On the Penrose inequality along null hypersurfaces

TL;DR

This paper develops a new approach to the null Penrose inequality using a functional on surfaces along null hypersurfaces, proving a general inequality and exploring conditions under which it holds, with applications to specific spacetime cases.

Contribution

Introduces the Renormalized Area Method and establishes conditions for the null Penrose inequality, extending previous results and applying to shear-free, vacuum, and Minkowski spacetime cases.

Findings

Proves a general Penrose-type inequality involving Hawking energy limits.

Shows existence of Geodesic Asymptotic Bondi (GAB) foliations approaching large spheres.

Derives a bound on the area of quasi-local black holes in terms of asymptotic quantities.

Abstract

The null Penrose inequality, i.e. the Penrose inequality in terms of the Bondi energy, is studied by introducing a funtional on surfaces and studying its properties along a null hypersurface $\Omega$ extending to past null infinity. We prove a general Penrose-type inequality which involves the limit at infinity of the Hawking energy along a specific class of geodesic foliations called Geodesic Asymptotic Bondi (GAB), which are shown to always exist. Whenever, this foliation approaches large spheres, this inequality becomes the null Penrose inequality and we recover the results of Ludvigsen-Vickers and Bergqvist. By exploiting further properties of the functional along general geodesic foliations, we introduce an approach to the null Penrose inequality called Renormalized Area Method and find a set of two conditions which implies the validity of the null Penrose inequality. One of the conditions involves a limit at infinity and the other a condition on the spacetime curvature along the flow. We investigate their range of applicability in two particular but interesting cases, namely the shear-free and vacuum case, where the null Penrose inequality is known to hold from the results by Sauter, and the case of null shells propagating in the Minkowski spacetime. Finally, a general inequality bounding the area of the quasi-local black hole in terms of an asymptotic quantity intrinsic of $\Omega$ is derived.