Searching for nonminimally coupled scalar hairs

We study the asymptotically flat, static, and spherically symmetric black-hole solutions of the theory described by the action S=∫ ${\mathit{d}}^{\mathit{nx}}$√-g{(1-ξ${\mathrm{\phi }}^2$)R-${\mathit{g}}^{\mathrm{\mu }\nu }$${\mathrm{\partial }}_{\mathrm{\mu }}$φ∂ ${}_{\nu }{}^{}{}_{}{}^{}\mathrm{\}}$, with n≳3 and arbitrary ξ. We demonstrate the absence of scalar hairs for ξ<0. For ξ≳${\xi }_{\mathit{c}}$=(n-2)/4(n-1), we show that there is no scalar hair obeying |φ(r)|<1/√ξ or |φ(r)|≳1/√ξ. For 0<ξ<${\xi }_{\mathit{c}}$, we prove the absence of scalar hairs such that |φ(r)|<1/√ξ or 1/ξ<${\mathrm{\phi }}^2$(r)<${\xi }_{\mathit{c}}$/ξ(${\xi }_{\mathit{c}}$-ξ). © 1996 The American Physical Society.