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The Fradkin-Schwinger functional methods to represent a Green function in anexternal gravitational field are used to study the eikonal and thenext-to-eikonal limit, including the nonlinear gravitational interactions, ofthe scattering amplitudes of an ultra-relativistic scalar particle on a staticsuper-massive scalar target in the nearly forward limit. The functionalapproach confirms the exponentiation of the leading eikonal which also appliesto the first non-leading power in the energy of the light particle, moreoverincludes the interaction at impact parameter much larger than the Schwarzschildradius associated with the center of mass energy in the ultra-relativisticlimit.

A functional approach to the next to eikonal approximation of high energy gravitational scattering