Stationary solutions to the Einstein field equations

Stationary solutions of interacting system usually play a significant role in physics as (potential) attractors of the evolution, i.e. equilibrium states. In the case of the Einstein field equations, which in the simplest case describe the interaction of the gravitational field with itself "in empty space", finding the stationary solutions is a rather non-trivial mathematical problem. In spacetime dimension four, the celebrated uniqueness theorem (a culmination of separate arguments due to Carter, Hawking, Robinson, Bunting, and others) states that the so called Kerr solution is the only such solution describing a black hole in empty space. Here I review some of the arguments leading to this conclusion and present a generalisation to higher dimensional spacetimes.