Causal classification of pathological Misner-type spacetimes

Abstract

We investigate three causality-violating spacetimes: Misner space (including Kip Thorne's "moving wall" model), the pseudo-Schwarzschild spacetime, and a new model introduced here, the pseudo-Reissner-Nordstr\"{o}m spacetime. Despite their different physical origins -- ranging from a flat vacuum solution to a black-hole-type vacuum solution to a non-vacuum model requiring exotic matter -- all three share a common warped-product structure, $2$-dimensional cylindrical base metrics of Eddington-Finkelstein type, and fundamental causal features such as Cauchy and chronology horizons, acausal regions, and analogous geodesic behaviour. Building on a conjecture first proposed in 2016, we present a formal proof that the three models are pairwise isocausal on their universal covers and on suitable causally regular regions of their compactified forms. The proof is constructive, providing explicit causal bijections on the covers and identifying a concrete deck-equivariance criterion governing descent to the compactified spacetimes: if the equivariance degree satisfies $|k|=1$ the models are globally isocausal, whereas if $|k|>1$ or equivariance fails, then at most a one-way causal relation holds between the compactified models. These results supply a rigorous causal classification linking these spacetimes, placing them within a unified Misner-type family and providing a framework for extending the classification to other causality-violating solutions.