Black Uniqueness Theorems
Pawel O. Mazur
TL;DR
The paper surveys the development of black hole uniqueness and no-hair theorems for stationary, axisymmetric spacetimes in Einstein-Maxwell theory, emphasizing a harmonic map reformulation via Ernst potentials. It shows that the Einstein-Maxwell equations reduce to a nonlinear sigma model with a negatively curved target space, enabling robust divergence-identity techniques to prove uniqueness. The core result is that stationary electrovacuum black holes are uniquely described by the Kerr-Newman family (extended to include magnetic charge when present), with the method underpinning potential extensions to broader nonlinear sigma models on symmetric spaces. The approach highlights the power of geometric and group-theoretic methods in global gravitational problems and suggests possible generalizations beyond axial symmetry.
Abstract
I review the black hole uniqueness theorem and the no hair theorems established for physical black hole stationary states by the early 80'. This review presents the original and decisive work of Carter, Robinson, Mazur and Bunting on the problem of no bifurcation and uniqueness of physical black holes. Its original version was written only few years after my proof of the Kerr-Newman et al. black hole uniqueness theorem has appeared in print. The proof of the black hole uniqueness theorem relies heavily on the positivity properties of nonlinear sigma models on the Riemannian noncompact symmetric spaces with negative sectional curvature. It is hoped that the first hand description of the original developments leading to our current understanding of the black hole uniqueness will be found useful to all interested in the subject.