ar X iv : g r - qc / 9 40 20 32 v 1 1 7 Fe b 19 94 “ No Hair ” Theorems – Folklore , Conjectures , Results

Various assumptions underlying the uniqueness theorems for black holes are discussed. Some new results are described, and various unsatisfactory features of the present theory are stressed. 1 Folklore, Conjectures A classical result in the theory of black holes, known under the name of “no–hair Theorem”, is the following: Theorem 1.1 Let (M, g) be a good electro–vacuum space–time with a non– empty black hole region and with a Killing vector field which is timelike in the asymptotic regions. Then (M, g) is diffeomorphically isometric to a Kerr– Newman space–time or to a Majumdar–Papapetrou space–time. This Theorem is known to be true under various definitions of “good space– time” (all of which actually imply that it cannot be a Majumdar–Papapetrou space–time), and the purpose of this paper is to discuss various problems related to the as–of–today–definition of “good space–time” needed above. Clearly, one would like to have a definiton of “good space–time” as weak as possible. Moreover one would like this definition to have some degree of verifiability and “controllability”, and to be compatible with our knowledge of the structure of the theory gained by some perhaps completely different investigations. We shall focus here on uniqueness theory of stationary electro–vacuum space– times. It is, however, worthwile mentioning that substantial progress has been made recently in the understanding of various other models. In particular one should mention various results about uniqueness of perfect fluid models [58, 10], σ–models [46, 47, 48], Einstein–Yang–Mills solutions [12, 11, 70, 71], and dilatonic black holes [57], cf. also [11, 40]. Many of the questions raised here as well as some of the results presented here are also relevant to those other models. One of the purposes of this paper is to give a careful definition of “good space–time” under which Theorem 1.1 holds, let us therefore start with the