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

Eric Poisson

doi:10.1103/PhysRevD.102.064059

paper

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

arXiv:2007.01678 · doi:10.1103/PhysRevD.102.064059

Abstract

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 $Ω$ , the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $Ω^{0} = 1$ . 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 $Ω$ leads to the promotion of the Love number to a Love tensor $k_{j k}^{pq}$ , a four-index object that relates the body's current quadrupole moment $S_{j k}$ to the gravitomagnetic tidal moment $B_{pq}$ . The tensorial nature of this quantity has to do with the fact that each $e^{im ϕ}$ 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 $k_{j k}^{pq}$ .

20 pages, 3 figures, new appendix, matches published version

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

Abstract

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