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

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.

41 pages, 5 figures

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• Effective two-body approach to the hierarchical three-body problem: quadrupole to 1PN
• Conservative Scattering of Reissner-Nordström Black Holes at Third Post-Minkowskian Order
• Observables and Unconstrained Spin Tensor Dynamics in General Relativity from Scattering Amplitudes
• The hierarchical three-body problem at $\mathcal{O}(G^2)$