The Two-Sheeted Topology of Extended Kerr-Type Spacetimes and a Parity-of-Crossings Property for Ring-Traversing Geodesics (2512.06549v1)

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 paper establishes a rigorous covering-space construction of extended Kerr-type spacetimes by excising the ring singularity and analyzing π1, confirming the two-sheeted topology.
• It derives a parity-of-crossings theorem showing that an even number of ring-crossings returns a geodesic to its original sheet, while an odd number leads to a sheet exchange.
• The study imposes discrete topological constraints on admissible histories, with implications for causal dynamics in black holes and potential applications in quantum gravity.

The Two-Sheeted Covering Space Structure of Extended Kerr-Type Spacetimes

Background and Motivation

The analytic extension of the Kerr spacetime, a solution to the vacuum Einstein equations with mass $M$ and angular momentum parameter $a$, reveals a complex internal topology featuring a ring-shaped curvature singularity at $\Sigma=0$ (that is, $r=0$, $\theta=\pi/2$). The maximal extension includes two asymptotically flat regions ($r\to+\infty$ and $r\to-\infty$), connected via regions behind the event and Cauchy horizons, with the ring singularity acting as a “branch locus” for analytic continuation. Prior work has invoked the two-sheetedness of the Kerr geometry—geodesics traversing the ring transition from the “positive $r$” ($r>0$) to the “negative $r$” ($r<0$) sheet—but usually only through patching arguments or limiting constructions. A rigorous, global, covering-space construction clarifying this picture had remained absent.

Topological Properties and Covering Space Construction

The analysis begins by considering the elementary analytic extension of Kerr, $\mathcal{M}$, and excising the ring singularity $\mathcal{S}$ to yield a smooth Lorentzian manifold $$\mathcal{M}_{\mathrm{exc}} = \mathcal{M} \setminus\mathcal{S}$$. A detailed local coordinate analysis shows that neighborhoods near the ring are diffeomorphic to $\mathbb{R}^3$ minus an embedded circle, i.e., $\mathbb{R}^3\setminus S^1$. Classical algebraic topology results, confirmed via deformation retraction and van Kampen’s theorem, demonstrate that $\pi_1(\mathcal{M}_{\mathrm{exc}}) \cong \mathbb{Z}$. Thus, every non-contractible loop around the ring singularity defines a generator of the first homotopy group.

Accordingly, there exists a unique connected double cover of $\mathcal{M}_{\mathrm{exc}}$, specified by the surjective homomorphism from $\mathbb{Z}\to\mathbb{Z}_2$ that sends a single generator to $1$. The deck group is $\mathbb{Z}_2$, the non-trivial element of which exchanges the two sheets. This formalizes the oft-cited but informal statement that Kerr’s extension is “two-sheeted,” with the ring acting as a branch locus and analytic continuation around/through the ring performing a sheet exchange.

This construction generalizes naturally to the case of $N$ disjoint Kerr-type ring singularities. The fundamental group becomes the free group $F_N$, and connected double covers are classified by homomorphisms $F_N\to\mathbb{Z}_2$. Of the $2^N-1$ non-trivial possibilities, the physically relevant case is when each ring singularity maps to the non-trivial deck element, yielding a uniform $\mathbb{Z}_2$ involution that acts as a sheet exchange upon crossing any ring.

Geodesic Continuation and Parity-of-Crossings

The analysis of admissible geodesics is central. These are geodesics that avoid the singularity $\mathcal{S}$ (i.e., do not intersect $r=0$, $\theta=\pi/2$) but may cross the hypersurface $r=0$ at $\theta\neq\pi/2$, where curvature remains finite. The geodesic equations are regular at such crossings, and geodesics pass smoothly from $r>0$ to $r<0$ (or vice versa), with the sign of $r$ strictly reversing in an analytic neighborhood of the crossing.

Crucially, each such crossing of $r=0$ by an admissible geodesic implements the non-trivial deck transformation: it takes the geodesic from one sheet of the double cover to the other. This yields the fundamental parity-of-crossings theorem: an admissible geodesic that crosses the ring an even number of times returns to its initial sheet, while an odd number of crossings moves it to the opposite sheet. This parity depends only on the total number of traversals, not on the order or which rings are crossed.

This parity argument extends to the maximal analytic extension of Kerr (the infinite Carter–Penrose chain). Defining the two global sheets as the unions of all the asymptotic $r>0$ and $r<0$ regions, the parity-of-crossings property holds globally—providing a rigorous, topological underpinning for the “two universes” language of earlier Carter and Boyer–Lindquist analyses.

Causality, Chronology Violation, and Discrete Self-Consistent Histories

Kerr and Kerr-type spacetimes are notorious for containing regions with closed timelike curves (CTCs), particularly in the $r<0$ domain and near the ring singularity. This produces potential chronology violations and challenges to global determinism.

Within the new covering-space framework, any admissible geodesic or evolution must be “sheet-consistent”: after all traversals of the ring(s), the physical fields and worldlines must be compatible with the identification imposed by the sheet-exchange involution. This leads to a discrete selection rule akin to the Novikov self-consistency principle. The space of globally consistent histories decomposes into discrete sectors, each labeled by the vector of ring-crossing parities for all admissible causal curves. Thus, not all conceivable evolutions are allowed—only those compatible with the parity structure enforced by the topology of the excised manifold and its double cover.

This discrete, topological restriction on admissible histories persists even though the underlying field equations are continuous. It bears indirect analogy to quantum superselection rules, with parity information acting as a discrete global constraint. This is a strong, nontrivial claim: permissible histories in these spacetimes are quantized by topological sector.

Implications and Outlook

The formalism developed in this work rigorously establishes the branched covering structure of extended Kerr and Kerr-like spacetimes with ring singularities. The identification of the parity-of-crossings law has immediate implications for the global causal and topological dynamics of geodesics, classical fields, and possibly semiclassical phenomena in such geometries.

Major theoretical implications include:

• Sheet-exchange as a selection rule: The deck transformation symmetry directly dictates the possible global evolution sectors, with physically distinguishable consequences for observers traversing the interior regions of rotating black holes.
• Constraints on Cauchy data and predictability: In chronology-violating regions, evolution is not arbitrary. The sheet-exchange structure quantizes allowed histories, sharpening the typical breakdown of predictability seen in CTC-containing spacetimes.
• Potential applications to quantum gravity: The discrete parity structure may be relevant to any future quantum theory incorporating topology change, exotic matter, or sign-changing mass scenarios, as suggested by analysis of the maximal extension and negative mass “mirror" sheets.
• A foundation for field theoretic investigations: The covering-space approach opens the way for systematic analysis of classical and quantum field equations (including boundary conditions and mode quantization) in the context of two-sheeted or multi-sheeted topologies.

Anticipated future developments include a detailed study of matter evolution and semiclassical effects on the double cover, the interaction of boundary conditions with sheet-exchange, and the role of such discrete topological data in black hole microphysics or information-theoretic models.

Conclusion

This work provides a mathematically precise and topologically robust account of the two-sheeted extension of Kerr-type metrics. The parity-of-crossings law for ring-traversing geodesics is rigorously established via covering space theory. The structure generalizes to any collection of disjoint ring singularities and extends to the infinite maximal extension. The associated global self-consistency condition discretizes the set of permissible classical or semiclassical histories, ensuring physical evolution respects both the geometry and the fundamental $\mathbb{Z}_2$ sheet-exchange symmetry. These results form a rigorous and extensible foundation for further mathematical and physical analysis of extended rotating black hole spacetimes.

Reference: "The Two-Sheeted Topology of Extended Kerr-Type Spacetimes and a Parity-of-Crossings Property for Ring-Traversing Geodesics" (2512.06549)