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

We prove three theorems in general relativity which rule out classical scalar hair of static, spherically symmetric, possibly electrically charged black holes. We first generalize Bekenstein’s no–hair theorem for a multiplet of minimally coupled real scalar fields 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 field nonminimally coupled to gravity to the preceding problem, thus establishing a no–hair theorem for the cases with nonminimal coupling parameter $\xi <0$ or $\xi \ge \frac{1}{2}$. The proof also makes use of a causality requirement on the field configuration. Finally, from the required behavior of the fields at the horizon and infinity we exclude hair of a charged black hole in the form of a charged self–interacting scalar field nonminimally coupled to gravity for any $\xi$.