Classical Love number for quantum black holes

We present a method for comparing the classical and quantum calculations of the electric quadrupolar Love number k2 and show that our previous derivation of the quantum Love number of a quantum black hole matches exactly the classical calculation of k2 when quantum expectation values are replaced by the corresponding classical quantities, as dictated by the Bohr correspondence principle. The standard derivation of k2 for classical relativistic stars relies on fixing boundary conditions on the surface of the star for the Einstein equations in the presence of an external perturbing field. An alternative method for calculating k2 uses properties of the spectrum of the nonrelativistic fluid modes of the star. We adopt this alternative method and use it to derive an effective description of the interior modes in terms of a collection of driven harmonic oscillators characterized by different frequencies and amplitudes. We compare these two classical methods and find that most of the interior information can be integrated out, reducing the problem of calculating k2 to fixing a single boundary condition for the perturbed Einstein equations on the surface of the deformed star. We then determine this single boundary condition in terms of the spectrum of the object and proceed to identify the relationship between classical quantities and quantum expectation values for the case of a quantum black hole and to verify the agreement between the results of the effective classical calculation and the quantum calculation.