# Post-Minkowskian Hamiltonians in Modified Theories of Gravity

## TL;DR

This paper develops a method to compute post-Minkowskian Hamiltonians in modified gravity theories, specifically $ ext{R}^3$ modifications, and applies it to binary black hole scattering, connecting amplitude relations with gravitational dynamics.

## Contribution

It introduces a novel approach to derive second-order post-Minkowskian Hamiltonians in $ ext{R}^3$ gravity using amplitude-Hamiltonian relations, extending previous results.

## Key findings

- Derived the second-order post-Minkowskian potential for $ ext{R}^3$ gravity.
- Calculated the scattering angle for binary black holes in this framework.
- Confirmed consistency with known bending angles in the non-relativistic limit.

## 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.