The most general action for a scalar field coupled to gravity that leads to second order field equations for both the metric and the scalar -Horndeski's theory -is considered, with the extra assumption that the scalar satisfies shift symmetry. We show that in such theories the scalar field is forced to have a nontrivial configuration in black hole spacetimes, unless one carefully tunes away a linear coupling with the Gauss-Bonnet invariant. Hence, black holes for generic theories in this class will have hair. This contradicts a recent no-hair theorem, which seems to have overlooked the presence of this coupling.PACS numbers: 04.70. Bw, 04.50.Kd In general relativity, black hole spacetimes are described by the Kerr metric, so long as they are stationary, asymptotically flat, and devoid of any matter in their surroundings [1]. Stationarity is a reasonable assumption for black holes that are thought to be quiescent as endpoints of gravitational collapse. Astrophysical black holes are certainly not asymptotically flat, but one can invoke separation of scales in order to argue that the cosmological background should not seriously affect local physics and hence the structure of black holes. Finally, black holes can also carry an electromagnetic charge in the presence of an electromagnetic field. It has been conjectured that they cannot carry any other charges, which are colloquially referred to as hair [2][27]. The no-hair conjecture was inspired by the uniqueness theorems for black hole solutions in general relativity [4][5][6][7].Hawking has proven that black holes cannot carry scalar charge, provided that the scalar couples to the metric minimally or as described by Brans-Dicke theory [8]. This result has been generalised to standard scalar-tensor theories [9] (see also earlier work by Bekenstein with the extra assumption of spherical symmetry [10,11]).All of these proofs actually demonstrate that the scalar has to be constant in a black hole spacetime, which is a stronger statement. Indeed, in principle, the scalar could have a nontrivial configuration without the black hole carrying an extra (independent) charge. This is sometime referred to as "hair of the second kind". The distinction is important if one is interested in the number of parameters that fully characterise the spacetime. But, a nontrivial configuration of the scalar is usually enough to imply that the black hole spacetime will not be a solution to Einstein's equations in vacuum, and hence it differs from the black holes of general relativity.The known proofs do not apply to theories with more general coupling between the metric and the scalar, or derivative self-interactions of the scalar. Hence, they do not cover the most general scalar-tensor theory that leads to second-order field equations, known as Horndeski theory [12]. Restricting attention to theories with second order field equations is justified, as higher order derivative models are generically plagued by the Ostrogradski instability [13]. Models that belong to this class have lately ...

In a recent Letter we have shown that in shift-symmetric Horndeski theory the scalar field is forced to obtain a nontrivial configuration in black hole spacetimes, unless a linear coupling with the Gauss-Bonnet invariant is tuned away. As a result, black holes generically have hair in this theory. In this companion paper, we first review our argument and discuss it in more detail. We then present actual black hole solutions in the simplest case of a theory with the linear scalar-GaussBonnet coupling. We generate exact solutions numerically for a wide range of values of the coupling and also construct analytic solutions perturbatively in the small coupling limit. Comparison of the two types of solutions indicates that non-linear effects that are not captured by the perturbative solution lead to a finite area, as opposed to a central, singularity. Remarkably, black holes have a minimum size, controlled by the length scale associated with the scalar-Gauss-Bonnet coupling. We also compute some phenomenological observables for the numerical solution for a wide range of values of the scalar-Gauss-Bonnet coupling. Deviations from the Schwarzschild geometry are generically very small.