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 $\mathrm{\Omega }$, the body’s angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like ${\mathrm{\Omega }}^0=1$. 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 $\mathrm{\Omega }$ 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 ${\mathcal{B}}_{pq}$. The tensorial nature of this quantity has to do with the fact that each $e^{im\varphi }$ 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 ${k_{jk}}^{pq}$.

Figure 1

Love number ${\tilde{k}}^m$ for $m=2$, plotted as a function of $\omega /\mathrm{\Omega }$, computed for the density models of Eq. (8.7). Red dash-dotted line: $b=0$. Black solid line: $b=1$. Blue short-dashed line: $b=2$. Green long-dashed line: $b=3$. The relevant inertial mode for $m=2$ is an $r$-mode, whose eigenfrequency $(4/3)\mathrm{\Omega }$ is independent of the body’s density profile. The resonance is indicated by the vertical line.

Figure 2

Love number ${\tilde{k}}^m$ for $m=1$, plotted as a function of $\omega /\mathrm{\Omega }$, computed for the density models of Eq. (8.7). Red dash-dotted line: $b=0$. Black solid line: $b=1$. Blue short-dashed line: $b=2$. Green long-dashed line: $b=3$. There are two relevant inertial modes for $m=1$, one with positive eigenfrequency, the other with negative eigenfrequency. The resonances are indicated by vertical lines.

Figure 3

Love number ${\tilde{k}}^m$ for $m=0$, plotted as a function of $\omega /\mathrm{\Omega }$, computed for the density models of Eq. (8.7). Red dash-dotted line: $b=0$. Black solid line: $b=1$. Blue short-dashed line: $b=2$. Green long-dashed line: $b=3$. The relevant inertial modes for $m=0$ are a pair of complex-conjugate modes with positive and negative eigenfrequencies of equal magnitude. The resonances are indicated by vertical lines.