From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist)

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: ∆Φ(J, E) = χ(J, E) + χ(−J, 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'. Our findings here and in [1910.03008] imply that the deflection angle already encodes vast amount of physical information for bound orbits, encouraging independent derivations using numerical and/or self-force methodologies.

Sixth post-Newtonian local-in-time dynamics of binary systems

“…(6.26) This result is obtained by analytically continuing the quasi-Keplerian representation of the hyperbolic motion used above [79] back to the elliptic-motion case (expressing all quantities in terms of γ and j and analytically continuing γ from γ hyperbolic > 1 to γ elliptic < 1). The result (6.26) is consistent with the analytic-continuation link between the scattering angle and the periastron precession [37], as is easily seen in view of the link [52] used above between the tail contribution to the scattering angle and the time integral of the gravitational-wave energy loss. The functional structure of ∆E elliptic GW (γ, j) is much simpler than that of ∆E hyperbolic GW (γ, j).…”

Extremal black hole scattering at $$ \mathcal{O} $$(G3): graviton dominance, eikonal exponentiation, and differential equations

“…Two pieces of evidence were provided in support of this conjecture: first, the absence of precession for the full O(G 2 ) dynamics, which directly follows from an analog of the "no-triangle" hypothesis [80][81][82][83][84][85] for massive scattering; and second, various all-orders-in-G calculations in the probe limit for different charge configurations. It is known that O(G 3 ) (or any odd power of G) corrections to the conservative dynamics cannot yield precession [86,87]. Instead we will use the scattering angle at O(G 3 ) to test this conjecture, and see that it deviates from the integrable Newtonian result at this order.…”

Classical gravitational scattering at $$ \mathcal{O} $$(G3) from Feynman diagrams

“…These include the study of the effect of these new 3PM corrections on the binding energy of a binary inspiral in comparison with numerical relativity [92]. Currently, the 5PN term of the 3PM result has been verified [93], while other methods for EFT matching have also been devised [16][17][18][19]. The case of massless scattering has also received more recent attention with new computations in supergravity as well as in Einstein gravity [94][95][96][97].…”

“…The breakthrough observation of gravitational waves at LIGO/Virgo [1,2] has triggered immense interest in bridging developments from the modern scattering amplitudes program to the physics of gravitational waves. Building on past work on the inspiral problem based on graviton effective field theory (EFT) [3][4][5][6][7] and matching to a classical potential [8,9,[11][12][13], many developments have now emerged which exploit classic methods [14][15][16][17][18][19][20][21] as well as recent amplitudes advances [22][23][24][25] to investigate systems with [26][27][28][29][30][31][32][33][34][35][36][37][38] and without spin [39,40].…”