Static response and Love numbers of Schwarzschild black holes

We derive the quadratic action for physical degrees of freedom massless spin-0, spin-1, and spin-2 perturbations on a Schwarzschild--(A)dS background in arbitrary dimensions. then use these results to compute static response asymptotically flat Schwarzschild black holes external fields. Our analysis reproduces known facts about hole Love numbers, particular that they vanish all types perturbation four spacetime dimensions, but also leads new results. For instance, we find neutral polarize presence an electromagnetic any number dimensions except four. Moreover, calculate first time numbers vector-type gravitational higher generically do not vanish. Along way, shed some light apparent discrepancy between previous literature, clarify aspects matching perturbative calculations point-particle effective theory