Possible obstructions to global Wick-rotation in distinguishing spacetimes

Question

Take the time-orientable $3+1$ dimensional spacetime $(M,g)$ that is locally Wick-rotatable at any arbitrary point $p \in M$ to a Riemannian manifold $(N,h)$.

Local Wick-rotatability of $(M,g)$ implies that its Riemann curvature tensor and Weyl tensor are both locally purely electric, which means the magnetic part of the Riemann curvature tensor acts as an obstruction to local Wick-rotation.

Since the manifold is time-oriented, one can take a globally defined time-like vector field $u$ over the manifold.

In $3+1$ dimensions one can equivalently define Pure Electricity by introduction of the tensor: $$H_{ab} = \frac{1}{2} \epsilon_{acef} {C^{ef}}_{bd} u^c u^d$$

as vanishing of the magnetic part of the Weyl tensor $C_{-}$ locally and so in our case everywhere locally (globally):

$${(C_{-})^{ab}}_{cd} = 2 \epsilon^{abef} u_{e} u_{[c} H_{d]f} + 2 \epsilon_{cdef} u^e u^{[a} H^{b]f}=0 $$

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Suppose the causal structure introduced by the Lorentzian metric $g$ is that of a "both past and future distinguishing" spacetime.

What is the nature of a possible obstruction to the global Wick-rotation of the manifold $(M,g)$ to $(N,h)$?

Can one give a few examples of locally but not globally Wick-rotatable spacetimes that are distinguishing (both past and future together), at least?

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PS: Whatever the obstruction might be, it should appear on the way of gluing the locally Wick-rotatable neighbourhoods to each other to form a global Wick-rotation. Which to me sounds like sheaf theory or Algebraic topology stuff. Though I am not by any measure an expert.

This post imported from StackExchange MathOverflow at 2024-11-12 21:45 (UTC), posted by SE-user Bastam Tajik