NONSINGULAR CHARGED BLACK HOLES À LA PALATINI

We argue that the quantum nature of matter and gravity should lead to a discretization of the allowed states of the matter confined in the interior of black holes. To support and illustrate this idea, we consider a quadratic extension of general relativity (GR) formulated à la Palatini and show that nonrotating, electrically charged black holes develop a compact core at the Planck density which is nonsingular if the mass spectrum satisfies a certain discreteness condition. We also find that the area of the core is proportional to the number of charges times the Planck area.

Pregroups and length functions

Pregroups were defined by Stallings[7] who showed that the elements of the group they define have a normal form up to an equivalence called interleaving. Recently Rimlinger[5] has shown that subject to a discreteness and a boundedness condition any pregroup P defines a graph of groups. We show here that closer analysis of P makes the boundedness condition superfluous. In § 1 we give results of Stallings and Rimlinger and prove some key lemmas. In §2 we show that the discreteness condition gives an integer-valued length function in the sense of Lyndon [4]. It follows from the work of Chiswell [2] and Serre [6] that this defines a graph of groups. I would like to thank the referee for his careful reading and useful comments on this paper.

Comments On a Discreteness Condition for Subgroups of SL(2, C)

SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then