Reduction of topological invariants on null hypersurfaces

TL;DR

This work addresses how gravitational topological invariants reduce to null hypersurfaces and how their boundary terms relate to helicity observables. It shows that the Pontryagin term vanishes at future null infinity under standard fall-off conditions, while the Nieh–Yan term in the teleparallel equivalent of GR yields a nontrivial gravitational helicity flux density via $H=\Lambda^2\int_{\mathcal I^+} \theta^a \wedge T_a$ with $\Lambda^2=1/(8\pi G)$, once a regularized spin connection is chosen. Extending the analysis to near-horizon (Carrollian) geometry reveals that the Pontryagin term can produce a nontrivial horizon observable—interpreted as a Carrollian fluid helicity with a precise identification $f=\sqrt{\kappa}$ and $\beta_A=-\sqrt{2/\kappa}\,\partial_A\varphi$—while the Nieh–Yan term does not yield a nontrivial horizon observable in TEGR. Overall, the paper clarifies how different topological invariants encode radiative gravitational helicity information on null infinity and horizons, and connects horizon Carrollian dynamics to Carrollian helicity through explicit geometric data.

Abstract

Gravitational helicity flux density represents the angular distribution of helicity flux in general relativity. In this work, we explore its relationship to the reduction of topological invariants at future null infinity. Contrary to initial expectations, the Pontryagin term, which contributes to the gravitational chiral anomaly, is not related to the gravitational helicity flux density. Instead, the Nieh-Yan term, another topological invariant within the teleparallel equivalent of general relativity (TEGR), can reproduce this flux density. We also reduce these two topological invariants to null hypersurfaces describing near-horizon geometry. In this near-horizon context, the Pontryagin term yields a non-trivial quantity that may characterize a Carrollian fluid helicity relevant to near-horizon physics.