Stability issues of black hole in non-local gravity

We discuss stability issues of Schwarzschild black hole in non-local gravity. It is shown that the stability analysis of black hole for the unitary and renormalizable non-local gravity with $\gamma_2=-2\gamma_0$ cannot be performed in the Lichnerowicz operator approach. On the other hand, for the unitary and non-renormalizable case with $\gamma_2=0$, the black hole is stable against the metric perturbations. For non-unitary and renormalizable local gravity with $\gamma_2=-2\gamma_0={\rm const}$ (fourth-order gravity), the small black holes are unstable against the metric perturbations. This implies that what makes the problem difficult in stability analysis of black hole is the simultaneous requirement of unitarity and renormalizability around the Minkowski spacetime.Comment: 1+15 pages, one figure, version to appear in Physics Letters