Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings

TL;DR

This work shows that at first post-Minkowskian order, the dynamics of two spinning black holes, to all orders in spin and multipoles, reduces to a scattering problem whose gauge-invariant content can be captured by simple mappings to spinning test bodies in Kerr and Kerr geodesics. Using a worldline-skeleton action and the MPD equations in harmonic gauge, the authors derive compact closed-form expressions for the net $O(G)$ changes in momenta and spins, demonstrating a factorization that effectively replaces the interaction with a Kerr-like single-spin structure. They then establish explicit EOB mappings between the two-body and test-body scattering results, including energy maps and new spin-mapping relations, and construct canonical Hamiltonians that reproduce these 1PM results and align with PN results at linear order in $G$. The findings provide a structurally transparent framework for binary BH dynamics at 1PM order, with direct implications for EOB models and gravitational-wave data analysis, and hint at extensions to higher orders. Crucially, the results illuminate a deep equivalence between two-body spinning BH motion and geodesic motion in Kerr, via gauge-invariant holonomies and EOB mappings, enabling resumation of spin effects to all orders in spin within the 1PM approximation.

Abstract

We demonstrate equivalences, under simple mappings, between the dynamics of three distinct systems---(i) an arbitrary-mass-ratio two-spinning-black-hole system, (ii) a spinning test black hole in a background Kerr spacetime, and (iii) geodesic motion in Kerr---when each is considered in the first post-Minkowskian (1PM) approximation to general relativity, i.e. to linear order $G$ but to all orders in $1/c$, and to all orders in the black holes' spins, with all orders in the multipole expansions of their linearized gravitational fields. This is accomplished via computations of the net results of weak gravitational scattering encounters between two spinning black holes, namely the net $O(G)$ changes in the holes' momenta and spins as functions of the incoming state. The results are given in remarkably simple closed forms, found by solving effective Mathisson-Papapetrou-Dixon-type equations of motion for a spinning black hole in conjunction with the linearized Einstein equation, with appropriate matching to the Kerr solution. The scattering results fully encode the gauge-invariant content of a canonical Hamiltonian governing binary-black-hole dynamics at 1PM order, for generic (unbound and bound) orbits and spin orientations. We deduce one such Hamiltonian, which reproduces and resums the 1PM parts of all such previous post-Newtonian results, and which directly manifests the equivalences with the test-body limits via simple effective-one-body mappings.

Figures (2)

• Figure 1: Spacetime diagrams, with the future being upward, of the zeroth-order scattering states. Top: The two body case, with the bodies' momenta $\vec{p}_1$ and $\vec{p}_2$ pointing along their worldlines, and with the impact parameter $\vec{b}$ and relative momenta $\pm\vec{p}_{\perp}$ contained in the plane orthogonal to the cm-frame velocity $\vec{u}_\mathrm{cm}$. Bottom: The test-body case, with the test and background bodies' momenta $\vec{p}_\mathrm{t}$ and $\vec{p}_\mathrm{b}$ pointing along their worldlines, and with $\vec{b}_\mathrm{t}$ and $\vec{p}_{\mathrm{t}{\perp}}$ contained in the plane orthogonal to the background velocity $\vec{u}_\mathrm{b}$.
• Figure 2: The twisted-radial congruence of straight lines in Euclidean 3-space, given by the lines of constant $\tilde{\theta}$ and $\tilde{\phi}$ in the TOS (twisted oblate spheroidal) coordinates $(\tilde{r},\tilde{\theta},\tilde{\phi})$, shown on the surfaces of constant $\tilde{\theta}$ which are half one-sheeted hyperboloids (instead of the half cones $\theta=\mathrm{const.}$). The ring, at $\rho=a$, $z=0$ or $\tilde{r}=0$, $\tilde{\theta}=\pi/2$, through which the waists of all the hyperboloids are threaded, is not pictured above, but is the inner boundary of the half hyperboloid $\tilde{\theta}=\pi/2$ shown below, which is the equatorial plane minus the disk $\rho<a$, $z=0$ or $\tilde{r}=0$.