Strong Field Scattering of Two Black Holes: Exploring Gauge Flexibility

Abstract

Recent advances in post-Minkowskian (PM) gravity provide new avenues for the high precision modeling of compact binaries. In conjunction with the effective one body (EOB) formalism, highly accurate PM informed models of binary black holes on scattering trajectories have emerged. Several complementary approaches currently exist, in particular the SEOB-PM model, the $w$_{\rm EOB} framework and the recent Lagrange-EOB (LEOB) approach. These models incorporate PM results in fundamentally different ways, employing distinct resummation schemes and gauge choices. Notably, both SEOB-PM and LEOB have been used to compute gravitational waves of bound systems, showing excellent agreement with numerical relativity (NR). The essential component to all of the models is the EOB mass-shell condition describing the dynamics of the two-body spacetime. In this work we will investigate how this mass-shell condition is constructed, paying particular attention to the impact of gauge choices and how they interact with different resummation schemes, showing that there is a strong dependence on both coordinate choice and EOB gauge. For the region of parameter space considered, we find that the best performing gauges coincide with the choices made in SEOB-PM and $w$_{\rm EOB}, with other choices exhibiting worse performance. The case of spinning black holes is also considered, where the current techniques for spinning EOB-PM are reviewed and compared. We also introduce a new gauge based upon the centrifugal radius, which improves upon previous models, particularly for large and negative spins. This offers a promising avenue for further resummation of spin information within the EOB-PM framework.

Figures (17)

• Figure 1: A comparison of the effective potentials at 4PM for an equal mass configuration at fixed energy ($\Gamma_1 \approx 1.02$) and angular momentum ($p_{\phi} \approx 4.4$). The left panel compares the two EOB gauges for Schwarzschild coordinates and the right for isotropic coordinates.
• Figure 2: A comparison of effective potentials across the different PM orders considered in all combinations of gauge choices. The left panels are the potentials with a Schwarzschild radial coordinate, and the right have an isotropic radial coordinate. Note that the PM expanded potentials ($w_{\rm eob}$ type) in PS and PS* are equivalent. Note also the similarity between the PS* resummed (SEOB-PM model) and the isotropic PM expanded potentials ($w_{\rm eob}$ model). All figures have energy $\Gamma_1$ and dimensionless angular momentum $p_{\phi}\approx4.4$. The dashed line is the dimensionless quantity $p_{\infty}^2 = \gamma^2 - 1$, if the potential never rises above the line then that model will always plunge for this value of $l$. Numerical relativity simulations at this energy and angular momentum Swain:2024ngs show that the system scatters.
• Figure 3: A comparison of nonspinning SEOB-type (resummed potential) models with the numerical relativity data from Swain:2024ngs. We vary the coordinate gauge and EOB-gauge for an equal mass nonspinning configuration, at 4PM order. Left: the lower energy comparison at $\Gamma_1\approx 1.02$. Right: the higher energy comparison at $\Gamma_7 \approx 1.22$. The curves are truncated when the model predicts a plunge.
• Figure 4: Comparing the PM expanded (E) and resummed (R) potential models across the four gauge choices, comparing at the lower energy $\Gamma_1$ for an equal mass system.
• Figure 5: Plots showing the performance of the 5PM-1SF information across the two best combinations of gauge and $w$-potential definition. The left panel shows an equal mass configuration at the lower energy $\Gamma_1$, with the right plot at the higher energy $\Gamma_7$. We see that in both cases the $5$PM-$1$SF information is slightly worse than the corresponding 4PM results. As in previous plots, the curves either asymptote or terminate when the model predicts a plunge.