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

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 [1]). 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 S n . 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.

A toy Penrose inequality and its proof

“…where the integral is performed over a closed curve γ, rather than a closed surface. This inequality does in fact hold true for the relevant class of curves, as can be shown using Theorem 2 in [11]. However, it does not contain the area (or rather the length) of the curve-in accordance with the fact that mass is dimensionless in this context.…”

“…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].…”

On the Penrose inequality along null hypersurfaces

“…Proof. The proof is immediate if we use the relation between u and the support function h, see [17]. We provide an alternative proof here for the sake of self-consistency.…”

“…In terms of the support function h of S 0 (see [17] for its definition in the present context), this inequality can be rewritten (after some manipulations) in the form…”

“…The first one was the existence of a quasi-local object defined on surfaces which enjoyed monotonicity properties along past directed null geodesic foliations. This functional was introduced by Bergqvist [4] ant it has been called sometimes Bergqvist mass in the literature [16], [17]. The second fact was a suitable upper bound for the area of the weakly outer trapped surface S 0 .…”