From Boundary Data to Bound States II: Scattering Angle to Dynamical Invariants (with Twist)

TL;DR

This work deepens the boundary-to-bound dictionary by proving a simple, powerful relation in the conservative sector: $\Delta\Phi(J,\mathcal{E}) = \chi(J,\mathcal{E}) + \chi(-J,\mathcal{E})$, established via analytic continuation in angular momentum and binding energy and shown to hold to all PM orders. By recasting the radial action in terms of the (analytically continued) scattering angle, the authors derive compact PM expressions for all bound-state dynamical invariants, including $\Delta\Phi$, $\Omega_r$, $\Omega_\phi$, and the redshift, and verify consistency with PN results up to two-loop order. The framework is extended to aligned-spin configurations, where the same mapping applies with the total angular momentum, and PN agreement is demonstrated up to 3.5PN; one-loop resummations further reveal nonperturbative structure. The results offer a streamlined, gauge-invariant pathway to extract two-body gravitational observables directly from scattering data, with implications for waveform modeling and insights into the spin-dependent structure of gravity, while suggesting future extensions to generic spin orientations and spinning-body generalizations of the impetus formula.

Abstract

We recently introduced in [1910.03008] a "boundary-to-bound" dictionary between gravitational scattering data and observables for bound states of non-spinning bodies. In this paper, we elaborate further on this (holographic) map. We start by deriving the following -- remarkably simple -- formula relating the periastron advance to the scattering angle: $ΔΦ(J,{\cal E}) =χ(J,{\cal E}) + χ(-J,{\cal E})$, via analytic continuation in angular momentum and binding energy. Using explicit expressions from [1910.03008], we confirm its validity to all orders in the Post-Minkowskian (PM) expansion. Furthermore, we reconstruct the radial action for the bound state directly from the knowledge of the scattering angle. The radial action enables us to write compact expressions for dynamical invariants in terms of the deflection angle to all PM orders, which can also be written as a function of the PM-expanded amplitude. As an example, we reproduce our result in [1910.03008] for the periastron advance, and compute the radial and azimuthal frequencies and redshift variable to two-loops. Agreement is found in the overlap between PM and Post-Newtonian (PN) schemes. Last but not least, we initiate the study of our dictionary including spin. We demonstrate that the same relation between deflection angle and periastron advance applies for aligned-spin contributions, with $J$ the (canonical) total angular momentum. Explicit checks are performed to display perfect agreement using state-of-the-art PN results in the literature. Using the map between test- and two-body dynamics, we also compute the periastron advance up to quadratic order in the spin, to one-loop and to all orders in velocity. We conclude with a discussion on the generalized "impetus formula" for spinning bodies and black holes as "elementary particles".

Figures (2)

• Figure 1: The geometry of the scattering problem. The motion of the bodies traces two hyperbolas, which are separated by $\tilde{r}_-$ at the point of closest approach. See paper I for details.
• Figure 2: Bound elliptic motion in the center of mass frame. The black ellipses mark the individual paths. The heavier body lies on the focus of the green dashed ellipse. The latter describes the worldline of the lighter body in the companion's frame. The dotted circle of radius $a$ defines the eccentric anomaly, $u$, with $e$ the eccentricity. See paper I for details.