Existence of black holes due to concentration of angular momentum

Marcus A Khuri

doi:10.1007/jhep06(2015)188

Abstract

We present a general sufficient condition for the formation of black holes due to concentration of angular momentum. This is expressed in the form of a universal inequality, relating the size and angular momentum of bodies, and is proven in the context of axisymmetric initial data sets for the Einstein equations which satisfy an appropriate energy condition. A brief comparison is also made with more traditional black hole existence criteria based on concentration of mass.

Highlights

Where G is the gravitational constant, c is the speed of light, and represents an order of magnitude; the precise constant depends on the choice of radius R
We present a general sufficient condition for the formation of black holes due to concentration of angular momentum
This is expressed in the form of a universal inequality, relating the size and angular momentum of bodies, and is proven in the context of axisymmetric initial data sets for the Einstein equations which satisfy an appropriate energy condition

Summary

Heuristic evidence and precise formulation

In [7], Dain introduced an inequality relating the size and angular momentum for General. Axisymmetry is imposed primarily to obtain a suitable and well-defined notion of angular momentum for bodies. Note that in this setting gravitational waves have no angular momentum, so all the angular momentum is contained in the matter sources. Without this assumption quasi-local angular momentum is difficult to define [21]. In analogy with [7], we define the radius that appears in (1.2) by With these definitions of angular momentum J and radius R, we obtain a precise formulation of inequality (1.2), save for the universal constant C to be described below. Apparent horizons may be thought of as quasi-local notions of event horizons, and assuming Cosmic Censorship, they must generically be contained inside black holes [23]

Inequality between size and angular momentum for bodies

Criterion for the existence of black Holes