How to Wick rotate generic curved spacetime

TL;DR

This paper challenges the traditional view of Wick rotation as a complex time deformation, proposing instead that it should be understood as a complex metric deformation, which has significant implications for quantum gravity.

Contribution

It introduces a new perspective on Wick rotation as a complex deformation of the spacetime metric rather than the time coordinate, impacting quantum gravity approaches.

Findings

Wick rotation is better viewed as a metric deformation.

This reformulation affects the understanding of Euclidean quantum gravity.

Flat space physics remains unaffected by this new perspective.

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.