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

TL;DR

This paper develops a framework for signature-changing semi-Riemannian manifolds with smooth degenerate metrics, revealing the existence of pseudo-timelike loops that challenge traditional notions of causality and time direction.

Contribution

It introduces new definitions and tools for analyzing signature-changing manifolds, demonstrating the existence of locally time-reversing loops and pseudo-timelike paths in such geometries.

Findings

Existence of pseudo-timelike loops at every point on the hypersurface.

Presence of closed pseudo-timelike paths reversing time.

Implication of loops as particle-antiparticle creation near the hypersurface.

Abstract

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.