arXiv subjects

Nathalie E. Rieger

Publications and source records attributed to Nathalie E. Rieger.

4 recordsLinked to original sources

arXiv2026

Topological Obstructions to Riemannian-Lorentzian Continuation

Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $χ$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$χ(\overline{M_R})=χ(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.

math.DG↗

arXiv2026

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

Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics, with particular emphasis on global topological obstructions. Using explicit geometric constructions based on the topology of the Möbius strip, we produce examples of crosscap manifolds where the gluing junction coincides with the locus of signature change. Our main result shows that on non-orientable compact surfaces, the radical of such metrics cannot be everywhere transverse along the hypersurface of signature change. In particular, metrics arising from the transformation prescription $\tilde{g}=g+fV^{\flat}\otimes V^{\flat}$, with $g$ a Lorentzian metric and $f$ a smooth interpolation function, necessarily fail to satisfy the transversality condition. This obstruction is of purely global origin and is closely related to topological invariants such as the Euler characteristic and the non-existence of nowhere-vanishing vector fields. These results demonstrate that non-orientability imposes intrinsic limitations on the class of admissible signature-type changing metrics.

math.DG↗

arXiv2026

Causal classification of pathological Misner-type spacetimes

We investigate three causality-violating spacetimes: Misner space (including Kip Thorne's "moving wall" model), the pseudo-Schwarzschild spacetime, and a new model introduced here, the pseudo-Reissner-Nordström spacetime. Despite their different physical origins -- ranging from a flat vacuum solution to a black-hole-type vacuum solution to a non-vacuum model requiring exotic matter -- all three share a common warped-product structure, $2$-dimensional cylindrical base metrics of Eddington-Finkelstein type, and fundamental causal features such as Cauchy and chronology horizons, acausal regions, and analogous geodesic behaviour. Building on a conjecture first proposed in 2016, we present a formal proof that the three models are pairwise isocausal on their universal covers and on suitable causally regular regions of their compactified forms. The proof is constructive, providing explicit causal bijections on the covers and identifying a concrete deck-equivariance criterion governing descent to the compactified spacetimes: if the equivariance degree satisfies $|k|=1$ the models are globally isocausal, whereas if $|k|>1$ or equivariance fails, then at most a one-way causal relation holds between the compactified models. These results supply a rigorous causal classification linking these spacetimes, placing them within a unified Misner-type family and providing a framework for extending the classification to other causality-violating solutions.

gr-qc↗

arXiv2025

A Conceptual Introduction To Signature Change Through a Natural Extension of Kaluza-Klein Theory

We propose an extension of basic Kaluza-Klein theory in which the higher-dimensional Lorentzian manifold develops a Cauchy horizon rather than remaining globally hyperbolic as in the conventional framework. In this setting, the $U(1)$-generating Killing field, assumed to exist in Kaluza-Klein theory, undergoes a transition in its causal character, from spacelike in the globally hyperbolic region to timelike in an acausal extension through a horizon. This yields a (lower-dimensional) quotient manifold whose metric changes signature from Lorentzian to Riemannian. In this way, one observes a singular, signature changing transition emerging rather naturally from the projection of a globally smooth, even analytic, Lorentzian geometry ``up in the bundle''. This reveals a ``signature change without signature change'' scenario -- a phrasing inspired by John Wheeler -- and extends the usual Kaluza-Klein framework in a conceptually natural direction.

gr-qc↗