Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory

TL;DR

This work tackles black-hole scattering near the transition to plunge by leveraging a scalar-field self-force toy model to probe strong-field behavior and to inform a resummation of the post-Minkowskian expansion. The authors derive the leading self-force divergence $A_1(v)$ at the separatrix, decompose the SF into conservative and dissipative parts, and construct a resummed scattering-angle formula that reproduces both the geodesic logarithmic divergence and the $1/\delta b$ SF divergence. They deploy a novel hybrid time-domain/frequency-domain numerical scheme to compute $A_1(v)$ across a range of velocities, validating the resummation against high-precision SF data and showing uniform accuracy from weak to strong fields. The results demonstrate that SF-informed PM resummation significantly extends the regime of validity of PM predictions and offer a computationally efficient pathway to improved BH scattering models, with clear potential to translate to gravitational self-force calculations.

Abstract

Geodesic scattering of a test particle off a Schwarzschild black hole can be parameterized by the speed-at-infinity $v$ and the impact parameter $b$, with a "separatrix", $b=b_c(v)$, marking the threshold between scattering and plunge. Near the separatrix, the scattering angle diverges as $\sim\log(b-b_c)$. The self-force correction to the scattering angle (at fixed $v,b$) diverges even faster, like $\sim A_1(v)b_c/(b-b_c)$. Here we numerically calculate the divergence coefficient $A_1(v)$ in a scalar-charge toy model. We then use our knowledge of $A_1(v)$ to inform a resummation of the post-Minkowskian expansion for the scattering angle, and demonstrate that the resummed series agrees remarkably well with numerical self-force results even in the strong-field regime. We propose that a similar resummation technique, applied to a mass particle subject to a gravitational self-force, can significantly enhance the utility and regime of validity of post-Minkowskian calculations for black-hole scattering.

Figures (8)

• Figure 1: Comparison of successive PM approximations with "exact" numerical SF values for the conservative (blue) and dissipative (orange) contributions to the 1SF scattering angle at $v=0.5$. The inset shows the relative difference between the analytical PM approximation and the numerical SF values (interpolated over $b$). The PM expressions accurately describe the weak-field behavior, with 4PM conservative results being within the numerical error of the data for $b\gtrsim 100M$. The PM expressions break down when approaching the separatrix, here at $b\simeq 8.807M$ (vertical dashed line), where $\chi^{\rm 1SF}$ diverges. The numerical values were generated using the time-domain code detailed in Sec. \ref{['sec:TD']}. The numerical errors are too small to be resolved on the scale of the main plot.
• Figure 2: Comparison of plain and resummed 4PM expressions for the geodesic scattering angle at $v=0.5$. The exact geodesic expression $\chi^{\rm 0SF}$ from Eq. (\ref{['eq:geodesic_scatter_angle']}) is shown for reference in black dashed line. The plain 4PM expression (blue) fails to capture the logarithmic divergence near the separatrix. In the resummed version (orange) we force the correct singular behavior, and this dramatically improves the performance of the PM model everywhere. The inset shows the relative difference between the full expression and the plain 4PM expression (blue) and between it and the resummed one (orange).
• Figure 3: Regularized $\ell$-mode contributions to $\nabla_t \Phi^R$ at positions $r_p = 4.7M$ and $r_p = 4.2M$ along the outbound legs of the orbits with parameters $(v,b) = (0.8, 6.07387M)$ and $(0.2, 20.3825M)$, respectively (both orbits have $\delta b \approx 0.0005M$). Also plotted are the analytical predictions for the large-$\ell$ asymptotics of these contributions. The dips near $\ell=13$ correspond to a sign change in both datasets. At high velocity, the $\ell$-mode contributions take longer to settle down to the predicted asymptotic $\ell^{-6}$ decay, with distinct large-$\ell$ peaks visible at intermediate radii. Such large-$\ell$ spectral "bumps", which may be a signature of radiation beaming, complicate computations at large $v$.
• Figure 4: The time component of the SF for the orbit $(v,b)=(0.7,6.71307M)$ (with $\delta b \approx 0.001M$), as calculated with the FD (blue) and TD (orange) codes. The visible difference between the two codes near periastron is due to the omission in the TD calculation of beaming features at $\ell>15$, beyond the reach of our TD code. The hybrid model is shown in the black dashed line, with $r_{\rm switch}\approx 5.28M$ marked by the vertical black lines. The model uses the TD data for $r>r_{\rm switch}$ and the FD data for $r<r_{\rm switch}$. The noisy features visible in the FD data are artifacts of our dynamical mode-sum truncation algorithm; they occur mostly at $r>r_{\rm switch}$ where they do not impact our hybrid model.
• Figure 5: Strong-field numerical results for the dissipative (orange), conservative (blue), and total (green) pieces of the scattering angle for an initial velocity $v=0.5$. The values were generated with the hybrid model, with error bars too small to be discerned on the scale of this plot. The solid lines are the extracted divergences of the form $\sim A_1 (b_c/\delta b)$. The fitted values of $A_1$ are given in Table \ref{['tab:A1_vs_v']}.