How to Wick rotate generic curved spacetime

Abstract

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.