Unique Carrollian manifolds emerging from Einstein spacetimes

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.

💡 Research Summary

The paper provides a complete classification of shear‑free null hypersurfaces (NHSs) embedded in four‑dimensional Einstein spacetimes, including vacuum asymptotically flat solutions. Starting from the Newman‑Penrose formalism, the authors introduce a complex co‑frame ((M,\bar M,\ell,k)) together with the associated connection one‑forms (\Gamma_{ab}). They write down the first and second Cartan structure equations and describe how Lorentz transformations preserving the null direction (k) act on the co‑frame and connection.

A null hypersurface (N) is defined by an embedding (\phi:C\to M) with (\phi^{*}L=0). Pull‑back of the Cartan equations onto (N) forces the imaginary part of the spin coefficient (\rho) and the coefficient (\kappa) to vanish, implying that the null generator (k) is hypersurface‑orthogonal and geodesic. The induced degenerate metric on (N) is (h=2M\bar M); together with the fundamental vector field (k) (dual to a real one‑form (K)) this defines a Carrollian structure. The equivalence class ((M,K)\sim(e^{i\phi}M,,A