Tidal deformation of a slowly rotating material body: Interior metric and Love numbers

Abstract

The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the response of a barotropic perfect fluid to spin-coupled tidal perturbations is described by two rotational-tidal Love numbers, which I calculate explicitly for polytropes. Two other rotational-tidal Love numbers identified in prior work are found to have a fixed, universal value for all barotropes. Equipped with the complete interior solution, I calculate the amplitude of the time-varying internal currents induced by the gravitomagnetic part of the tidal field. For a typical neutron star in an equal-mass binary system, the size of the equatorial velocity perturbation is on the order of kilometers per second.