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

TL;DR

The paper develops a rigorous framework for signature-changing semi-Riemannian manifolds with a transverse radical, introducing pseudo-timelike curves and a generalized affine parameter to handle crossing the degenerate hypersurface $\mathcal{H}$. It proves local and global loop theorems showing time-reversing pseudo-timelike loops exist near $\mathcal{H}$ and, under global hyperbolicity, through every point in $M$, implying chronology-violating structures with a particle–antiparticle interpretation near the hypersurface. The results extend Lorentzian causal analysis to singular geometries and offer a precise mathematical lens on Hartle–Hawking-type no-boundary scenarios. Overall, the work clarifies how signature change interacts with causality, time orientation, and global spacetime structure in a rigorously defined setting.

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.

Figures (8)

• Figure 1: The curve $\gamma$ is not pseudo-timelike since it approaches a null vector at the locus of signature change. This curve is asymptotically lightlike.
• Figure 2: The curve defined by $t=\textrm{sgn}(x)\cdot\left(\mid\frac{3}{2}\tan x\mid\right)^{\frac{2}{3}}$.
• Figure 3: Riemannian and Lorentzian region in the Hartle-Hawking no-boundary model.
• Figure 4: In the left example the curves $\alpha$ and $\gamma$ are both future-directed. The curve $\beta$ runs within the edge that is twisted and identified with the left edge; therefore $\beta$ is neither future-directed nor past-directed. In the right example the curves $\alpha$, $\beta$ and $\gamma$ are future-directed. In both examples the loops around $\mathcal{H}$ are neither future-directed nor past-directed.
• Figure 5: For an event $p\in\mathcal{H}$ there exists a future-directed pseudo-timelike curve (as depicted) that connects the points $p$ and $q$ in $M$. Similarly any point in $M$ can be reached by such a future-directed pseudo-timelike curve from $p$. That is why for the pseudo-chronological future we have $\mathcal{I}^{+}(p)=\{q\in M\colon p\ll q\}=M$.

Theorems & Definitions (61)

• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Theorem 1
• Definition 3: Pseudo-timelike curve
• Definition 4: Pseudo-timelike
• Definition 5: Pseudo-time orientable
• Theorem 2: Local loops
• Theorem 3: Global loops