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Embedding Galilean and Carrollian geometries I. Gravitational waves

Kevin Morand

TL;DR

This work extends the ambient approach to embedding non-Riemannian geometries by broadening the class of gravitational waves beyond Bargmann–Eisenhart to Platonic and Dodgson waves, enabling a richer embedding of torsionfree Galilean and Carrollian manifolds. Galilean embeddings are realised on quotient spaces of Kundt or Platonic waves, recovering the Eisenhart lift in a generalized, Lichnerowicz-inspired form; Carrollian embeddings are achieved on lightlike foliations by invariant or pseudo-invariant leaves, with Dodgson waves providing the largest ambient class for such embeddings. Crucially, the work demonstrates that (A)dS spacetimes admit lightlike foliations by Carrollian manifolds of equal radius, and establishes a Carroll train as an ultrarelativistic analogue of the Eisenhart lift. The results offer a unifying geometric framework to relate non-Riemannian kinematics to higher-dimensional relativistic backgrounds, with potential applications to holography, flat holography, and ultrarelativistic limits of gravitational theories.

Abstract

The aim of this series of papers is to generalise the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this first part, we propose a generalisation of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalisation by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves - dubbed Dodgson waves - is introduced and geometrically characterised. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows to generalise the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds.

Embedding Galilean and Carrollian geometries I. Gravitational waves

TL;DR

This work extends the ambient approach to embedding non-Riemannian geometries by broadening the class of gravitational waves beyond Bargmann–Eisenhart to Platonic and Dodgson waves, enabling a richer embedding of torsionfree Galilean and Carrollian manifolds. Galilean embeddings are realised on quotient spaces of Kundt or Platonic waves, recovering the Eisenhart lift in a generalized, Lichnerowicz-inspired form; Carrollian embeddings are achieved on lightlike foliations by invariant or pseudo-invariant leaves, with Dodgson waves providing the largest ambient class for such embeddings. Crucially, the work demonstrates that (A)dS spacetimes admit lightlike foliations by Carrollian manifolds of equal radius, and establishes a Carroll train as an ultrarelativistic analogue of the Eisenhart lift. The results offer a unifying geometric framework to relate non-Riemannian kinematics to higher-dimensional relativistic backgrounds, with potential applications to holography, flat holography, and ultrarelativistic limits of gravitational theories.

Abstract

The aim of this series of papers is to generalise the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this first part, we propose a generalisation of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalisation by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves - dubbed Dodgson waves - is introduced and geometrically characterised. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows to generalise the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds.

Paper Structure

This paper contains 29 sections, 38 theorems, 31 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary and main results
  3. Intrinsic geometries
  4. Galilean
  5. Carrollian
  6. Geometry of gravitational waves
  7. Bargmannian structures
  8. Gravitational waves
  9. Kundt waves
  10. Bargmann--Eisenhart waves
  11. Platonic waves
  12. Walker waves
  13. Dodgson waves
  14. Summary
  15. Brinkmann coordinates
  16. ...and 14 more sections

Key Result

Proposition 2.4

Let $(\mathscr{M},\psi,h)$ be a Galilean structure with closed clock $\psi$.

Figures (3)

  • Figure 1: Foliation of a gravitational wave by wavefront worldvolumes
  • Figure 2: Hierarchy of gravitational waves
  • Figure 3: Summary of the embedding of torsionfree Galilean and Carrollian manifolds into gravitational waves

Theorems & Definitions (110)

  • Definition 2.1: Galilean structure
  • Remark 2.2
  • Definition 2.3: Torsionfree Galilean manifold
  • Proposition 2.4: Classification of torsionfree Galilean connections
  • Remark 2.5
  • Proposition 2.6
  • Definition 2.7: Newtonian manifold
  • Remark 2.8
  • Example 2.9: Galilean manifolds
  • Remark 2.10
  • ...and 100 more