A 2025 SpEC numerical-relativity (NR) study of binary-black-hole scattering (Long et al., Phys. Rev. D 2025) left a bundle of open modeling problems: the O(G5ν2)O(G^5\nu^2) (second-self-force) content of the post-Minkowskian scattering angle was missing, resummed and evolution effective-one-body (EOB) models misplace the scatter–capture separatrix for spinning unbound orbits, the newly measured asymmetric per-black-hole scattering-angle difference θ2−θ1\theta_2-\theta_1 was not quantitatively reproduced, and a call for broader cross-code NR comparison and unexplored extreme regimes remained. The open problem is: the fully dissipative 5PM/2SF sector and a settled, finite definition of the complete conservative 5PM/2SF scattering angle (with the spurious γ=3\gamma=3 divergence and dimensional-regularization pole demonstrably canceling between potential, memory, and tail regions), plus the incorporation of this complete 5PM(2SF) content into resummed EOB models that improve agreement with NR scattering observables — relative angles including near-separatrix spinning configurations, and the asymmetric angle difference — addressing the model deficiencies documented in Long et al. (2025).

Background

The gravitational two-body problem asks for the motion and radiation of two compact objects in general relativity. For unbound (hyperbolic, scattering) encounters a clean gauge-invariant observable exists: the scattering angle χ\chi, the net deflection of the relative trajectory, expressible as a function of the impact parameter bb and the asymptotic relative Lorentz factor γ\gamma. Two perturbative expansions describe it. The post-Minkowskian (PM) expansion is an expansion in powers of Newton's constant GG (equivalently in GM/bGM/b) valid at all velocities; the self-force (SF) expansion is an expansion in the symmetric mass ratio ν=m1m2/(m1+m2)2\nu=m_1m_2/(m_1+m_2)^2, exact in the small-mass-ratio limit where the smaller body is treated as a perturbation of the larger black hole; the post-Newtonian (PN) expansion is the complementary low-velocity expansion in powers of the relative speed v/cv/c, in which many radiative results are organized. The conservative two-body dynamics is completely known through 4PM order, and at 5PM (G5G^5) through first self-force order; the O(ν2)O(\nu^2) (2SF) piece at 5PM was unknown in Long et al. (2025). Effective-one-body (EOB) theory maps the two-body dynamics onto an effective particle moving in a deformed Kerr/Schwarzschild spacetime, with the perturbative information resummed rather than expanded; 'SEOB-PM' and wEOBw_{\rm EOB} are closed-form EOB Hamiltonians built from PM scattering data, while 'evolution' EOB models such as SEOBNRv5 and TEOBResumS evolve Hamilton's equations with radiation reaction (dissipative loss of energy and angular momentum through gravitational waves) and are the workhorses of gravitational-wave data analysis. Numerical relativity (NR) solves the full Einstein equations numerically; the scatter–capture separatrix is the critical initial angular momentum below which an unbound encounter is captured (plunges) rather than scatters, making it a sharp probe of the strong-field dynamics. When the two black holes are dynamically asymmetric (unequal masses or anti-parallel spins), gravitational-wave emission is asymmetric, so the two holes are deflected by different angles, θ1≠θ2\theta_1\neq\theta_2; their difference θ2−θ1\theta_2-\theta_1 is a sub-percent, high-order effect tied to recoil (the center-of-mass 'kick'). Long et al. (2025) presented 60 scattering simulations with SpEC (the Spectral Einstein Code, a high-accuracy pseudospectral NR code) up to mass ratio q=10q=10, performed the first direct scattering-angle comparison between NR codes (SpEC and the Einstein Toolkit (ETK), an independent community NR code, finding good agreement), measured θ2−θ1\theta_2-\theta_1 from NR for the first time, and showed that closed-form PM-based models track the NR scattering angle to within 5% except near the separatrix, while both evolution EOB models misplace the separatrix once spins are present and the new asymmetric SEOB-PM constructions reproduce only the qualitative trend of θ2−θ1\theta_2-\theta_1. In the year that followed, the potential-region conservative 5PM/2SF scattering angle was computed independently by two groups — via scattering amplitudes and the double copy (Bern et al. 2025) and via worldline quantum field theory (Driesse et al. 2026) — and the 5PM nonlocal-in-time tail contributions were obtained through tenth self-force order (Dlapa et al. 2026). Driesse et al. (2026) propose a γ\gamma–3 prescription for the conservative extraction, whose justification and uniqueness they leave open, and discuss a definition of the conservative sector based on the N^=−ilog⁡S^\hat N=-i\log\hat S operator.

Problem Statement

The open problem is to complete the second-self-force content of the two-body gravitational scattering angle at fifth post-Minkowskian order and to use it to make effective-one-body scattering models quantitatively reliable against numerical relativity. Concretely: (1) complete the missing O(G5ν2)O(G^5\nu^2) contributions for the non-spinning sector, with spin-dependent 5PM(2SF) coefficients and generic precessing-spin configurations eventual extensions rather than part of this question's scope (the spin-aligned and asymmetric-angle NR comparisons required in (3) are unaffected by this exclusion) — the radiative-region conservative memory pieces (the 5PM nonlocal-in-time tail contributions being already known through tenth self-force order) and the fully dissipative sector at all velocities, with the potential-region conservative 5PM/2SF results of Bern et al. (2025) and Driesse et al. (2026) and the tail contributions of Dlapa et al. (2026) as inputs to this question rather than its target — and settle a definition of the conservative sector under which the complete 5PM/2SF scattering angle is finite, the spurious γ=3\gamma=3 (equivalently v/c=8/3v/c=\sqrt{8}/3) divergence and the dimensional-regularization pole (a spurious 1/ϵ1/\epsilon pole from continuing the loop integrals to 4+ϵ4+\epsilon spacetime dimensions) cancel between the potential, memory, and tail regions, and the definition is unique up to a criterion, or derivable from first principles; (2) incorporate the complete 5PM(2SF) information into resummed EOB Hamiltonians and radiation-reaction prescriptions for unbound orbits, taking into account the documented gauge- and resummation-scheme sensitivity of the SEOB-PM/wEOBw_{\rm EOB}/LEOB mass-shells (Clark and Pratten, arXiv:2508.12126), which must be acknowledged when interpreting strong-field predictions; and (3) demonstrate, on the published SpEC scattering datasets of Long et al. (the 60-simulation set, including spin-aligned and unequal-mass configurations up to q=10q=10 and the near-separatrix slices), that the resulting closed-form and evolution EOB models improve agreement with the NR scattering angles relative to the deficiencies documented in Long et al. (2025) — in particular the misplacement of the scatter–capture separatrix by evolution models for spinning unbound orbits, where current evolution models fail — and improve the prediction of the NR-measured asymmetric per-black-hole angle difference θ2−θ1\theta_2-\theta_1, which the current asymmetric SEOB-PM constructions reproduce only qualitatively. These three sub-goals jointly serve one objective: PM/self-force-informed EOB models of unbound encounters that are quantitatively reliable against NR in the strong-field, comparable-mass regime. The comparison is made against the existing published NR scattering data of Long et al.; generating new simulations in extreme regimes (ultra-high energy, near-extremal spins, q>10q>10) and systematic multi-code NR cross-validation campaigns are adjacent infrastructure problems outside the scope of this question.

References

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