The Einstein equation for signature type changing spacetimes. (English) Zbl 0820.53070

Kossowski, Marek; Kriele, Marcus

Proc. R. Soc. Lond., Ser. A 446, No. 1926, 115-126 (1994).

A signature type changing spacetime is an \(m\)-dimensional manifold \(M\) with a “metric” tensor field \(g\) whose signature changes from Riemannian to Lorentzian at a hypersurface \(D\). (The physical motivation to consider such objects comes from quantum gravity.) In the paper under review the authors restricted themselves to the case that \(g\) is everywhere smooth and that at all points of \(D\) the kernel of \(g\) is transverse to \(D\). For Einstein’s field equation with initial data given at \(D\), local existence and uniqueness theorems are proven in the analytic category. The first theorem refers to a dust energy momentum tensor, the second one to a scalar field. Moreover, it is shown that in either case all solutions to the initial value problem are “resolvable”, i.e., that they can be constructed by analytically joining two solutions of the non-signature-changing initial value problems.

Reviewer: V.Perlick (Berlin)

MSC:

53Z05	Applications of differential geometry to physics
83C75	Space-time singularities, cosmic censorship, etc.

Keywords:

signature change; signature type change