Abstract
Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much electric charge. In this article, we are interested in bounds on the angular momentum and electromagnetic charges, in terms of total mass and size. We are mainly concerned with inequalities for black holes and ordinary objects. The former are the most studied systems in this context in General Relativity, and where most results have been found. Ordinary objects, on the other hand, present numerous challenges and many basic questions concerning geometrical estimates for them are still unanswered. We present the many results in these areas. We make emphasis in identifying the mathematical conditions that lead to such estimates, both for black holes and ordinary objects.
Highlights
It is interesting that in Theorem 1 a rigidity statement is obtained even in the non-electrovacuum case. It is not known whether an analogous rigidity result holds for axially symmetric matter fields with non-trivial angular momentum in which the equality in (146) is achieved
Cabrera-Munguia et al (2010) work on the Tomimatsu and Dietz–Hoenselaers solution describing two Kerr black holes, one of which has negative mass. They find that there is a rank in the parameters of the individual black holes such that the total mass is smaller than the total angular momentum, that is M2
The only extra assumptions we make in this result is that the functions α and f, fk entering the stability criteria for minimal surfaces and marginally outer trapped surfaces (MOTSs) respectively must be axially symmetric
Summary
The other is the maximum charge and/or angular momentum a black hole can have, beyond which it becomes a naked singularity This problem arose after Reissner (1916) and Nordström (1918) found the solution describing a static, spherically symmetric, electrically charged object. These thresholds in physical parameters values can be identified as limit cases of appropriate geometrical inequalities. We address systems containing ordinary material objects and/or black holes The latter, and the frontier between black holes and naked singularities, is the original and main motivation for the geometrical inequalities presented here.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
