Abstract
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. This leads to considerations of signature-type changing spacetimes, wherein the “initially” Riemannian manifold, characterized by a positive definite metric, undergoes a signature-type change, ultimately transitioning into a Lorentzian universe without boundaries or singularities. A metric with such a signature-type change is inherently degenerate or discontinuous at the locus of the signature change.
We present a coherent framework for signature-type changing manifolds characterized by a degenerate yet smooth metric. We adapt well-established Lorentzian tools and results to the signature-type changing scenario. Subsequently, we explore global issues, specifically those related to the causal structure, in singular semi-Riemannian manifolds. We introduce new definitions that carry unforeseen causal implications. A noteworthy consequence is the presence of locally closed time-reversing loops through each point on the hypersurface. By imposing the constraint of global hyperbolicity on the Lorentzian region, we demonstrate that throughout every point in M, there always exists a pseudo-timelike loop. Or put another way, there always exists a closed pseudo-timelike path in M around which the direction of time reverses, and a consistent designation of future-directed and past-directed vectors cannot be defined.
Moreover, we present a method for converting any arbitrary Lorentzian manifold (M,g) into a transverse type-changing semi-Riemannian manifold \ensuremath{(M,\tilde{g})}. Then we establish the Transformation Theorem, asserting that, conversely under certain conditions, such a metric (M,\tilde{g}) can be obtained from a Lorentz metric g through the aforementioned transformation procedure.
Rieger, Nathalie