No-hair theorems in general relativity and scalar–tensor theories

In this paper, we consider the status of the classification of the vacuum, stationary and asymptotically flat black holes in scalar–tensor gravity. Contrary to the similar problem in general relativity, the black hole classification in scalar–tensor theories is much more challenging due to the very complicated character of the field equations and the very complex mathematical structure of the scalar–tensor gravity as a whole. We review most of the known no-hair results, and where possible new ones, as well demonstrate some of the difficulties that appear in our attempts to classify the black holes within scalar–tensor gravity. The proofs of the theorems and the underlying mathematical techniques are given in sufficient detail. To make the review self-contained, we also present the vacuum black hole uniqueness theorems in general relativity and their proofs.