Geometric aspects of covariant Wick rotation.

We discuss some generic geometric properties of metrics ${\hat g}_{ab}$ constructed from Lorentzian metric $g_{ab}$ and a nowhere vanishing, hypersurface orthogonal, timelike direction field $u^a$. The metric ${\hat g}_{ab}$ has Euclidean signature in a certain domain, with the transition to Lorentzian signature occurring at some hypersurface $\Sigma$ orthogonal to $u^a$. Geometry associated with ${\hat g}_{ab}$ has recently been shown to yield remarkable new insights for classical and quantum gravity. In this work, we prove several general results that would be applicable in physically relevant spacetimes for congruences $u^i$ with non-zero acceleration $a^i$. In particular, we consider examples of dynamical spherically symmetric spacetimes and maximally symmetric spacetimes. We also discuss holonomy of loops when part of the loop lies in the Euclidean regime. We show that the contribution of the Euclidean domain to holonomy is completely determined by extrinsic curvature $K_{ab}$ of $\Sigma$ and acceleration $a^i$. We also discuss entropy associated with euclidean lagrangians obtained by replacing $g_{ab} \to \widehat{g}_{ab}$.