In this chapter we present the uniqueness theorem for the Kerr–Newman metric. The latter describes the only asymptotically flat, stationary and axisymmetric electrovac black hole solution with regular event horizon. The proof of this fact involves the following steps: First, one has to establish circularity of the domain of outer communications as a consequence of the symmetry properties of the electromagnetic field. The Einstein–Maxwell equations are then reduced to a 2–dimensional elliptic boundary–value problem for the complex Ernst potentials E and Λ. One then takes advantage of the symmetries of the Ernst equations to derive a divergence identity for the difference of two solutions. Since the boundary and regularity conditions are completely parametrized in terms of the total mass, angular momentum and charge, Stokes' theorem finally yields the desired result.

The chapter is organized as follows: The first section gives a short outline of the reasoning. In the second section we parametrize the Ernst potentials in terms of the hermitian matrix Φ, describing the nonlinear sigma model on the symmetric space G/H = SU(1, 2)/S(U(1) × U(2)) (or G/H = SU(1, 1)/U(1) in the vacuum case). We then establish the variational equation for Φ and derive an identity for the difference of two solutions to this equation. Evaluating the expressions in a circular spacetime, we obtain the Mazur identity (Mazur 1982) in the third section. This identity - or a related identity found by Bunting in 1983 - must be considered the key to the uniqueness theorems for rotating black holes.