Scattering Amplitudes and N-Body Post-Minkowskian Hamiltonians in General Relativity and Beyond

TL;DR

This paper develops a framework to derive classical N-body Hamiltonians in general relativity from scattering amplitudes, introducing new methods for complex integrals and confirming results with known calculations and black hole scattering scenarios.

Contribution

It introduces a novel approach to compute N-body Hamiltonians from scattering amplitudes, including new techniques for non-trivial integrals and subtraction methods for iteration contributions.

Findings

Derived the 3-body potential at order G^2 in general relativity.

Confirmed agreement with previous post-Newtonian results up to G^2 v^4.

Validated the framework with black hole scattering in extremal backgrounds.

Abstract

We present a general framework for calculating post-Minskowskian, classical, conservative Hamiltonians for $N$ non-spinning bodies in general relativity from relativistic scattering amplitudes. Novel features for $N>2$ are described including the subtraction of tree-like iteration contributions and the calculation of non-trivial many-body Fourier transform integrals needed to construct position space potentials. A new approach to calculating these integrals as an expansion in the hierarchical limit is described based on the method of regions. As an explicit example, we present the $\mathcal{O}\left(G^2\right)$ 3-body momentum space potential in general relativity as well as for charged bodies in Einstein-Maxwell. The result is shown to be in perfect agreement with previous post-Newtonian calculations in general relativity up to $\mathcal{O}\left(G^2 v^4\right)$. Furthermore, in appropriate limits the result is shown to agree perfectly with relativistic probe scattering in multi-center extremal black hole backgrounds and with the scattering of slowly-moving extremal black holes in the moduli space approximation.