  Eikonal exponentiation in QFT describes the emergence of classical physics at
long distances in terms of a non-trivial resummation of infinitely many
diagrams. Long ago, 't Hooft proposed a beautiful correspondence between
ultra-relativistic scalar eikonal scattering and one-to-one scattering in a
background shockwave space-time, bypassing the need to resum. In this spirit,
we propose here a covariant method for computing one-to-one amplitudes in
curved background space-times which gives rise what we conjecture to be a
general expression for the eikonal amplitude. We show how the one-to-one
scattering amplitude for scalars on any stationary space-time reduces to a
boundary term that captures the long-distance behavior of the background and
has the structure of an exponentiated eikonal amplitude. In the case of scalar
scattering on Schwarzschild, we recover the known results for gravitational
scattering of massive scalars in the eikonal regime. For Kerr, we find a
remarkable exponentiation of the tree-level amplitude for gravitational
scattering between a massive scalar and a massive particle of infinite spin.
This amplitude exhibits a Kawai-Lewellen-Tye-like factorization, which we use
to evaluate the eikonal amplitude in momentum space, and study its analytic
properties.
