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This is a contribution on 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 to vanish (strong signature change). Led by a Lagrangian point of view, we write down eight candidate action functionals S 1 ,. . . 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| 1/2 R[g]. The other four actions are constructed as sums of integrals over singe-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 * Work supported by the Austrian Academy of Sciences in the framework of the "Austrian Programme for Advanced Research and Technology". 0 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 favoured over strong one, because it requires lessà priori restrictions for the class of off-shell metrics. In addition, a proposal for the use of the Lagrangian framework in cosmology is made.

The complete class of cosmic string solutions in flat space that undergo rigid rotation around a fixed axis is exhibited. It consists of spiral-type curves and plane cycloids and possesses three free parameters.PACS number(s): 98.80.Cq, 04.20.Jb

It is shown -- using a FRW model with ${\bf S}^3 \times {\bf S}^6$ as spatial sections and a positive cosmological constant -- that classical signature change implies a new compactification mechanism. The internal scale factor is of the order $\Lambda^{-1/2}$, and the solutions are stable against small perturbatons. In the case of compactified ${\bf S}^6$, it is shown that the effective four-dimensional space-time metric has Lorentzian signature, undergoes exponential inflation in ${\bf S}^3$ and is unique. Speculations concerning relations to quantum cosmology and conceivable modifications are added.Comment: 31 pages, LaTeX, 4 Figures are available on request as hard copies. Some clarifications are included concerning the particular model of signature change considered; misprints corrected and some references adde

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