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Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation

Donato Bini, Thibault Damour

Abstract

We compute the rotations, during a scattering encounter, of the spins of two gravitationally interacting particles at second-order in the gravitational constant (second post-Minkowskian order). Following a strategy introduced in Phys. Rev. D {\bf 96}, 104038 (2017), we transcribe our result into a correspondingly improved knowledge of the spin-orbit sector of the Effective One-Body (EOB) Hamiltonian description of the dynamics of spinning binary systems. We indicate ways of resumming our results for defining improved versions of spinning EOB codes which might help in providing a better analytical description of the dynamics of coalescing spinning binary black holes.

Gravitational spin-orbit coupling in binary systems at the second post-Minkowskian approximation

Abstract

We compute the rotations, during a scattering encounter, of the spins of two gravitationally interacting particles at second-order in the gravitational constant (second post-Minkowskian order). Following a strategy introduced in Phys. Rev. D {\bf 96}, 104038 (2017), we transcribe our result into a correspondingly improved knowledge of the spin-orbit sector of the Effective One-Body (EOB) Hamiltonian description of the dynamics of spinning binary systems. We indicate ways of resumming our results for defining improved versions of spinning EOB codes which might help in providing a better analytical description of the dynamics of coalescing spinning binary black holes.

Paper Structure

This paper contains 30 sections, 284 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scattering holonomy
  3. Evaluation of the spin holonomy
  4. The two-body metric at the 2PM order
  5. 1PM-accurate orbits of the two bodies
  6. Regular terms in the 2PM metric, and its derivative
  7. Computation of $\Lambda_{(1)}^x{}_y =\Lambda_{(1)}^1{}_2$
  8. Final result for the holonomy map
  9. EOB computation of the spin-holonomy rotation angle $\theta_1$
  10. EOB transcription of the spin-holonomy rotation angle $\theta_1$
  11. Comparison with PN-expanded results
  12. High-energy (HE) behavior, strong-field behavior and resummation
  13. High-energy (HE) behavior
  14. Strong-field behavior and resummation
  15. Concluding remarks
  16. ...and 15 more sections

Figures (2)

  • Figure 1: Panel (a). The behavior of $\tilde{c}^1_{S}$ is shown as a function of $\gamma$ for selected values of $\nu=[0.1 \,\hbox{(black online)},0.15 \, \hbox{(blue online)},0.25\, \hbox{(red online)}]$. The dotted line corresponds to the asymptotic value $\tilde{c}^1_{S}{}^\infty=35/8$. Panel (b). The behaviour of $\tilde{c}^1_{S*}$ is shown as for $\tilde{c}^1_{S}$ with asymptotic value $\tilde{c}^1_{S*}{}^\infty=5$.
  • Figure 2: Panel (a). The behavior of $g_S(\gamma,\nu,u)$ as given in Eq. \ref{['resum_gsstar']} (with known 3PM terms included) is plotted as a function of $u$ for different values of $\gamma=1,10$ and $\nu=0.1\, \hbox{(black online)},0.25\, \hbox{(red online)}$. Panel (b). $g_{S*}(\gamma,\nu,u)$ is plotted as a function of $u$ for the same parameter choice of panel (a).