There are a few viable ways to formulate Wick rotatability that preserve distinct features.
One is mentioned in the post:
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
There's also the paper by Helleland: Wick rotations and real GIT
By this answer one knows that strong causality and pure electricity are two orthogonal conditions at each generic point $p$ of a manifold $M$.
My personal question is
if any of these definitions of Wick Rotatability(specially that of Helleland) imply that a Wick Rotatable spacetime is Strongly Causal.
Bearing in mind that a Wick rotatable metric, is necessarily purely electric
one can oppositely ask for a counter example where:
if there's any Wick rotatable spacetime(so also purely electric) that is not strongly causal as a counter example.
I'd be more curious about the 3+1 dimensional case which is realistic.
A weaker version of the question would be:
Can a CTC contaning spacetime be Wick Rotated?
PS: I am trying to check if the obstruction to the Wick rotation can be of topological nature, since there's a topological mismatch between manifold topology and the Alexandrov topology(observable topology) in case of time-oriented spacetimes that are not strongly causal.
