The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces

TL;DR

This paper analyzes the asymptotic behavior of Hawking energy along broad classes of null hypersurfaces, establishing its limit in terms of background foliations and invariance properties, generalizing known results about Bondi energy.

Contribution

It introduces a general framework for understanding the limit of Hawking energy without requiring geodesic or large sphere foliations, extending previous asymptotic analyses.

Findings

Hawking energy converges to a limit described by background foliation and a positive function.

The limit expression has covariance and invariance properties under foliation changes.

Standard Bondi energy results are recovered within this generalized setting.

Abstract

In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces $\Omega$ satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference background geodesic foliation approaching large spheres and a positive function, constant along the null generators on $\Omega$, which describes the relation between the two foliations at infinity. The integrand in the limit expression has interesting covariance and invariance properties with respect to change of background foliation. The standard result that the Hawking energy tends to the Bondi energy under suitable circumstances is recovered in this framework.