When are geodesically generated surfaces everywhere spacelike?

Suppose that $\langle M, g\rangle$ is a Lorentzian manifold, and that $\xi$ is a timelike vector in $T_pM$, at some point $p \in M$.

Let $S$ be a surface consisting of points that lie on some geodesic passing through $p$ orthogonal to $\xi$. In general, I believe, this surface will be space-like in at most a neighborhood of $p$.

I would like to know: if $M$ is globally hyperbolic (i.e. it contains at least one Cauchy surface), does it follow that $S$ is everywhere spacelike?

Also, I'd like to know if $S$ is everywhere spacelike, does it follow that $S$ is a Cauchy surface?

I'd be interested to know what counterexamples to these claims are like, if there are any. Any help would be greatly appreciated!

• Do you mean "passing through $p$ orthogonally to $\xi$"? – Robert Israel May 8 '15 at 0:31
• Yes I did. I've edited the question, thanks! – Andrew Bacon May 8 '15 at 3:08