No-Hair Theorems and Black Holes with Hair

TL;DR

The paper revisits the no-hair theorems for black holes by testing the resilience of the uniqueness program against selfgravitating scalar and non-Abelian fields. It develops and uses scaling arguments, energy-condition bounds, conformal methods, and divergence identities to map when Kerr–Newman uniqueness extends and when hair persists, notably highlighting the Kerr uniqueness in the harmonic-mapping sector via Ernst and Mazur techniques. It shows that harmonic mappings with nonnegative potentials admit no solitons and no static hair under spherical symmetry, while non-Abelian Yang–Mills hair can evade no-hair conclusions, as demonstrated by soliton solutions and the breakdown of simple circularity. The work also provides tools such as the Mazur identity and Bogomol’nyi-type relations to connect horizon data, charges, and global parameters, clarifying the conditions under which Kerr remains the unique stationary, axisymmetric BH and outlining avenues for further generalizations.

Abstract

The critical steps leading to the uniqueness theorem for the Kerr-Newman metric are reexamined in the light of the new black hole solutions with Yang-Mills and scalar hair. Various methods -- including scaling techniques, arguments based on energy conditions, conformal transformations and divergence identities -- are reviewed, and their range of application to selfgravitating scalar and non-Abelian gauge fields is discussed. In particular, the no-hair theorem is extended to harmonic mappings with arbitrary Riemannian target manifolds. (This paper is an extended version of an invited lecture held at the Journées Relativistes 96.)