Two-body gravitational spin-orbit interaction at linear order in the mass ratio

We analytically compute, to linear order in the mass ratio, the “geodetic” spin-precession frequency of a small spinning body orbiting a large (nonspinning) body to the eight-and-a-half post-Newtonian order, thereby extending previous analytical knowledge which was limited to the third post-Newtonian level. These results are obtained applying analytical gravitational self-force theory to the first-derivative level generalization of Detweiler’s gauge-invariant redshift variable. We compare our analytic results with strong-field numerical data recently obtained by Dolan et al. [Phys. Rev. D 89, 064011 (2014)]. Our new, high-post-Newtonian-order results capture the strong-field features exhibited by the numerical data. We argue that the spin precession will diverge as $\approx -0.14/(1-3y)$ as the light ring is approached. We transcribe our kinematical spin-precession results into a corresponding improved analytic knowledge of one of the two (gauge-invariant) effective gyrogravitomagnetic ratios characterizing spin-orbit couplings within the effective-one-body formalism. We provide simple, accurate analytic fits both for spin precession and the effective gyrogravitomagnetic ratio. The latter fit predicts that the linear-in-mass-ratio correction to the gyrogravitomagnetic ratio changes sign before reaching the light ring. This strong-field prediction might be important for improving the analytic modeling of coalescing spinning binaries.

Figure 1

The quantity $\delta \psi$ either from numerical GSF or from PN theory is plotted as a function of $y\in [0,1/3[$. The boxes are the points taken from the numerical results of [24] while the dotted curve corresponds to the fit (5.11). The other curves are the PN approximants of order $2\le n\le 8.5$. The curves are distinguishable from their values when $y\ge 0.2$ where they are ordered from top to bottom as follows: 2PN, 3PN, 4PN, 6PN, 5PN, 6.5PN, 7PN, 7.5PN, 8.5PN, 8PN.

Figure 2

The quantity $g_{S_{*}}^{1\mathrm{SF}}$ either exacted from numerical GSF or approximated by using the data fits is plotted as a function of $u\in [0,1/3]$. The used data fits are obtained from our model (5.11) for $\delta \psi$, and from model $\#14$ of Ref. [19] for $a_{1\mathrm{SF}}(y)$.

Figure 3

The full function $g_{S_{*}}^{\text{eff},\text{circ}}(u;\nu )$ is plotted as a function of $u$, for selected values of $\nu =[0,0.05,0.1,0.15,0.2,0.25]$.