We describe the computation of post-Minkowskian Hamiltonians in General Relativity from scattering amplitudes. Using a relativistic Lippmann-Schwinger equation, we relate perturbative amplitudes of massive scalars coupled to gravity to the post-Minkowskian Hamiltonians of classical General Relativity to any order in Newton's constant. We illustrate this by deriving an Hamiltonian for binary black holes without spin up to 2nd order in the post-Minkowskian expansion and demonstrate explicitly the equivalence with the recently proposed method based on an effective field theory matching. PACS numbers: 04.60.-m, 04.62.+v, 04.80.Cc 1 arXiv:1906.01579v1 [hep-th] 4 Jun 2019daring angle of attack is to treat the bound state problem as not expanded in momentum while still expanding to fixed order in Newton's constant. Such an approach has recently been proposed by Cheung, Rothstein and Solon [8], and it has already been pushed one order higher in the expansion in Newton's constant [9] (and compared to the post-Newtonian expansions in [10]). Here the method of effective field theory is used to extract the interaction Hamiltonian: the underlying Einstein-Hilbert action coupled to matter produces the classical part of the scattering amplitude while an effective theory of two massive objects define the interaction Hamiltonian. The correct matching between the two theories is performed

Using the implicit function theorem we demonstrate that solutions to the classical part of the relativistic Lippmann-Schwinger equation are in one-to-one correspondence with those of the energy equation of a relativistic two-body system. A corollary is that the scattering angle can be computed from the amplitude itself, without having to introduce a potential. All results are universal and provide for the case of general relativity a very simple formula for the scattering angle in terms of the classical part of the amplitude, to any order in the post-Minkowskian expansion.

We extract the long-range gravitational potential between two scalar particles with arbitrary masses from the two-to-two elastic scattering amplitude at 2nd Post-Minkowskian order in arbitrary dimensions. In contrast to the four-dimensional case, in higher dimensions the classical potential receives contributions from box topologies. Moreover, the kinematical relation between momentum and position on the classical trajectory contains a new term which is quadratic in the tree-level amplitude. A precise interplay between this new relation and the formula for the scattering angle ensures that the latter is still linear in the classical part of the scattering amplitude, to this order, matching an earlier calculation in the eikonal approach. We point out that both the eikonal exponentiation and the reality of the potential to 2nd post-Minkowskian order can be seen as a consequence of unitarity. We finally present closed-form expressions for the scattering angle given by leading-order gravitational potentials for dimensions ranging from four to ten.