On the Penrose inequality along null hypersurfaces

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 Ω 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 Ω is derived.

Interior of a Binary Black Hole Merger

“…out using event horizons is seen as strong support for cosmic censorship. Thus far, the inequality has been established rigorously for time symmetric initial data for an arbitrary number of BHs [13,14,26] (see [1,15,31,33] for some alternate approaches, and [32] for a review). For a BBH system, the Penrose inequality implies…”

Proof of a Null Geometry Penrose Conjecture

“…1´| v| 2 . In the general setting, considering an asymptotically flat null cone Ω, it follows that E H approaches a measure of total energy along an asymptotically round foliation called a Bondi Energy, E B (see [11]). Returning to Schwarzschild, we verify that the standard null…”

A toy Penrose inequality and its proof

“…Lately progress has been made in the original setup [11,12]. The original idea has also been generalized to other spacetime geometries-including Minkowski as a special case-for instance Schwarzschild spacetime [13], and asymptotically flat spacetimes satisfying the dominant energy condition in general [14]. Taken together all of these results have shown that the inequality holds for a large class of surfaces.…”