  By combining the KMOC-formalism with the exponential representation of the
scattering matrix we show that the two-body scattering angle is given by the
corresponding matrix element of the exponential representation. This holds to
all orders in the Post-Minkowskian expansion of gravity when restricted to the
conservative sector. Once gravitational radiation is taken into account new
terms correcting this relationship appear starting at fourth Post-Minkowskian
order. A systematic expansion of the momentum kick is provided to any order,
thus illustrating the iterative structure that partly recycles terms from lower
orders in the Post-Minkowskian expansion. We provide explicit results for this
computation to fourth Post-Minkowskian order, the first complete calculation at
this order based on scattering amplitudes.
