arXiv subjects

N. E. Rieger

Publications and source records attributed to N. E. Rieger.

4 recordsLinked to original sources

arXiv2026

Coverings and Non-Hausdorff Extensions of Misner Spacetime

Misner spacetime is obtained by quotienting a timelike wedge of two-dimensional Minkowski spacetime by a discrete boost. The familiar Hausdorff extensions and the Hawking--Ellis non-Hausdorff extension are classical, but the passage from covering constructions of the punctured Minkowski plane to genuine extensions of Misner spacetime is subtler than is often stated. In this article we separate systematically the notions of covering and extension, classify the connected coverings of the punctured model that are compatible with the boost action, construct the induced quotient spacetimes, and exhibit explicit embeddings of Misner spacetime into each of them. This yields a natural family consisting of the Hawking--Ellis extension, its universal-cover analogue, and the intermediate finite cyclic coverings. We prove a precise non-Hausdorffness statement for the punctured quotient, formulate and prove a classification theorem for the resulting family within the covering-compatible class, and identify a causal adjacency invariant distinguishing the finite-sheeted and universal-cover cases. Finally, we compare these spacetimes with two-dimensional Schwarzschild-type metrics from the viewpoint of isocausality.

gr-qc↗

arXiv2026

A Transformation Theorem for Transverse Signature-Type Changing Semi-Riemannian Manifolds

In the early eighties Hartle and Hawking put forth that signature-type change may be conceptually interesting, paving the way to the so-called 'no boundary' proposal for the initial conditions for the universe. Such singularity-free universes have no beginning, but they do have an origin of time. In mathematical terms, we are dealing with signature-type changing manifolds where a Riemannian region (i.e., a region with a positive definite metric) is smoothly joined to a Lorentzian region at the surface of transition where time begins. We present a transformation prescription to transform an arbitrary Lorentzian manifold into a singular signature-type changing manifold. Then we establish the Transformation Theorem, asserting that, conversely, under certain conditions, such a metric $(M,\tilde{g})$ can be obtained from some Lorentz metric $g$ through the aforementioned transformation procedure. By augmenting the assumption by certain constraints, mutatis mutandis, the global version of the Transformation Theorem can be proven as well. In conclusion, we make use of the Transformation Prescription to demonstrate that the induced metric on the hypersurface of signature change is either Riemannian or a positive semi-definite pseudo metric.

math.DG↗

arXiv2025

Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical

In 1983, Hartle and Hawking proposed the no-boundary proposal, suggesting that the universe has no beginning in the sense of a spacetime singularity or boundary. Nevertheless, there is an origin of time. Mathematically, this involves signature-type changing manifolds in which a Riemannian region smoothly transitions to a Lorentzian region across the hypersurface $\mathcal{H}$ where time begins. We develop a coherent framework for signature changing manifolds with a degenerate yet smooth metric. Established Lorentzian tools and results are then adapted to this setting, and new definitions are introduced that carry unforeseen causal implications. A noteworthy consequence is the presence of locally time-reversing loops through every point on the hypersurface. Imposing global hyperbolicity on the Lorentzian region, we prove that for every point $p \in M$ there exists a pseudo-timelike loop self-intersecting at $p$. Equivalently, $M$ always admits a closed pseudo-timelike path around which the time direction reverses, preventing any consistent distinction between future- and past-directed vectors. To an observer near $\mathcal{H}$, such loops may appear as the creation of a particle-antiparticle pair at two distinct points.

math.DG↗

arXiv2025

Embedding Signature-Changing Manifolds: A Braneworld and Kaluza-Klein Perspective

We investigate a class of semi-Riemannian manifolds characterized by smooth metric signature changes with a transverse radical. This class includes spacetimes relevant to cosmological models such as the Hartle-Hawking "no boundary" proposal, where a Riemannian manifold transitions smoothly into a Lorentzian spacetime without boundaries or singularities. For this class, we prove the existence of global isometric embeddings into higher-dimensional pseudo-Euclidean spaces. We then strengthen this result by demonstrating that a specific type of global isometric embedding, which we term an $\mathcal{H}$-global embedding, also exists into both Minkowski space and Misner space. For the canonical $n$-dimensional signature-changing model, we explicitly construct a full global isometric embedding into $(n+1)$-dimensional Minkowski and Misner spaces, a significantly stronger result than an $\mathcal{H}$-global embedding for this specific case. This embedding framework provides new geometric tools for studying signature change and braneworlds through the geometry of submanifolds embedded in a bulk, thus presenting a mathematically well-defined approach to these phenomena.

math.DG↗