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Potential Carroll Structures and Special Carrollian Manifolds

Samuel Blitz, Gabriel Herczeg, David McNutt

TL;DR

The paper investigates intrinsic geometries on null hypersurfaces by formalizing two Carrollian frameworks: special Carrollian manifolds (SCMs), where a principal Ehresmann connection fixes the ambient affine connection with minimal torsion, and potential Carroll structures (PCS), where a 1-form acts as a metric potential. It establishes minimal data conditions that uniquely determine the associated connections for both SCMs and PCS and derives explicit curvature- and torsion-based criteria that govern when one structure can be transformed into the other via modifications of the Ehresmann connection. The results reveal strong geometric constraints and obstructions to converting SCMs to PCS and vice versa, highlighting how Carrollian structures encode the intrinsic geometry of null boundaries relevant to holography. Overall, the work clarifies the relationship between SCMs and PCS and provides a principled framework for analyzing null-hypersurface geometry in holographic contexts.

Abstract

It is well-known that unlike space-like and time-like hypersurfaces, null hypersurfaces in Lorentzian manifolds do not naturally inherit an affine connection from the spacetime in which they are embedded. On the other hand, recent developments in flat-space holography motivate the study of the intrinsic geometry of null hypersurfaces such as null infinity and black hole event horizons. Here we initiate the study of potential Carroll structures, a candidate for an intrinsic description of null hypersurfaces which may be particularly useful in settings where conformal isometries are of interest, and we explore their relationship to another such candidate intrinsic geometry, the special Carrollian manifolds.

Potential Carroll Structures and Special Carrollian Manifolds

TL;DR

The paper investigates intrinsic geometries on null hypersurfaces by formalizing two Carrollian frameworks: special Carrollian manifolds (SCMs), where a principal Ehresmann connection fixes the ambient affine connection with minimal torsion, and potential Carroll structures (PCS), where a 1-form acts as a metric potential. It establishes minimal data conditions that uniquely determine the associated connections for both SCMs and PCS and derives explicit curvature- and torsion-based criteria that govern when one structure can be transformed into the other via modifications of the Ehresmann connection. The results reveal strong geometric constraints and obstructions to converting SCMs to PCS and vice versa, highlighting how Carrollian structures encode the intrinsic geometry of null boundaries relevant to holography. Overall, the work clarifies the relationship between SCMs and PCS and provides a principled framework for analyzing null-hypersurface geometry in holographic contexts.

Abstract

It is well-known that unlike space-like and time-like hypersurfaces, null hypersurfaces in Lorentzian manifolds do not naturally inherit an affine connection from the spacetime in which they are embedded. On the other hand, recent developments in flat-space holography motivate the study of the intrinsic geometry of null hypersurfaces such as null infinity and black hole event horizons. Here we initiate the study of potential Carroll structures, a candidate for an intrinsic description of null hypersurfaces which may be particularly useful in settings where conformal isometries are of interest, and we explore their relationship to another such candidate intrinsic geometry, the special Carrollian manifolds.
Paper Structure (12 sections, 12 theorems, 83 equations)

This paper contains 12 sections, 12 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Carrollian Structures, Connections, and Torsions
  4. Minimal features of special Carrollian manifolds
  5. Principal 1-form implies connection
  6. Connection implies 1-form
  7. Minimal Features of Potential Carroll Structures
  8. 1-form implies connection
  9. Connection implies 1-form
  10. Relating special Carrollian manifolds to potential Carroll structures
  11. From potential Carroll structures to SCMs
  12. From SCMs to potential Carroll structures

Key Result

Lemma 2.1

Let $(M,g, \ell)$ be a Carrollian structure equipped with an affine connection $\nabla$ satisfying $\nabla g = \nabla \ell = 0$. Then the torsion tensor, $T$, associated with the connection satisfies:

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 23 more