Gravitational scattering of two black holes at the fourth post-Newtonian approximation

We compute the (center-of-mass frame) scattering angle $\chi$ of hyperboliclike encounters of two spinning black holes, at the fourth post-Newtonian approximation level for orbital effects, and at the next-to-next-to-leading order for spin-dependent effects. We find it convenient to compute the gauge-invariant scattering angle (expressed as a function of energy, orbital angular momentum and spins) by using the effective-one-body formalism. The contribution to scattering associated with nonlocal, tail effects is computed by generalizing to the case of unbound motions the method of time localization of the action introduced in the case of (small-eccentricity) bound motions by Damour et al. [Phys. Rev. D 91, 084024 (2015)].

Figure 1

The behavior, as a function of $ϵ=1/e$, of the additional constant $C(ϵ=1/e)$ in Eq. (167) is plotted using both the data points of Table 1, and the simple cubic fit (183) for the normalized version (182) of $C(ϵ)$.