No-Hair Theorems and Black Holes with Hair

TL;DR

This paper reviews and extends no-hair theorems for black holes, analyzing conditions under which black holes can have scalar or gauge field hair, and discusses various mathematical methods used in these proofs.

Contribution

It broadens the no-hair theorem to include harmonic mappings with arbitrary Riemannian target manifolds and reviews multiple proof techniques.

Findings

No-hair theorem extended to harmonic mappings with arbitrary Riemannian targets

Various methods like energy conditions and divergence identities are applicable to non-Abelian gauge fields

Black holes can possess scalar or gauge hair under certain conditions

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 Journ\'ees Relativistes 96.)