No hair for spherical black holes: Charged and nonminimally coupled scalar field with self-interaction.

Avraham E. Mayo and Jacob D. Bekenstein The Racah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel (March 1, 1996) We prove three theorems in general relativity which rule out scalar hair of static, spherically symmetric, possibly electrically charged black holes. We rst generalize Bekenstein's no{hair theorem for a multiplet of minimally coupled real scalar elds with not necessarily quadratic action to the case of a charged black hole. We then use a conformal map of the geometry to convert the problem of a charged (or neutral) black hole with hair in the form of a neutral self{interacting scalar eld nonminimally coupled to gravity to the preceding problem, thus establishing a no{hair theorem for the cases with nonminimal coupling parameter < 0 or 1 2 . The proof also makes use of a causality requirement on the eld con guration. Finally, from the analytic behavior of the elds at the horizon we exclude hair of a charged black hole in the form of a charged self{interacting scalar eld nonminimally coupled to gravity for any . 04.70.-s, 04.70.Bw, 11.15.Ex, 95.30.Tg, 97.60.Lf