Causality in the maximally extended Reissner--Nordström spacetime with identifications

Abstract

The maximally extended Reissner--Nordstr\"{o}m (RN) spacetime with $e^2 < m^2$ can be interpreted either as an infinite chain of asymptotically flat regions connected by tunnels between timelike singularities or as a set of just one asymptotically flat region and one tunnel; the repetitions of this set in the infinite chain being identified. The second interpretation gives rise to the suspicion of acausality, i.e. the possibility of sending messages to one's own past. A numerical investigation of this problem was carried out in this paper and gave the following result. Let E be the initial point of a radial timelike future-directed ingoing geodesic G, lying halfway between the outer horizon and the image of the null infinity in the maximally extended RN spacetime. Let E$'$ be the first future copy of E. It was verified whether the turning point of G will lie to the future or to the past from the past light cone (PLC) of E$'$. In the second case the breach of causality does occur. It turned out that the acausality is present when $V_E$, the timelike coordinate of E, is negative with a sufficiently large $|V_E|$, and is absent with a sufficiently large $V_E > 0$. In between these values there exists a $\widetilde{V}_E$, dependent on the initial data for the geodesic, for which the turning point lies on the PLC. So, the identification does lead to acausality. Nonradial timelike and null geodesics were also investigated, and a few hitherto unknown properties of the maximal extension were revealed. For example, the singularity arc at $r = 0$ may be convex or concave, depending on the values of $m$ and $e$.