Recent progress on the tidal deformability of spinning compact objects

We give a short review on recent progress in the theory of tidal deformability of a slowly spinning compact object. A rotating object immersed in a quadrupolar, electric tidal field can acquire some induced mass, spin, quadrupole, octupole and hexadecapole moments to second order in the spin. Angular momentum introduces couplings between electric and magnetic distortions and new classes of spin-induced, tidal Love numbers emerge. All tidal Love numbers of a Kerr black hole were proved to be exactly zero to first order in the spin and also to second order in the spin, at least in the axisymmetric case. The tidal Love numbers of a neutron star depend strongly on the equation of state. Spin-tidal couplings deteriorate some approximate universal relations that exist for neutron stars in the static case. For a binary system close to the merger, various components of the tidal field become relevant. Preliminary results suggest that spin-tidal couplings can introduce important corrections to the gravitational waveforms of spinning neutron-star binaries approaching the merger.