| Summary: | 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 O $$ \mathcal{O} $$ (G 2) 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 O $$ \mathcal{O} $$ (G 2 v 4). 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.
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