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

TL;DR

This paper develops a holographic map linking scattering data to bound state observables in gravitational systems, deriving simple formulas and extending the framework to include spin effects with high-order precision.

Contribution

It introduces a simple analytic continuation formula relating periastron advance to scattering angle and extends the boundary-to-bound dictionary to spinning bodies, providing new tools for gravitational dynamics analysis.

Findings

Derived a simple relation between periastron advance and scattering angle.

Reconstructed the radial action from scattering angle data.

Extended the boundary-to-bound map to include spin effects.

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: $\Delta \Phi(J,{\cal E}) =\chi(J,{\cal E}) + \chi (-J,{\cal E})$, 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".