On the Penrose inequality for dust null shells in the Minkowski spacetime of arbitrary dimension

TL;DR

This paper investigates the Penrose inequality for null shells in Minkowski spacetime across arbitrary dimensions, reformulating it geometrically and establishing its validity for specific classes of surfaces.

Contribution

It reformulates the Penrose inequality in geometric terms for null shells in Minkowski spacetime and proves its validity in certain cases and dimensions.

Findings

Validity of the inequality for null shells in any dimension when the surface lies in the past null cone of a point.

A sufficient condition for the inequality's validity in four-dimensional Minkowski spacetime.

The inequality holds for a large class of convex surfaces in Minkowski spacetime.

Abstract

A particular, yet relevant, particular case of the Penrose inequality involves null shells propagating in the Minkowski spacetime. Despite previous claims in the literature, the validity of this inequality remains open. In this paper we rewrite this inequality in terms of the geometry of the surface obtained by intersecting the past null cone of the original surface S with a constant time hyperplane and the "time height" function of S over this hyperplane. We also specialize to the case when S lies in the past null cone of a point and show the validity of the corresponding inequality in any dimension (in four dimensions this inequality was proved by Tod). Exploiting properties of convex hypersurfaces in Euclidean space we write down the Penrose inequality in the Minkowski spacetime of arbitrary dimension n+2 as an inequality for two smooth functions on the sphere. We finally obtain a sufficient condition for the validity of the Penrose inequality in the four dimensional Minkowski spacetime and show that this condition is satisfied by a large class of surfaces.