Black Holes and Generalized Scalar Field

We study the possibility of occurrence of scalar hair with a non-canonical kinetic term for a static, spherically symmetric asymptotically flat black hole spacetime. We first obtain a general equation for this purpose and then consider various examples for the kinetic term $F(X)$ with $X=-\frac{1}{2}{\mathrm{\partial }}^{\mu }\varphi {\mathrm{\partial }}_{\mu }\varphi$. Our study shows that for a tachyon field with a positive potential, which naturally arises in open string theory, asymptotically flat a static black hole solution does not exist.