Bopp-Podolsky electrodynamics is generalized to curved space-times. The equations of motion are written for the case of static spherically symmetric black holes and their exterior solutions are analyzed using Bekensteins method. It is shown the solutions split-up into two parts, namely a non-homogeneous (asymptotically massless) regime and a homogeneous (asymptotically massive) sector which is null outside the event horizon. In addition, in the simplest approach to Bopp-Podolsky black holes, the non-homogeneous solutions are found to be Maxwells solutions leading to a Reissner-Nordstrom black hole. It is also demonstrated that the only exterior solution consistent with the weak and null energy conditions is the Maxwells one. Thus, in light of energy conditions, it is concluded that only Maxwell modes propagate outside the horizon and, therefore, the no-hair theorem is satisfied in the case of Bopp-Podolsky fields in spherically symmetric space-times.
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