Gödel dust
The Gödel dust spacetime was the first spacetime with closed timelike curves recognized as such where the closed timelike curves stemmed from the geometry and not the topology (otherwise, the timelike cylinder or the van Stockum dust are older).
1. History
2. Topology
The Gödel spacetime has the topology $\mathbb{R}^4$.
3. Metrics and coordinates
$$ds^2 = \frac{1}{2\omega^2} \left[ -(dt + e^x dz)^2 + dx^2 + dy^2 + \frac{1}{2} e^{2x} dz^2 \right] $$ $$ds^2 = 2 \omega^{-2} (-dt^2 + dr^2 - (\sinh^4 r - \sinh^2 r) d\varphi^2 + 2 \sqrt{2} \sinh^2 r d\varphi dt)$$4. Tensor quantities
5. Symmetries
Its Killing vectors are $\partial_t$, $\partial_y$, $\partial_z$, $\partial_x - z \partial_z$ and $-2 e^{-x} \partial_t + z \partial_x + (e^{-2x} - \frac{z^2}{2}) \partial_z$.
6. Stress-energy tensor
7. Curves
8. Equations
9. Causal structure
The Gödel metric is totally vicious but causally closed. It has no singularities.