Coverings and Non-Hausdorff Extensions of Misner Spacetime

Abstract

Misner spacetime is obtained by quotienting a timelike wedge of two-dimensional Minkowski spacetime by a discrete boost. The familiar Hausdorff extensions and the Hawking--Ellis non-Hausdorff extension are classical, but the passage from covering constructions of the punctured Minkowski plane to genuine extensions of Misner spacetime is subtler than is often stated. In this article we separate systematically the notions of covering and extension, classify the connected coverings of the punctured model that are compatible with the boost action, construct the induced quotient spacetimes, and exhibit explicit embeddings of Misner spacetime into each of them. This yields a natural family consisting of the Hawking--Ellis extension, its universal-cover analogue, and the intermediate finite cyclic coverings. We prove a precise non-Hausdorffness statement for the punctured quotient, formulate and prove a classification theorem for the resulting family within the covering-compatible class, and identify a causal adjacency invariant distinguishing the finite-sheeted and universal-cover cases. Finally, we compare these spacetimes with two-dimensional Schwarzschild-type metrics from the viewpoint of isocausality.