Wicked metrics

There are various ways of defining the Wick rotation in a gravitational context. In order to preserve the manifold structure, it would be preferable to view it as an analytic continuation of the metric, instead of the coordinates. We focus on one very general definition and show that it is not always compatible with the additional requirements of preserving the field equations and the symmetries at global level. The counterexamples are the de Sitter and Schwarzschild metrics: requiring that their Euclidean continuations satisfy Einstein's equations and have the same number of Killing vectors, we find that the Euclidean metric cannot be defined on the original Lorentzian manifold but only on a submanifold. This phenomenon seems to be due to the existence of horizons.