Post-Minkowskian Hamiltonians in Modified Theories of Gravity

TL;DR

This work extends the post-Minkowskian framework to modified gravity by computing the 2PM Hamiltonians for R3 (cubic) corrections to General Relativity. It defines a PM potential V2PM in cubic theories from massive-scalar scattering amplitudes and derives explicit expressions for V2PM^{I1} and V2PM^{G3}, including their nonrelativistic limits that reproduce known post-Newtonian results. The authors then obtain the fully relativistic scattering angle at 2PM, χ2PM^α, by solving the Hamilton-Jacobi equation and regularizing divergences with Hadamard finite parts, confirming agreement with bending-angle results in the NR limit and highlighting a distinct G3 contribution in the relativistic two-body case. Overall, the paper provides a practical route to incorporate higher derivative gravity corrections into PM analyses of binary black hole dynamics, with consistency checks against PN results and known bending-angle limits.

Abstract

The aim of this note is to describe the computation of post-Minkwoskian Hamiltonians in modified theories of gravity. Exploiting a recent relation between amplitudes of massive scalars and Hamiltonians for relativistic point-particles, we define a post-Minkowskian potential at second order in Newton's constant arising from $\mathcal{R}^3$ modifications in General Relativity. Using this result we calculate the associated contribution to the scattering angle for binary black holes at second post-Minkowskian order, showing agreement in the non relativistic limit with previous results for the bending angle of a massless particle around a static massive source in $\mathcal{R}^3$ theories.