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Carrollian manifolds and null infinity: A view from Cartan geometry

Yannick Herfray

TL;DR

This work unifies conformal Carrollian perspectives on null infinity within the Cartan geometry framework. It shows that characteristic gravity data at null infinity correspond to Cartan geometries modelled on the Poincaré group, while gauge choices interpolate between Carrollian, conformally Carrollian, and Bondi-gauge pictures; gravitational radiation appears as the obstruction to identifying a global Poincaré subgroup inside the BMS group, encoded by the curvature of the Cartan connection. The paper develops a detailed tractor- and null-tractor formalism, introduces the Poincaré operator, and clarifies how Bondi gauge alters the available symmetry and connection data. The results provide a structural, coordinate-free understanding of null infinity and its symmetries, with implications for memory effects, radiative data, and the geometric interpretation of asymptotic symmetries across dimensions. Together, these insights offer a coherent Cartan-geometric language for the boundary geometry of asymptotically flat spacetimes.

Abstract

We discuss three different (conformally) Carrollian geometries and their relation to null infinity from the unifying perspective of Cartan geometry. Null infinity \emph{per se} comes with numerous redundancies in its intrinsic geometry and the two other Carrollian geometries can be recovered by making successive choices of gauge. This clarifies the extent to which one can think of null infinity as being a (strongly) Carrollian geometry and we investigate the implications for the corresponding Cartan geometries. The perspective taken, which is that characteristic data for gravity at null infinity are equivalent to a Cartan geometry for the Poincaré group, gives a precise geometrical content to the fundamental fact that ``gravitational radiation is the obstruction to having the Poincaré group as asymptotic symmetries''.

Carrollian manifolds and null infinity: A view from Cartan geometry

TL;DR

This work unifies conformal Carrollian perspectives on null infinity within the Cartan geometry framework. It shows that characteristic gravity data at null infinity correspond to Cartan geometries modelled on the Poincaré group, while gauge choices interpolate between Carrollian, conformally Carrollian, and Bondi-gauge pictures; gravitational radiation appears as the obstruction to identifying a global Poincaré subgroup inside the BMS group, encoded by the curvature of the Cartan connection. The paper develops a detailed tractor- and null-tractor formalism, introduces the Poincaré operator, and clarifies how Bondi gauge alters the available symmetry and connection data. The results provide a structural, coordinate-free understanding of null infinity and its symmetries, with implications for memory effects, radiative data, and the geometric interpretation of asymptotic symmetries across dimensions. Together, these insights offer a coherent Cartan-geometric language for the boundary geometry of asymptotically flat spacetimes.

Abstract

We discuss three different (conformally) Carrollian geometries and their relation to null infinity from the unifying perspective of Cartan geometry. Null infinity \emph{per se} comes with numerous redundancies in its intrinsic geometry and the two other Carrollian geometries can be recovered by making successive choices of gauge. This clarifies the extent to which one can think of null infinity as being a (strongly) Carrollian geometry and we investigate the implications for the corresponding Cartan geometries. The perspective taken, which is that characteristic data for gravity at null infinity are equivalent to a Cartan geometry for the Poincaré group, gives a precise geometrical content to the fundamental fact that ``gravitational radiation is the obstruction to having the Poincaré group as asymptotic symmetries''.
Paper Structure (33 sections, 6 theorems, 130 equations, 1 figure, 4 tables)

This paper contains 33 sections, 6 theorems, 130 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction/Summary
  2. Carrollian spacetimes
  3. Homogeneous model
  4. Tetrad formalism adapted to Carrollian geometry
  5. Strongly Carrollian geometry
  6. The Carrollian Cartan connection
  7. "Carrollian good-cuts"
  8. BMS coordinates
  9. Carrollian good-cuts
  10. Covariantly constant sections
  11. Null Infinity
  12. Homogeneous model
  13. Conformal geometry of null infinity
  14. Bundle of scale
  15. Vertical connection
  16. ...and 18 more sections

Key Result

Theorem 1.1

Fundamental theorem of Cartan geometry (see e.g. sharpe_differential_1997) Let $\left(\mathcal{G} \to M , \omega\right)$ be a Cartan geometryLet us recall to the reader that a Cartan geometry modelled on $G/H$ is the data of a $H$-principal bundle $\mathcal{G} \to M$ together with a $\mathfrak{g}$-v

Figures (1)

  • Figure 1: Galilean (non-relativistic) versus Carrollian (ultra-relativistic) limit: As the speed of light goes to infinity/zero the tangent null cones degenerate into lines/hypersurfaces of constant absolute space/time

Theorems & Definitions (8)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1