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Unique Carrollian manifolds emerging from Einstein spacetimes

S. Blitz, D. McNutt, P. Nurowski

TL;DR

The paper tackles the problem of classifying shear-free null hypersurfaces embedded in Einstein spacetimes by explicitly solving the Cartan structure equations projected onto the hypersurface. It shows that such NHSs are equivalent to Carrollian manifolds equipped with a unique pair of coframe and connection, and it derives two principal cases (NEH-like and non-NEH) with closed-form data, thereby providing a canonical geometric bridge from NHSs to Carrollian geometry. This mapping fixes the Carrollian frame and yields a well-defined intrinsic connection, enabling analysis of horizon physics (e.g., Carrollian hydrodynamics and Brown–York charges) and potential holographic applications. The work thus resolves how to canonically fix the Ehresmann connection on these degenerate geometries and opens avenues for generalizations to broader spacetimes and additional fields.

Abstract

We explicitly determine all shear-free null hypersurfaces embedded in an Einstein spacetime, including vacuum asymptotically flat spacetimes. We characterize these hypersurfaces as oriented 3-dimensional manifolds where each is equipped with a coframe basis, a structure group and a connection. Such manifolds are known as null hypersurface structures (NHSs). The coframe and connection one-forms for an NHS appear as solutions to the projection of the Cartan structure equations onto the null hypersurface. We then show that each NHS corresponds to a Carrollian structure equipped with a unique pair of Ehresmann connection and affine connection.

Unique Carrollian manifolds emerging from Einstein spacetimes

TL;DR

The paper tackles the problem of classifying shear-free null hypersurfaces embedded in Einstein spacetimes by explicitly solving the Cartan structure equations projected onto the hypersurface. It shows that such NHSs are equivalent to Carrollian manifolds equipped with a unique pair of coframe and connection, and it derives two principal cases (NEH-like and non-NEH) with closed-form data, thereby providing a canonical geometric bridge from NHSs to Carrollian geometry. This mapping fixes the Carrollian frame and yields a well-defined intrinsic connection, enabling analysis of horizon physics (e.g., Carrollian hydrodynamics and Brown–York charges) and potential holographic applications. The work thus resolves how to canonically fix the Ehresmann connection on these degenerate geometries and opens avenues for generalizations to broader spacetimes and additional fields.

Abstract

We explicitly determine all shear-free null hypersurfaces embedded in an Einstein spacetime, including vacuum asymptotically flat spacetimes. We characterize these hypersurfaces as oriented 3-dimensional manifolds where each is equipped with a coframe basis, a structure group and a connection. Such manifolds are known as null hypersurface structures (NHSs). The coframe and connection one-forms for an NHS appear as solutions to the projection of the Cartan structure equations onto the null hypersurface. We then show that each NHS corresponds to a Carrollian structure equipped with a unique pair of Ehresmann connection and affine connection.
Paper Structure (7 sections, 5 theorems, 56 equations)

This paper contains 7 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Cartan Structure equations and their pullback to a null hypersurface
  3. Field equations for null hypersurfaces in Einstein solutions
  4. Proof of Proposition \ref{['Prop:NEHCarroll']}
  5. Proof of Proposition \ref{['Prop:NotNEHCarroll']}
  6. Shear-free Null hypersurface structures as Carrollian manifolds
  7. Conclusions

Key Result

Proposition 1

Any solution to the Einstein structure equations NullCartan2Einstein on a null hypersurface $\mathcal{N}$ satisfying $\rho = \sigma = 0$ is gauge equivalent to M = e^wd ζ, K =d r + (X + r w̅_,ζ)d ζ + (X̅ + r w_,ζ̅)d ζ̅, with the following connection coefficients and their complex conjugates: Here, $w = w(\zeta, \bar{\zeta}), X=X(\zeta, \bar{\zeta})$ are arbitrary complex-valued functions, and

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • Remark