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