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

Sabbir A. Rahman

TL;DR

The paper formalizes the two-sheeted topology of extended Kerr-type spacetimes by identifying the excised Kerr manifold as a branched double cover with the ring singularity as branch locus and a deck transformation that exchanges sheets. It proves a parity-of-crossings rule for admissible geodesics crossing ring singularities and extends the construction to spacetimes with $N$ disjoint rings, where the fundamental group is the free group $F_N$ and connected double covers correspond to $ abla F_N o abla Z_2$, with a symmetric cover implementing global sheet exchange. The results extend to the maximal analytic extension, yielding global two-sheeted structure and a globally valid parity rule, applicable to the Carter–Penrose chain. The authors then couple this topological picture to causality via a Novikov-style self-consistency principle, showing that globally consistent histories are discretized by ring-crossing parities. The framework paves the way for studying field dynamics and quantum-classical interfaces in two-sheeted Kerr-type spacetimes and suggests connections between topology, causality, and discrete histories.

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.

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

TL;DR

The paper formalizes the two-sheeted topology of extended Kerr-type spacetimes by identifying the excised Kerr manifold as a branched double cover with the ring singularity as branch locus and a deck transformation that exchanges sheets. It proves a parity-of-crossings rule for admissible geodesics crossing ring singularities and extends the construction to spacetimes with disjoint rings, where the fundamental group is the free group and connected double covers correspond to , with a symmetric cover implementing global sheet exchange. The results extend to the maximal analytic extension, yielding global two-sheeted structure and a globally valid parity rule, applicable to the Carter–Penrose chain. The authors then couple this topological picture to causality via a Novikov-style self-consistency principle, showing that globally consistent histories are discretized by ring-crossing parities. The framework paves the way for studying field dynamics and quantum-classical interfaces in two-sheeted Kerr-type spacetimes and suggests connections between topology, causality, and discrete histories.

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 -action that exchanges the two sheets ( and ) 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 disjoint ring singularities, we prove that the fundamental group of the excised manifold is the free group 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.

Paper Structure

This paper contains 36 sections, 11 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Kerr and its Analytic Extension
  3. The Kerr metric in Boyer--Lindquist coordinates
  4. The ring singularity and the excised manifold
  5. Admissible geodesics
  6. Topology of the Excised Kerr Manifold
  7. Geometric neighbourhood of the ring singularity
  8. The fundamental group of $\mathcal{M}_{\mathrm{exc}}$
  9. Construction of the double cover
  10. Interpretation as a branched double cover
  11. Geodesic Continuation Through the Ring and the Sheet-Exchange Map
  12. Geodesic equations near the ring
  13. Lifting geodesics to the double cover
  14. Parity-of-crossings for the single-ring case
  15. The Multi-Ring Case: Topology, Covering Spaces, and Geodesic Parity
  16. ...and 21 more sections

Key Result

Lemma 3.1

Let $\mathcal{M}_{\mathrm{exc}}$ be the excised Kerr manifold (the elementary extended block) with a single ring singularity removed. Then there exists a deformation retraction of a neighbourhood of the ring in $\mathcal{M}_{\mathrm{exc}}$ onto $\mathbb{R}^3\setminus S^1$. In particular,

Theorems & Definitions (31)

  • Definition 2.1: Excised Kerr manifold
  • Definition 2.2: Admissible geodesic
  • Lemma 3.1
  • proof
  • Theorem 3.2: Existence of a unique connected double cover
  • proof
  • Lemma 4.1: Smooth continuation across $r=0$
  • proof
  • Proposition 4.2: Ring traversal induces sheet exchange
  • proof
  • ...and 21 more