Spacelike reduction of the gravitational topological terms and associated helicity densities

Abstract

Helicity in gravity is a multifaceted concept, and several inequivalent definitions exist in the literature. In this work, we reduce the topological Pontryagin and Nieh-Yan terms to a constant time spacelike hypersurface. Remarkably, upon applying the SVT decomposition at the linearized level, we obtain four distinct helicity densities. The helicity density arising from the Nieh-Yan term is termed the spin-1 and spin-2 gravitomagnetic helicity density, depending on whether it originates from vector or tensor modes. Meanwhile, the helicity density arising from the Pontryagin term is termed the spin-1 and spin-2 gravito-current helicity density, according to the corresponding mode contributions. We study the mode and multipole expansions of these helicity functionals and apply them to leading order Newtonian two-body systems, weak field boosted Kerr, a locally defined slowly varying gyratonic pp-wave model, and a linearized gravitational Hopfion.