We revisit the classical stability of the Schwarzschild black hole in fourth-order theories of gravity. Contrary to the stability of this black hole, it turns out that the linearized perturbations exhibit unstable modes featuring the Gregory-Laflamme instability of the five-dimensional black string. This shows clearly an instability of the black hole in fourth-order gravity with α=-β/3.
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Yun Soo Myung (2013) studied this question.
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