Inequalities Between Size, Mass, Angular Momentum, and Charge for Axisymmetric Bodies and the Formation of Trapped Surfaces

Abstract

Dedicated to the memory of our late friend and colleague Sergio Dain.Abstract. We establish inequalities relating the size of a material body to its mass, angular momentum, and charge, within the context of axisymmetric initial data sets for the Einstein equations. These inequalities hold in general without the assumption of the maximal condition, and use a notion of size which is easily computable. Moreover, these results give rise to black hole existence criteria which are meaningful even in the time-symmetric case, and also include certain boundary effects.

Self-gravitating perfect-fluid tori around black holes: Bifurcations, ergoregions, and geometrical properties

“…It is common in mathematical relativity to define the geometric size of an axially symmetric body in terms of the largest circumference of a circle that can be embedded within this body. Equivalently, one can search for the maximum of η µ η µ in the region occupied by the body [29]. The result can be quite surprising, as the circle with the largest circumference does not have to be the outermost one.…”

Two mass conjectures on axially symmetric black hole-disk systems

“…The two statements of (11) and (12) can be useful in estimating the amount of angular momentum within a fixed volume. There is already a formidable work done in this direction [29,30], although results are far from being precise [14]. We think that (11) and (12) are true and they would lead to substantial improvements of present estimates.…”

Bekenstein bounds, Penrose inequalities, and black hole formation

“…It is a basic folklore belief that if enough matter/energy is concentrated in a sufficiently small region, then gravitational collapse must ensue. This is typically referred to as the hoop conjecture or trapped surface conjecture [26,30], and is quite difficult to formulate precisely, see the references in [23]. One of the most general results in this direction is due to Schoen and Yau [28], who exploited the techniques developed in their proof of the positive mass theorem [27] to prove the existence of apparent horizons whenever matter density is highly concentrated.…”

“…This will be rigorously proven in spherical symmetry, and motivation will be given to indicate why the result should hold in generality. Related results concerning black hole existence due to concentration of angular momentum or charge have been given in [18,19,23], using different methods.…”