general relativity - Wick Rotation in Curved space

So over time I have learned to do exhaustive searches before asking things here. Wick rotations are cool if you are trying to work in qft and make statements about the thermodynamics of some physical thing you are probing. You want to talk about some thermodynamics and who knows may be statistics, and like most people . . . oh Wick rotation is what you would think. Now we are in curved space, do you just throw away this tactic in the dust bin? And yes, I have read up a few things, but it seems I need a proper pointer to where to find how to attack this thing properly. Just really curious.

Answer

Matt Visser's How to Wick rotate generic curved spacetime is a great reference on this subject, which basically summarizes a lot of folklore on the subject.

Addendum (Summary of Paper). This turns out to be an important problem in quantum gravity and QFT in curved spacetime for the obvious reason ("How do we know the usual tricks still work in curved spacetime?").

Visser re-frames the Wick rotation in a more coordinate independent way (the naive $t\mapsto it$ prescription gives incorrect solutions even for de Sitter spacetime).

The more general Wick rotation analytically continues the metric while leaving the local coordinate charts invariant. When we restrict our attention back to flat spacetime, this approach recovers the usual QFT-textbook prescription for Wick rotations.

What does it look like? Well, suppose $g_{L}(-,-)$ is the metric tensor (using MTW-style notation), and $V$ is a non-vanishing timelike vector field. So in $-+++$ signature, $g_{L}(V,V)<0$. The Wick rotation amounts to swapping out this Lorentzian metric for: $$\tag{1}g_{\epsilon} = g_{L} + i\epsilon\frac{V\otimes V}{g_{L}(V,V)}$$ and using this $g_{\epsilon}(-,-)$ metric everywhere instead.

How do we recover the usual Wick rotation? Well, use flat spacetime, so $g_{L}\mapsto\eta_{L}$ and $V\mapsto(1,0,0,0)$. Then the propagator for the scalar field, for example, becomes $$\tag{2}\Delta_{F}(P) = \frac{-i}{\eta_{\epsilon}(P,P)+m^{2}}$$ where $\eta_{\epsilon}$ is the Wick rotated metric tensor for flat spacetime. Eq (2) precisely what you'd find in any generic textbook on QFT. So, good, this generalized procedure --- i.e., Eq (1) --- recovers the usual results we want.