Signature change in general relativity

Abstract

This work gives various classical results of relevance to quantum cosmology. The junction conditions are derived for matching a region of Lorentz-signature spacetime to a region of Riemannian-signature space across a spatial surface according to the vacuum Einstein or Einstein-Klein-Gordon equations. In particular, the spatial metric and the Klein-Gordon field must be instantaneously stationary at the junction. Real, globally valid fields and equations are found for the general Cauchy problem where signature change is allowed. An affine parameter is defined for geodesics crossing the junction orthogonally. For homogeneous and isotropic cosmological-constant cosmologies, the signature-changing solution is unique up to a constant, non-singular and inflationary, with a flatness parameter which is ultimately unity. The hypothesis of initially Riemannian signature gives a prediction of a highly restricted class of observable cosmologies for a given matter field. In particular, inflation, initially zero entropy and partial isotropy are predicted.