Causality in the maximally extended extreme Reissner--Nordstr\"{o}m spacetime with identifications

TL;DR

This paper investigates whether causality violations occur in the maximally extended extreme Reissner--Nordström spacetime with identified asymptotic regions, concluding that such identifications do not lead to causality breaches based on numerical evidence.

Contribution

It provides numerical analysis showing that causality violations do not occur in the extreme RN spacetime with identified asymptotic regions, unlike in the non-extreme case.

Findings

Timelike and null geodesics cannot reach the causal past of the emitter’s future copy.

Ingoing radial null geodesics hit the singularity at r=0 and stop.

Identification does not lead to causality breaches in the extreme case.

Abstract

In continuation of the similarly titled paper on the $e^2 < m^2$ Reissner -- Nordstr\"{o}m (RN) metric (arXiv 2409.03786), in this paper it was verified whether it is possible to send (by means of timelike and null geodesics) messages to one's own past in the maximally extended {\it extreme} ($e^2 = m^2$) RN spacetime with the asymptotically flat regions being identified. Numerical examples show that timelike and nonradial null geodesics originating outside the horizon have their turning points to the future of the past light cone of the future copy of the emitter. This means that they cannot reach the causal past of the emitter's future copy. Ingoing radial null geodesics hit the singularity at $r = 0$ and stop there. So, unlike in the $e^2 < m^2$ case, identification of the asymptotically flat regions does not lead to causality breaches. A formal mathematical proof of this thesis (as opposed to the numerical examples given in this paper) is still lacking and desired.