Given any two points p and q in Gödel spacetime, there is a smooth, future-directed timelike curve that runs from p and q. (Hence, since we can always combine timelike curves that run in the two directions and smooth out the joints, there is a smooth, closed timelike curve that contains p and q.)
A time traveler in Gödel spacetime can start at any point p, return to that point, and stop off at any other desired point q along the way. To see why this holds, consider the figure above. It gives, at least, a rough, qualitative picture of Gödel spacetime with one dimension suppressed. We may as well take the central line to be the axis A and take p to be a point on A. (By homogeneity, there is no loss in generality in doing so.) Notice first that given any other point p′ on A, no matter how “far down,” there is a smooth, future-directed timelike curve that runs from p to p′. We can think of it as arising in three stages. (i) By moving “radially outward and upward” from p (i.e., along a future-directed timelike curve whose tangent vector field is of the form t ̃a + αra, with α positive), we can reach a point p1 with coordinate value r > rc. At that radius, we know, φa is timelike and future-directed. So we can find an ε > 0 such that (−εt ̃a + φa) is also timelike and future-directed there. (ii) Now consider the maximally extended, future-directed timelike curve γ through p1 whose tangent is everywhere equal to (−εt ̃a + φa) (for that value of ε). It is a spiral-shaped curve of fixed radius, with “downward pitch.” By following γ far enough, we can reach a point p2 that is well “below” p′. Now, finally, (iii) we can reach p′ by working our way upward and inward from p2 via a curve whose tangent vector is the form t ̃a + αra, but now with α negative. It remains only to smooth out the “joints” at intermediate points p1 and p2 to arrive at a smooth timelike curve that, as required, runs from p to p′.
Now consider any point q. It might not be possible to reach q from p in the same simple way we went from p to p1 – i.e., along a future-directed timelike curve that moves radially outward and upward – p might be too “high” for that. But we can get around this problem by first moving to an intermediate point p′ on A sufficiently “far down” – we have established that that is possible – and then going from there to q.
Other interesting features of Gödel spacetime are closely related to the existence of closed timelike curves. So, for example, a slice (in any relativistic spacetime) is a spacelike hypersurface that, as a subset of the background manifold, is closed. We can think of it as a candidate for a “global simultaneity slice.” It turns out that there are no slices in Gödel spacetime. More generally, given any relativistic spacetime, if it is temporally orientable and simply connected and has smooth closed timelike curves through every point, then it does not admit any slices.