Spinning-black-hole scattering and the test-black-hole limit at second post-Minkowskian order

TL;DR

This work analyzes spinning black-hole scattering in the post-Minkowskian framework, proposing and validating a $2$PM mapping that relates the conservative dynamics of a real aligned-spin binary to the scattering of a spinning test-BH in a Kerr background. By comparing against known PN results, spin-orbit and spin-squared terms, and exact test-BH–Kerr calculations, the authors show that the $2$PM angle can be expressed as a mass-weighted test-BH angle with an overall energy factor, up to higher PN corrections. The results unify PN and PM descriptions, enabling a compact, gauge-invariant way to encode the conservative dynamics and offering a route to improved EOB models for spinning binaries. They also discuss implications for amplitudes-based approaches and the role of tail effects, outlining future directions for extending the mapping beyond the current orders. The study thus provides a principled, nonperturbative handle on spin effects at $2$PM that can feed into waveform modeling for gravitational-wave astronomy.

Abstract

Recently, the gravitational scattering of two black holes (BHs) treated at the leading order in the weak-field, or post-Minkowskian (PM), approximation to General Relativity has been shown to map bijectively onto a simpler effectively one-body process: the scattering of a test BH in a stationary BH spacetime. Here, for BH spins aligned with the orbital angular momentum, we propose a simple extension of that mapping to 2PM order. We provide evidence for the validity and utility of this 2PM mapping by demonstrating its compatibility with all known analytical results for the conservative local-in-time dynamics of binary BHs in the post-Newtonian (weak-field and slow-motion) approximation and, separately, in the test-BH limit. Our result could be employed in the construction of improved effective-one-body models for the conservative dynamics of inspiraling spinning binary BHs.

Figures (2)

• Figure 1: A schematic diagram of an aligned-spin two-BH system and its limits as discussed in Sec. \ref{['sec:SEOB']}. We depict in green the spinning BHs' ring singularities with radii $a=|\boldsymbol a|$, and in black the BH horizons. To obtain the limit of a spinning test BH, we take its mass $m_\mathrm t$ to be negligible, $m_\mathrm t/m_\mathrm B\to 0$, while keeping its rescaled spin $\boldsymbol a_\mathrm t$ finite. Taking the spin of the test BH to zero yields a monopolar test point mass, following a geodesic in a Kerr background.
• Figure 2: Above: the (arbitrary-mass-ratio) two-body case. Below: the test-body case. Left: the Minkowskian geometry of the (incoming) zeroth-order state. Right: the spatial geometry of the scattering plane---in the center-of-mass frame above, and in the background frame below. Above left, the 4-momenta are decomposed as $p_1^\mu=E_1 u_\mathrm{cm}^\mu+\boldsymbol p^\mu$ and $p_2^\mu=E_2 u_\mathrm{cm}^\mu-\boldsymbol p^\mu$; $E=E_1+E_2$ is the total center-of-mass frame energy of Eq. (\ref{['Egamma']}). The test-body's momentum is decomposed according to $p_\mathrm t^\mu=E_\mathrm t u_\mathrm B^\mu+\boldsymbol p_\mathrm t^\mu$. The magnitudes of the "spatial" momenta are $|\boldsymbol p|=m_1m_2\gamma v/E$ and $|\boldsymbol p_\mathrm t|=m_\mathrm t\gamma v$, so that Eqs. (\ref{['J']}) and (\ref{['Jt']}) can be rewritten as $J=|\boldsymbol p|b$ and $J_\mathrm t=|\boldsymbol p_\mathrm t|b$, respectively.