In a connected Lorentzian spacetime that is Ricci flat and also nice (say smooth, global hyperbolic, etc), can a global Killing vector field be null in an open subset and timelike (or spacelike) in another open subset? Any examples?
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Take a cylinder $\mathbb{R} \times S^1$ with coordinates $(t,u)$ and consider the metric $$g = f(u) dt^2 + 2 h(u) dt \, du + du^2.$$ The vector field $\partial_t$ is obviously a Killing vector for this metric. The norm $g(\partial_t, \partial_t) = f(u)$ can be freely specified, so it can be chosen to be 0 for some open interval of $u$ to make it null there, and non-zero on another open interval to make it say timelike (fixing the appropriate sign conventions). I think it is not hard to see that choosing a compensating $h(u)$ such that the metric is non-degenerate and $\partial_u$ is everywhere spacelike, the surface $t=0$ is spacelike and a small tubular neighborhood of it will be globally hyperbolic. A similar example could probably produce an example with a null/spacelike Killing vector.
I believe that this is an example of the behavior you were asking for.
answered Nov 27 at 22:00
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$\begingroup$ Thank you very much for the answer; it's a great example! I realized that I need to add the condition that spacetime is Ricci flat in order to impose constraints on the Killing vector, and it seems that the example doesn't satisfy this condition. I've edited the question accordingly. $\endgroup$– SeanCommented Nov 29 at 15:43