The Two-Sheeted Topology of Extended Kerr-Type Spacetimes and a Parity-of-Crossings Property for Ring-Traversing Geodesics

Abstract

We revisit the global structure of the extended Kerr spacetime and of a broader class of Kerr-type spacetimes possessing ring singularities. By working with the elementary analytic extension (the union of the interior and exterior regions glued across the disk), we show that excising the ring singularity yields a domain that can be realised as a branched double cover of an exterior Kerr region. The branch locus is the ring itself, and the associated deck transformation defines a non-trivial $\mathbb{Z}_2$-action that exchanges the two sheets ($r>0$ and $r<0$) of the spacetime. We give a covering-space characterisation of this double-sheeted structure and show that admissible geodesics which cross the ring singularity implement the non-trivial deck transformation. In particular, we prove a parity-of-crossings property: any admissible geodesic that traverses an even number of ring singularities returns to its original sheet, while an odd number of traversals terminates on the opposite sheet. Generalising to $N$ disjoint ring singularities, we prove that the fundamental group of the excised manifold is the free group $F_N$ generated by simple loops around each ring, and we classify the associated double covers. Identifying the physically distinguished cover where every ring induces a sheet exchange, we extend the parity-of-crossings theorem to the multi-ring setting. We then formally extend these results to the maximal analytic extension (the infinite Carter--Penrose chain), proving that the sheet-exchange mechanism applies globally to this infinite structure. Finally, applying the Novikov self-consistency principle to this topological framework, we demonstrate that the requirement of global consistency restricts admissible histories to discrete sectors labelled by ring-crossing parities.