Wicked metrics

There are various ways of defining the Wick rotation in a gravitational context. There are good arguments to view it as an analytic continuation of the metric, instead of the coordinates. We focus on one very general definition and argue that it is incompatible with the requirement of preserving the field equations and the symmetries at global level: in some cases the Euclidean metric cannot be defined on the original Lorentzian manifold but only on a submanifold. This phenomenon is related to the existence of horizons, as illustrated in the cases of the de Sitter and Schwarzschild metrics.