Asymptotically flat solutions with scalar hair in a subclass of Horndeski theories
Black holes in general relativity have been long known to possess a limited number of charges known as hair. It has been recently known that this is also true in modified theories of gravity collectively known as Horndeski theories where a scalar field provides an additional degree of freedom other than the metric. We provide a more explicit proof for this no-hair theorem in a subclass of these theories where the derivatives of the shift-symmetric scalar field are coupled to the Einstein tensor. It is shown that a static, spherically symmetric, and asymptotically flat black hole cannot possess a nontrivial scalar field, and the resulting spacetime is the Schwarzschild solution.