Black hole stereotyping: induced gravito-static polarization

We discuss the black hole effective action and define its static subsector. We determine the induced gravito-static polarization constants (electric Love numbers) of static black holes (Schwarzschild) in an arbitrary dimension, namely the induced mass multipole as a result of an external gravitational field. We demonstrate that in 4d these constants vanish thereby settling a disagreement in the literature. Yet in higher dimensions these constants are non-vanishing, thereby disproving (at least in d > 4) speculations that black holes have no effective couplings beyond the point particle action. In particular, when l/(d-3) is half integral these constants demonstrate a (classical) renormalization flow consistent with the divergences of the effective field theory. In some other cases the constants are negative indicating a novel non-spherical instability. The theory of hypergeometric functions plays a central role.