Gravitational instability of Einstein–Gauss–Bonnet black holes under tensor mode perturbations

We analyse the tensor mode perturbations of static, spherically symmetric solutions of the Einstein equations with a quadratic Gauss–Bonnet term in dimension D > 4. We show that the evolution equations for this type of perturbation can be cast in a Regge–Wheeler–Zerilli form, and obtain the exact potential for the corresponding Schrödinger-like stability equation. As an immediate application we prove that for D ≠ 6 and α > 0, the sign choice for the Gauss–Bonnet coefficient suggested by string theory, all positive mass black holes of this type are stable. In the exceptional case D = 6, we find a range of parameters where positive mass asymptotically flat black holes, with regular horizon, are unstable. This feature is found also in general for α < 0.