Avoiding closed timelike curves with a collapsing rotating null dust shell

We present an idealised model of gravitational collapse, describing a collapsing rotating cylindrical shell of null dust in flat space, with the metric of a spinning cosmic string as the exterior. We find that the shell bounces before closed timelike curves can be formed. Our results also suggest slightly different definitions for the mass and angular momentum of the string. Introduction A stationary, cylindrically-symmetric rotating solution of the Einstein equations containing closed timelike curves (CTCs) was first published in [18]. Its possible role as a time-machine was discussed by Tipler [16] and Bonnor [1]. Other metrics with CTCs include those of Gödel [7], Newman-Unti-Tamburino [13, 12], Gott [8] and the rotating cosmic string [5]. In [2, 3] Bonnor has emphasised the need for a proper understanding for the formation of CTCs. Several attempts have been made to rule out the possibility of creating CTCs [17, 9, 6]. This would follow from a proof of the strong cosmic censorship conjecture, in the following sense: the domain of dependence of an appropriate initial surface, evolving under the Einstein equations with a reasonable matter content, is (by construction) globally hyperbolic, and hence contains no CTCs. If in addition it is (generically) inextendible, which is the usual formulation of strong cosmic censorship, then there are no CTCs at all. It would then follow that, if one took an initial surface with initial data set up so that CTCs could form, in fact they would not. It would be desirable to have non-stationary models which could provide test beds for this idea. In this paper we present a simplified example, obtained by matching a Minkowski interior across a collapsing, rotating cylindrical shell of null dust with positive energy density to a spinning cosmic string exterior. We find that the shell bounces before the critical radius at which CTCs would be formed can be reached. We follow the conventions of [19]. The solutions considered are four-dimensional but the z-coordinate is ignorable, and we will omit it from the calculations. 1 Interior We take the flat Minkowski interior, g = −dτ + dξ + dη, The simplest condition is to take it compact. A similar matching was done across a stationary cylindrical shell in [11]; thin shell collapse in 2+1 dimensions was studied in [14], [4].