Actions for signature change

This is a contribution to the controversy about junction conditions for classical signature change. The central issue in this debate is whether the extrinsic curvature on slices near the hypersurface of signature change has to be continuous (weak signature change) or has to vanish (strong signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals ${\mathit{S}}_1$,. . .,${\mathit{S}}_8$ as possible generalizations of general relativity and investigate to what extent each of these defines a sensible variational problem, and which junction condition is implied. Four of the actions involve an integration over the total manifold. A particular subtlety arises from the precise definition of the Einstein-Hilbert Lagrangian density ‖g${\mathrm{\Vert }}^{1/2}$R[g]. The other four actions are constructed as sums of integrals over single-signature domains. The result is that both types of junction conditions occur in different models, i.e., are based on different first principles, none of which can be claimed to represent the ‘‘correct’’ one, unless physical predictions are taken into account. From a point of view of naturality dictated by the variational formalism, weak signature change is slightly favored over a strong one because it requires less a priori restrictions for the class of off-shell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made.