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

TL;DR

This paper introduces a transformation theorem that links Lorentzian and signature-changing semi-Riemannian manifolds, providing a mathematical framework for models of the universe with signature change.

Contribution

It presents a transformation prescription and theorem that characterize how Lorentzian manifolds can be transformed into signature-changing manifolds, with implications for cosmological models.

Findings

A transformation prescription for Lorentzian to signature-changing manifolds.

Theorem establishing conditions for such transformations.

The induced metric on the signature change hypersurface is Riemannian or semi-definite.

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. 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.