Timelike cylinder
1. History
2. Topology
The timelike cylinder has the topology $S \times \mathbb{R}^{n-1}$.
3. Metrics and coordinates
The timelike cylinder has the same metric and coordinates as Minkowski space, that is, the cartesian coordinates $(t, x, y, z)$ and the metric
$$ds^2 = -dt^2 + \sum_{i = 1}^n (dx_i)^2$$Except that the timelike coordinate only runs on the interval $[0, a]$ and has the identification $(t, x, y, ...) \to (t + a, x, y, ...)$
4. Tensor quantities
5. Symmetries
6. Stress-energy tensor
Much like Minkowski space, assuming $\Lambda = 0$, the timelike cylinder has a $0$ stress-energy tensor which, assuming positive energy and pressure, implies that its fields are $0$ as well.
7. Curves
8. Equations
9. Causal structure
By construction, the timelike cylinder is totally vicious. It is causally closed and has no singularities.