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

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.