The response of a gravitating object to an external tidal field is encoded in its Love numbers, which identically vanish for classical black holes in 4 spacetime dimensions. I will explain why for quantum black holes, generically, the Love numbers should be nonvanishing and negative, and show that their magnitude depends on the lowest lying collective excited levels of the quantum spectrum of the black hole.
I will then discuss the standard derivation of the quadrupolar Love number k2 for classical relativistic stars, which relies on fixing boundary conditions on the surface of the star for the Einstein equations in the presence of an external perturbing field. I will show that calculating k2 reduces to fixing a single boundary condition for the perturbed Einstein equations on the surface of the deformed star which can be determined in terms of its fluid-modes spectrum.
By relating the classical quantities and quantum expectation values for the case of a quantum black hole I will verify the agreement on the resulting value of k2 between the effective classical calculation and the quantum calculation.
Finally, I will discuss the detectability of k2 in future gravitational-wave observations and show that, under favourable circumstances, its magnitude is large enough to imprint an observable signature on the gravitational waves emitted during the inspiral phase of two moderately spinning black holes.