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Möbius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds

Nathalie E. Rieger

TL;DR

This work analyzes signature-type change on non-orientable manifolds built from Möbius-strip topologies, focusing on transverse degeneracy loci $ mathcal{H}$ and the radical of the metric. It demonstrates three key phenomena: causal trapping in a rotating-Minkowski background after Transformation Prescription, topological obstructions to pseudo-space orientability via the Euler characteristic, and the emergence of mixed radicals at the change hypersurface that cannot arise from purely transverse transformations. Providing explicit models—the non-compact infinite Möbius strip, a compact Möbius strip with boundary, and the crosscap—reveals obstructions to applying standard prescriptions and motivates a generalized framework for signature change on non-orientable spaces. The results have implications for quantum cosmology and Wick-rotation-based approaches, highlighting the need for refined tools to handle non-orientable topology in singular semi-Riemannian settings.

Abstract

Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics. Using explicit constructions based on the topology of the Möbius strip, we produce examples of crosscap manifolds where the gluing junction serves as the locus of signature change. In another set of examples, we convert the Möbius strip into a singular signature-type changing manifold. For these resulting manifolds, we test whether the metric can be expressed as $\tilde{g}=g+fV^{\flat}\otimes V^{\flat}$, with $g$ a Lorentzian metric and $f$ a smooth interpolation function between the Lorentzian and Riemannian regions, separated by the signature change hypersurface $\mathcal{H}$. Our analysis reveals that the radical of the metric can transition from transverse to tangent at $\mathcal{H}$, pseudo-space orientability is obstructed by the Euler characteristic, and pseudo-time orientability may still hold. These examples illustrate subtle obstructions to applying standard transformation prescriptions for signature change and highlight novel phenomena in compact, non-orientable semi-Riemannian manifolds.

Möbius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds

TL;DR

This work analyzes signature-type change on non-orientable manifolds built from Möbius-strip topologies, focusing on transverse degeneracy loci and the radical of the metric. It demonstrates three key phenomena: causal trapping in a rotating-Minkowski background after Transformation Prescription, topological obstructions to pseudo-space orientability via the Euler characteristic, and the emergence of mixed radicals at the change hypersurface that cannot arise from purely transverse transformations. Providing explicit models—the non-compact infinite Möbius strip, a compact Möbius strip with boundary, and the crosscap—reveals obstructions to applying standard prescriptions and motivates a generalized framework for signature change on non-orientable spaces. The results have implications for quantum cosmology and Wick-rotation-based approaches, highlighting the need for refined tools to handle non-orientable topology in singular semi-Riemannian settings.

Abstract

Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics. Using explicit constructions based on the topology of the Möbius strip, we produce examples of crosscap manifolds where the gluing junction serves as the locus of signature change. In another set of examples, we convert the Möbius strip into a singular signature-type changing manifold. For these resulting manifolds, we test whether the metric can be expressed as , with a Lorentzian metric and a smooth interpolation function between the Lorentzian and Riemannian regions, separated by the signature change hypersurface . Our analysis reveals that the radical of the metric can transition from transverse to tangent at , pseudo-space orientability is obstructed by the Euler characteristic, and pseudo-time orientability may still hold. These examples illustrate subtle obstructions to applying standard transformation prescriptions for signature change and highlight novel phenomena in compact, non-orientable semi-Riemannian manifolds.
Paper Structure (15 sections, 7 theorems, 120 equations, 6 figures)

This paper contains 15 sections, 7 theorems, 120 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and fundamental definitions
  3. Orientability and pseudo-orientability
  4. Summary of main results
  5. Non-orientable models and signature change
  6. The Lorentzian background: rotating Minkowski metric
  7. Causal Structure and Causal Trapping
  8. Non-Compact Models: The Infinite Möbius strip
  9. General application of the Transformation Prescription
  10. Construction of the signature-changing infinite Möbius strip
  11. Compact models and topological obstructions
  12. Compact Möbius strip with boundary
  13. The crosscap manifold
  14. The real projective plane ($\mathbb{R}P^{2}$) construction
  15. Conclusion

Key Result

Proposition 1.8

Let $(M,g)$ be the two-dimensional manifold $M=\mathbb{R}^2$ with coordinates $(t,x)$ equipped with the "rotating Minkowski" metric For each $k\in\mathbb{Z}$ define the stationary stripe Then $\partial_t$ is a Killing field and is timelike on each $M_k$. Moreover, any future-directed causal curve which meets a stationary stripe $M_k$ at some point is entirely contained in $M_k$ (equivalently: cau

Figures (6)

  • Figure 1: The "rotating Minkowski" metric $g=-\cos(2\varphi)\,(dt)^{2}+2\sin(2\varphi)\,dt\,dx+\cos(2\varphi)\,(dx)^{2}$ on $\mathbb{R}^{2}$. The manifold has a "stripe-like causal pattern" consisting of alternating stationary and non-stationary stripes (regions), such that adjacent stripes are separated by lightlike curves.
  • Figure 2: The hypersurface of signature change can be either compact or non-compact: The very left example depicts a hypersurface that is not compact. The two right examples show a hypersurface that is compact. For the latter two, it should be noted that the inclinations of the tangents at $\mathcal{H}$ must also align with the identification.
  • Figure 3: The yellow unit square represents the quotient manifold with boundary $M=([0,1]\times[0,1])/\sim$, obtained by identifying the opposite sides via $(t,0)\sim(1-t,1)$.
  • Figure 4: The light cone structure in the Möbius strip.
  • Figure 5: Schematic light cone structure (not drawn to scale) for the metric $g=(1-t^{2})(dt)^{2}+2txdtdx+(1-x^{2})(dx)^{2}$. Pseudo-timelike curves pass through $\mathcal{H}$, then go through the disk which is the Riemannian regime, and finally re-emerge in the Lorentzian regime through $\mathcal{H}$ again.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Definition 1.2: Character of the radical
  • Definition 1.4
  • Definition 1.5: Pseudo-timelike
  • Definition 1.6: Pseudo-time orientable
  • Definition 1.7: Pseudo-space orientable
  • Proposition 1.8: Trapping of causal curves in stationary stripes
  • Proposition 1.9: Orientability
  • Proposition 1.10: Radical and signature change on the crosscap
  • Remark 2.1
  • Proposition
  • ...and 11 more