Comments on Penrose inequality with angular momentum for outermost apparent horizons

Abstract

In axially symmetric space-times it is expected that the Penrose inequality can be strengthened to include angular momentum. In a recent work [2] we have proved a weaker version of this inequality for minimal surfaces, using the monotonicity of the Geroch energy on 2-surfaces along the inverse mean curvature flow. In this article, using similar techniques and the same measure of size, we extend and improve the previous result for compact and connected general horizons. For this case we use the monotonicity of the Hawking energy, instead of the Geroch energy, along the inverse mean curvature flow, and assume different conditions on the extrinsic curvature. This type of relations constitutes an important test to evaluate the cosmic censorship conjecture. *

A Penrose-type inequality with angular momentum and charge for axisymmetric initial data

“…However, it turns out that including horizon area together with angular momentum is quite difficult. In fact, there appear to be only two results in the literature to date in this direction [1,2], and the approach taken in those articles is based on inverse mean curvature flow. In contrast, the present paper focuses on the techniques used to establish the massangular momentum inequalities, namely minimizing renormalized harmonic energies.…”

Recent Developments in the Penrose Conjecture

“…This result also assumes maximal initial data in order to have non-negative scalar curvature. Anglada has recently used similar techniques and the monotonicity properties of the Hawking rather than the Geroch mass to prove the same Penrose inequality for axially symmetric initial data such that the boundary is a compact connected outermost apparent horizon, as long as one assumes existence of a smooth inverse mean curvature flow of surfaces which starts with spherical topology [42].…”

On extensions of the Penrose inequality with angular momentum

“…Inequality (3) uses the size measure R(S) of the apparent horizon [16]. This quantity is difficult to calculate, but we can use a bound, that was shown in [16], that R(S) is not larger than √ 10M C , which in turn is approximated by √ 10M BH . Thus the necessary condition for the validity of (3) reads…”

“…here S, M BH and J BH are the area, quasilocal mass and quasilocal angular momentum of the outermost apparent horizon. Anglada [16] and Khuri [17] have proved other versions of (2) under the assumption of axial symmetry,…”