Gravitomagnetic Love tensor of a slowly rotating body: post-Newtonian theory

The gravitomagnetic tidal Love number of a slowly rotating body was calculated previously under the assumption that the velocity perturbation created by the tidal field consists of an induction piece proportional to the vector potential, and a rotational piece that scales with $\Omega$, the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $\Omega^0 = 1$. The previous calculations are therefore incorrect, and the purpose of this paper is to repair the mistake. To keep the technical difficulties to a minimum, the treatment here is restricted to a post-Newtonian expansion carried out to leading order -- previous calculations of the gravitomagnetic Love number were performed in full general relativity. On the other hand, the computation presented here is not restricted to a stationary tidal field. I show that the correct scaling of the velocity perturbation with $\Omega$ leads to the promotion of the Love number to a Love tensor $k_{jk}^{\ \ pq}$, a four-index object that relates the body's current quadrupole moment $S_{jk}$ to the gravitomagnetic tidal moment ${\cal B}_{pq}$. The tensorial nature of this quantity has to do with the fact that each $e^{im\phi}$ piece of the tidal force gives rise to an $m$-specific velocity perturbation, and therefore to a Love number that depends on $m$. The collection of these $m$-specific Love numbers makes up the Love tensor $k_{jk}^{\ \ pq}$.