Strong-Field Scattering of Two Black Holes: Numerics Versus Analytics

Abstract

We probe the gravitational interaction of two black holes in the strong-field regime by computing the scattering angle $χ$ of hyperbolic-like, close binary-black-hole encounters as a function of the impact parameter. The fully general-relativistic result from numerical relativity is compared to two analytic approximations: post-Newtonian theory and the effective-one-body formalism. As the impact parameter decreases, so that black holes pass within a few times their Schwarzschild radii, we find that the post-Newtonian prediction becomes quite inaccurate, while the effective-one-body one keeps showing a good agreement with numerical results. Because we have explored a regime which is very different from the one considered so far with binaries in quasi-circular orbits, our results open a new avenue to improve analytic representations of the general-relativistic two-body Hamiltonian.

Figures (3)

• Figure 1: Coordinate trajectories of the two BHs in hyperbolic-like encounters for four selected values of the impact parameter ${b}_{\rm NR}$.
• Figure 2: Comparing the NR scattering angle with various EOB and PN predictions. NR data and the state-of-the-art EOB model agree within their respective error bars.
• Figure 3: EOB effective potentials $H_{\rm EOB}(r,\bar{J},p_r=0)$ at various PN approximations. The values $(\bar{E},\bar{J})=\left(E_{\rm in}-\frac{1}{2} \Delta E^{\rm NR},J_{\rm in}^{\rm NR}-\frac{1}{2} \Delta J^{\rm NR}\right)$ correspond, in Table \ref{['tab:all_info']}, to ${b}_{\rm NR}=9.6\,M$. The EOB motion corresponds to a particle starting at large $r$ with negative radial momentum $p_{r}^{0}$ and moving towards the left at constant energy $\bar{E}$. Note that 1PN, 2PN and 3PN EOB models predict plunge instead of scattering.