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 $$ \mathcal{O} $$ O (G2) 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 $$ \mathcal{O} $$ O (G2v4). 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.

Poincaré invariance of spinning binary dynamics in the post-Minkowskian Hamiltonian approach

“…A detailed discussion for N-body PM Hamiltonian can be found e.g. in [28]. In particular, it would be interesting to see how one of our main findings, ⃗ X [1] = z 2 ⃗ x 1 + z 1 ⃗ x 2 , should be generalized in the N-body problem.…”

Conservative scattering of Reissner-Nordström black holes at third post-Minkowskian order

“…Modern on-shell amplitudes methods have been used to compute the fully relativistic scattering impulse and radiated momentum of extremal black holes at the 2PM and 3PM orders [59,60,101,[115][116][117] in the context of N = 8 supergravity. Amplitudes techniques have also been applied in Einstein-Maxwell theory to compute the two-body and three-body potentials [80,118] at 2PM order.…”

Absorptive effects and classical black hole scattering

“…Recent years have seen tremendous advances in these weak-field perturbative approaches. In particular, new and powerful calculational methods based on relativistic scattering amplitudes [24][25][26][27][28][29][30][31][32][33][34][35][36] and worldline effective field theories (EFT) [37][38][39][40][41][42][43][44][45][46], have allowed for rapid progress in understanding the PM scattering regime of the gravitational two-body problem. These approaches share the common strategy of initially framing the problem in quantum mechanical language, and then using the eventual classical or ℏ → 0 limit as an additional JHEP03(2024)015 expansion leading to vast simplifications in the required calculations.…”