Misner space
Misner space is a simple example of a spacetime with closed timelike curves and a compact Cauchy horizon, easy to deal with since it is just a quotient of Minkowski space.
1. History
2. Topology
Misner space has the topology $\mathbb R \times S$.
3. Metrics and coordinates
3.1. Minkowski coordinates
Since Misner space is simply a quotient of Minkowski space, the simplest metric is simply that of flat space
$$ds^2 = -dt^2 + dy^2$$with the coordinate identifications
$$(t, x) \to (t \cosh \pi + x \sinh \pi,x \cosh \pi + t \sinh \pi)$$Misner coordinates
$$ds^2 = -2dt d\varphi + t d\varphi$$4. Tensor quantities
4.1. In Minkowski coordinates
4.1.1. Christoffel symbols
$${\Gamma^\sigma}_{\mu\nu} = 0$$4.1.2. Riemann tensor
5. Symmetries
6. Stress-energy tensor
7. Curves
7.1. In Minkowski coordinates
7.1.1 Geodesics
8. Equations
8.1 Wave equation
8.1.1. In Minkowski coordinates
Being simply Minkowski space, the wave equation simply reduces to the usual flat space one
$$\Box f = 0$$The difference comes from the acceptable class of solutions, as the solutions are required to match on the identification
9. Causal structure
Misner space is geodesically complete, but admits a chronology-violating region after $t = 0$. As such it is at best non-totally vicious. It is also causally closed. It has a Cauchy horizon $H^+(M) = \{ (x,t) | t = 0 \}$.
10. Asymptotic structure
11. Energy conditions
Being just a quotient of Minkowski space, Misner space obeys all the same usual energy condition as Minkowski space.