Gravitational polarizability of black holes

The gravitational polarizability properties of black holes are compared and contrasted with their electromagnetic polarizability properties. The "shape" or "height" multipolar Love numbers h(l) of a black hole are defined and computed. They are then compared to their electromagnetic analogs h(l)(EM). The Love numbers h(l) give the height of the lth multipolar "tidal bulge" raised on the horizon of a black hole by faraway masses. We also discuss the shape of the tidal bulge raised by a test-mass m, in the limit where m gets very close to the horizon.