Carter spacetime
The Carter spacetime is the standard example of a spacetime that is distinguishing, but not strongly causal, and contains imprisonned curves that are not closed causal curves.
1. History
2. Topology
3. Metrics and coordinates
$$ds^2 = (\cosh(t) - 1)^2 (-dt^2 + dy^2) - 2 dt dy + dz^2$$The coordinates have the following identifications :
\begin{eqnarray} (t,y,z) \sim (t,y,z+1)\\ (t,y,z) \sim (t, y+1, z+a) \end{eqnarray}where $a$ is an irrational number between $0$ and $1$ ($a \in (0,1) \cap (\mathbb{R} \setminus \mathbb{Q}$).