Exact Solutions of Einstein's Field Equations: Notation

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.

Wicked metrics

It is an article of folklore that the collection of ideas identified as Euclidean quantum gravity may be derived from ordinary Lorentzian signature gravity by the procedure of Wick rotation. This note will attempt to shed some light on this relatively ill-understood procedure. I argue that it proves inappropriate and unhelpful to regard Wick rotation in terms of a complex deformation of the time coordinate. Rather, Wick rotation can more usefully be viewed as a complex deformation of the spacetime metric. This simple reformulation of the Wick rotation procedure, while it leaves flat space physics unaffected, has profound implications for quantum gravity.

How to Wick rotate generic curved spacetime

A Wick-rotatable metric is purely electric

Wick rotations and real GIT

Wick-rotations of pseudo-Riemannian Lie groups

We show that a metric of arbitrary dimension and signature which allows for a standard Wick-rotation to a Riemannian metric necessarily has a purely electric Riemann and Weyl tensor

Christer Helleland

Papers

A Wick-rotatable metric is purely electric

Wick rotations and real GIT

Wick-rotations of pseudo-Riemannian Lie groups

A Wick-rotatable metric is purely electric

Real GIT with applications to compatible representations and Wick-rotations