High-energy gravitational scattering and the general relativistic two-body problem

TL;DR

The paper derives a second post-Minkowskian (2PM) effective-one-body (EOB) Hamiltonian for a general two-body system in general relativity, extending previous 1PM results. It casts the 2PM dynamics in a deformed mass-shell form with a Q term, derives an explicit closed-form for q2(H_Schw, ν), and shows that the high-energy limit yields ν-independent Regge trajectories, connecting to Amati–Ciafaloni–Veneziano (ACV) results. The work also analyzes the self-force regime, the light-ring behavior, and the gauge choices that tame ultraviolet-like divergences, while proposing a program to relate quantum gravity amplitudes to classical dynamics by quantizing the EOB Hamiltonian and extracting higher-PM information from multiper-loop amplitudes. Overall, these results sharpen the bridge between classical two-body GR and quantum scattering techniques, with implications for high-energy gravitational scattering and gravitational-wave phenomenology. This framework paves the way for testing predictions via numerical simulations and deeper amplitude-based computations at higher PM orders.

Abstract

A technique for translating the classical scattering function of two gravitationally interacting bodies into a corresponding (effective one-body) Hamiltonian description has been recently introduced [Phys.\ Rev.\ D {\bf 94}, 104015 (2016)]. Using this technique, we derive, for the first time, to second-order in Newton's constant (i.e. one classical loop) the Hamiltonian of two point masses having an arbitrary (possibly relativistic) relative velocity. The resulting (second post-Minkowskian) Hamiltonian is found to have a tame high-energy structure which we relate both to gravitational self-force studies of large mass-ratio binary systems, and to the ultra high-energy quantum scattering results of Amati, Ciafaloni and Veneziano. We derive several consequences of our second post-Minkowskian Hamiltonian: (i) the need to use special phase-space gauges to get a tame high-energy limit; and (ii) predictions about a (rest-mass independent) linear Regge trajectory behavior of high-angular-momenta, high-energy circular orbits. Ways of testing these predictions by dedicated numerical simulations are indicated. We finally indicate a way to connect our classical results to the quantum gravitational scattering amplitude of two particles, and we urge amplitude experts to use their novel techniques to compute the 2-loop scattering amplitude of scalar masses, from which one could deduce the third post-Minkowskian effective one-body Hamiltonian.

Figures (5)

• Figure 1: Feynman-like diagrams for the classical gravitational scattering at first order in $G$.
• Figure 2: Some of the Feynman-like diagrams for the classical gravitational scattering at second order in $G$.
• Figure 3: Graph of the relation between the rescaled angular momentum $j$ and the rescaled effective energy ${{\widehat{H}}_{\rm eff}}$ for $\nu=0.25$, and the 2PM Hamiltonian.
• Figure 4: 2PM-accurate, equal-mass ($\nu=\frac{1}{4}$) rescaled effective Hamiltonian ${{\widehat{H}}_{\rm eff}}$ as a function of the inverse radial variable $u=GM/R$, for the rescaled angular momentum $j=30$. Note that radial infinity is at $u=0$ on the left. The horizontal line indicates the critical value of the effective energy for which the two-body system would end up (in absence of dissipation) in an infinite whirl motion.
• Figure 5: Graphs of the relations (for $\nu=0.2$ and the 2PM Hamiltonian) between the two redshifts $z_a$ and either the usual frequency parameter $x$ (leading to the two curves that turn back towards the left) or the EOB-motivated effective frequency parameter $x_{\rm eff}$. Only the latter choice defines functions $z_a(x_{\rm eff};\nu)$.