Stationary Black Holes: Uniqueness and Beyond
Piotr T. Chruściel, João Lopes Costa, Markus Heusler
TL;DR
The paper surveys four-dimensional stationary black holes and the uniqueness program, detailing how symmetry reductions turn Einstein–Maxwell theory into a 3D gravity coupled sigma-model (notably via Ernst potentials) and how divergence identities and harmonic-map methods yield Kerr–Newman uniqueness (under various hypotheses). It then extends the discussion to higher dimensions, Kaluza–Klein setups, and theories with non-Abelian fields, highlighting that hair and non-uniqueness phenomena arise outside the Abelian EM paradigm. Key techniques include the Robinson–Mazur and Bunting–Weinstein frameworks, the Varzugin–Neugebauer–Meinel inverse-scattering approach, and rigidity theorems that connect horizon structure to spacetime symmetries. The work underscores both the robustness of the Kerr–Newman paradigm in four dimensions under broad regularity assumptions and the rich landscape of nontrivial solutions beyond EM, with important implications for black-hole thermodynamics and gravitational physics across dimensions.
Abstract
The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.