On an exponential representation of the gravitational S-matrix

An exponential representation of the S-matrix provides a natural framework for understanding semi-classical limit scattering amplitudes. While sharing some similarities with eikonal formalism it differs from in details. Computationally, rules are simple because pieces that must be subtracted given by combinations unitarity cuts. Analyzing classical gravitational to third Post-Minkowskian order both maximal supergravity and Einstein gravity we find agreement other approaches, including contributions radiation reaction terms. The kinematical relation two-body problem isotropic coordinates follows immediately this procedure, again inclusion up order.