Abstract

We briefly review the notion of the intrinsic torsion of a $G$-structure and then go on to classify the intrinsic torsion of the $G$-structures associated with spacetimes: namely, galilean (or Newton-Cartan), carrollian, aristotelian and bargmannian. In the case of galilean structures, the intrinsic torsion classification agrees with the well-known classification into torsionless, twistless torsional and torsional Newton-Cartan geometries. In the case of carrollian structures, we find that intrinsic torsion allows us to classify Carroll manifolds into four classes, depending on the action of the Carroll vector field on the spatial metric, or equivalently in terms of the nature of the null hypersurfaces of a lorentzian manifold into which a carrollian geometry may embed. By a small refinement of the results for galilean and carrollian structures, we show that there are sixteen classes of aristotelian structures, which we characterise geometrically. Finally, the bulk of the paper is devoted to the case of bargmannian structures, where we find twenty-seven classes which we also characterise geometrically while simultaneously relating some of them to the galilean and carrollian structures. This paper is dedicated to Dmitri Vladimirovich Alekseevsky on his 80th birthday.

On the intrinsic torsion of spacetime structures

TL;DR

<3-5 sentence high-level summary>The paper develops a unified framework to study spacetime geometries via -structures and their intrinsic torsion, using the Spencer differential to identify torsion as the obstruction living in a cokernel. It then explicitly classifies four families of non-Lorentzian geometries—Galilean, Carrollian, Aristotelian and Bargmannian—into 3, 4, 16 and 27 intrinsic-torsion classes respectively, with geometric characterisations and interrelations (notably Bargmannian structures relating to both Galilean and Carrollian via null reductions and null hypersurfaces). The results illuminate how intrinsic torsion encodes physically meaningful data such as the derivatives of clock forms and Carrollian/ Bargmannian second fundamental forms, and they lay groundwork for realizing these structures in homogeneous and null-reduction settings. The work deepens understanding of non-Lorentzian spacetime geometries and provides a structured taxonomy that connects nonrelativistic limits, null reductions, and embedded Carrollian geometries.

Abstract

We briefly review the notion of the intrinsic torsion of a -structure and then go on to classify the intrinsic torsion of the -structures associated with spacetimes: namely, galilean (or Newton-Cartan), carrollian, aristotelian and bargmannian. In the case of galilean structures, the intrinsic torsion classification agrees with the well-known classification into torsionless, twistless torsional and torsional Newton-Cartan geometries. In the case of carrollian structures, we find that intrinsic torsion allows us to classify Carroll manifolds into four classes, depending on the action of the Carroll vector field on the spatial metric, or equivalently in terms of the nature of the null hypersurfaces of a lorentzian manifold into which a carrollian geometry may embed. By a small refinement of the results for galilean and carrollian structures, we show that there are sixteen classes of aristotelian structures, which we characterise geometrically. Finally, the bulk of the paper is devoted to the case of bargmannian structures, where we find twenty-seven classes which we also characterise geometrically while simultaneously relating some of them to the galilean and carrollian structures. This paper is dedicated to Dmitri Vladimirovich Alekseevsky on his 80th birthday.

Paper Structure

This paper contains 36 sections, 23 theorems, 132 equations, 8 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The intrinsic torsion of a $G$-structure
  3. $G$-structures
  4. Adapted connections
  5. Intrinsic torsion
  6. Galilean $G$-structures
  7. The group $G$ of a galilean structure
  8. The intrinsic torsion of a galilean structure
  9. Geometric characterisation
  10. Carrollian $G$-structures
  11. The group $G$ of a carrollian structure
  12. The intrinsic torsion of a carrollian structure
  13. Geometric characterisation of carrollian structures
  14. Aristotelian $G$-structures
  15. The group $G$ of an aristotelian structure
  16. ...and 21 more sections

Key Result

Proposition 2.1

Let $P \xrightarrow{\pi} M$ be a $G$-structure and $\omega \in \Omega^1(P,\mathfrak{g})$ the connection one-form of an Ehresmann connection with torsion two-form $\Theta \in \Omega^2_G(P,V)$. If $\omega' = \omega + \kappa$ is another Ehresmann connection, then its torsion two-form $\Theta' = \Theta defined by $\partial\kappa(v,w) = \kappa_v w - \kappa_w v$ for all $v,w \in V$.

Figures (8)

  • Figure 1: Action of boosts on $\mathfrak{so}(n-1)$-submodules of $Z^\perp \otimes V^*$
  • Figure 2: Hasse diagram of bargmannian structures
  • Figure 3: Hasse diagram of totally geodesic bargmannian structures
  • Figure 4: Hasse diagram of minimal bargmannian structures
  • Figure 5: Hasse diagram of totally umbilical bargmannian structures
  • ...and 3 more figures

Theorems & Definitions (42)

  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 32 more