  The open question of whether a black hole can become tidally deformed by an
external gravitational field has profound implications for fundamental physics,
astrophysics and gravitational-wave astronomy. Love tensors characterize the
tidal deformability of compact objects such as astrophysical (Kerr) black holes
under an external static tidal field. We prove that all Love tensors vanish
identically for a Kerr black hole in the nonspinning limit or for an
axisymmetric tidal perturbation. In contrast to this result, we show that Love
tensors are generically nonzero for a spinning black hole. Specifically, to
linear order in the Kerr black hole spin and the weak perturbing tidal field,
we compute in closed form the Love tensors that couple the mass-type and
current-type quadrupole moments to the electric-type and magnetic-type
quadrupolar tidal fields. For a dimensionless spin ~ 0.1, the nonvanishing
quadrupolar Love tensors are ~ 0.002, thus showing that black holes are
particularly "rigid" compact objects.
