Geometric Inequalities for Black Holes

Abstract

It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.

Penrose-like inequality with angular momentum for minimal surfaces

“…This result has also been extended to other cases in [1,16,18]. Thus inequality (5) is equivalent to (see [13], [32]):…”

Isoperimetric surfaces and area-angular momentum inequality in a rotating black hole in new massive gravity

“…where µ = 4GM , the angular momentum is given by J = M a, M is the mass measured with respect to the zero mass black hole and b is the hair parameter. The rotational parameter a satisfies −l ≤ a ≤ l and the extreme case is obtained when |a| = l. Before continuing, from (6) we see that σ can be taken with either sign. If we allow it to be negative, then we notice that making the change b → −b and σ → −σ takes the metric functions to themselves, that is N → N , N φ → N φ , F → F .…”

“…For these rotating solution we are interested in the search of geometric inequalities as the ones presented in [6] for the Kerr black hole.…”

“…The extreme rotating case of this NMG black hole can be included after making a change in the hair parameter as suggested in [5]. The extreme case is obtained when |a| = l. We are interested in the search of geometric inequalities as the one presented in [6] for the Kerr black hole. We do this by finding the isoperimetric surfaces and analyzing their stability.…”

Comments on Penrose inequality with angular momentum for outermost apparent horizons

“…Several versions of the problem have been studied, see the review articles [26,4], as well as the general approaches to it [13,6,5]. Moreover one can strengthen the Penrose heuristic argument to include charge and angular momentum (see [10] [8] [26] for more details). Good progress has been made in considering the case of a charged black hole without angular momentum, and different versions of an inequality relating the mass, the area of the horizon and the electric charge have been studied [30,12,21,22,23,24].…”