Unique Carrollian manifolds emerging from Einstein spacetimes

TL;DR

This paper classifies shear-free null hypersurfaces in Einstein spacetimes, revealing their structure as Carrollian manifolds with specific geometric connections, advancing understanding of spacetime boundaries.

Contribution

It explicitly characterizes shear-free null hypersurfaces in Einstein spacetimes as Carrollian manifolds with unique geometric structures, including connections.

Findings

All shear-free null hypersurfaces in Einstein spacetimes are characterized as Carrollian structures.

Each hypersurface admits a unique pair of Ehresmann and affine connections.

The geometric structures are solutions to projected Cartan structure equations.

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.