2024/09/04 by Rieger, N. E., Hasse, W. · 2 citations
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2409.02403
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 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 ∈ 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 H, such loops may appear as the creation of a particle-antiparticle pair at two distinct points.