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The projection formalism that associates to spacetimes with a timelike or spacelike everywhere Killing vector field the mapping $\psi : M \to S$, where $S$ is the set of all trajectories with tangents being the Killing field, is the basis on which the decomposition of metrics for Killing fields is done, ie, for a stationary spacetime with Killing vector $K$,

$$ds = \langle K, K \rangle dt^2 + h_{ab} dx^a dx^b$$

And similarly for spacelike Killing fields. But, from Geroch's original paper, there is a hidden assumption that $S$ must have the structure of a 3-manifold, which may fail if the spacetime has imprisoned curves.

Is this an actual obstacle for some spacetimes? Are there spacetimes with Killing vector fields where some of the trajectories are imprisoned curves? And does this prevent the metric from being written in this form?

The simplest example I can think of is the Clifford torus with the Minkowski metric,

$$ds^2 = -dt^2 + dx^2$$

which admits the geodesic with tangent vector $U = \alpha \partial_t + \beta \partial_x$, which is timelike for $\beta^2 < \alpha^2$, and if $\beta/\alpha \in \Bbb R \setminus \Bbb Q$, the curve will never join up with itself and fill the whole space. The vector field induced by the tangent vector of the curve is a Killing vector field, and the set of all trajectories does not have the structure of a $1$-manifold (it is a $0$-manifold with the singleton composed of the sole geodesic).

But of course despite this the metric still decomposes the usual way since there are other timelike Killing vector fields with that structure. Is there always such a vector field if there exists one timelike Killing vector field?

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  • $\begingroup$ There is something missing in the first sentence. Spacetimes with timelike or spacelike what? $\endgroup$ – MBN Apr 28 '17 at 13:52
  • $\begingroup$ Killing vector field $\endgroup$ – Slereah Apr 28 '17 at 13:53

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