Penrose inequality in anti–de Sitter space

Abstract

For asymptotically flat spacetimes the Penrose inequality gives an initial data test for the weak cosmic censorship hypothesis. We give a formulation of this inequality for asymptotically anti-deSitter (AAdS) spacetimes, and show that the inequality holds for time asymmetric data in spherical symmetry. Our analysis is motivated by the constant-negative-spatial-curvature form of the AdS black hole metric.Keywords: Cosmic Censorship, Penrose Inequality, anti-deSitter spacetime One of the most important unresolved questions in classical general relativity is Penrose's cosmic censorship hypothesis. The weak version of this hypothesis, Weak Cosmic Censorship (WCC), is the statement that a spacetime singularity arising from gravitational collapse of "reasonable" matter must be shrouded by an event horizon [1].Penrose also proposed an "initial data test" suggested by the WCC hypothesis. Consider an asymptotically flat solution of Einstein equations with matter satisfying the dominant energy condition. Then if a Cauchy slice of this solution contains an outer-trapped 2-surface S of area A(S), and if M is the Arnowitt-Deser-Misner (ADM) mass of the data on the slice, the conjectured inequality is(1)The intuition behind this inequality is that the mass m ah contained within the outermost apparent horizon cannot exceed the total mass M of the data. This is expected to be the case for an ongoing gravitational collapse since, in general, the region between this horizon and spatial infinity will contain uncollapsed (positive energy) matter. Thus the inequality may equivalently be written aswith equality in the quiescent black hole state, when all matter has fallen into the horizon. A review of the status of WCC and the Penrose inequality appears in Refs. [2,3] In this paper we study a generalization of this inequality for asymptotically anti-deSitter (AAdS) spacetimes in 4-spacetime dimensions. This requires definitions of AAdS data and quasi-local mass. The former is well known (see eg.[4]), and the latter requires a suitable generalization of one several mass definitions [5] to AAdS. The basic requirement is that any quasi-local mass should give the total mass of data in the asymptotic limit. This is guided by, and tested for, using the Schwarzschild-AdS spacetime.We will use the Hawking mass [6], which has been used to study the Penrose inequality in the asymptotically flat case [7]. This quantity is a measure of the mass contained within a closed 2-surface S in a spatial slice of spacetime. The definition iswhere A S is the area of S and θ ± are the future directed ingoing (−) and outgoing (+) null expansions of S. In terms of the ADM data (q ab , K ab ) these arewhere s a is the unit normal to S. As a prelude to introducing the class of data we will work with, let us note that the AdS black hole may be written in a form analogous to the Painleve-Gulstrand (PG) form, but rather than using the flat-slice PG form for AdS, it is more convenient to work with 3−slices of constant negative curvature. With such a choice of coord...

General bounds on holographic complexity

“…The corresponding AdS dual would be a proof that the Schwarzschild-AdS geometry minimizes the volume within some class of wormholes (see related work by [45]). Such a result about C F would in fact constitute a parallel with a less famous companion to the positive energy theorem, the Riemannian Penrose Inequality, a lower bound on the mass given the existence of a minimal area surface [46][47][48][49]. More precisely, the area of a minimal surface on a maximal Cauchy slice in a spacetime of a given mass is conjectured to be identical to that of Schwarzschild with the same mass if and only if the spacetime is identical to Schwarzschild.…”

“…Note that while some results are established for the Riemannian Penrose Inequality in AdS[49][50][51][52][53][54], the full statement remains conjectural. We bring it up here to draw a parallel.…”

The Hawking energy on photon surfaces

“…Of course it has to be admitted that the Schwarzschild spacetime is a very special spacetime. The Penrose inequality holds in spherical symmetry [20,21], and we are not able to offer any suggestions about how to move beyond that. But we have seen that the Hawking energy has many subtle properties.…”

Holography abhors visible trapped surfaces

“…Concrete evidence in favor of such a conclusion was found recently in e.g. [33], which used the holographic dictionary to prove the Penrose Inequality in AdS [34][35][36], a key result implied by the combination of two oft-quoted but unproven conjectures: (1) that trapped surfaces lie behind horizons, and (2) that black holes equilibrate. The proof of [33] assumed neither (1) nor (2) but instead made use of the holographic entanglement entropy proposal of Ryu-Takayanagi [37] and Hubeny-Rangamani-Takayanagi [38] (HRT)…”