All elastic amplitudes in the (black hole) eikonal phase

In this article we calculate the eikonal scattering amplitude for an arbitrary number of in- and out-particles, using covariant quantization in a spherical harmonics basis on the Schwarzschild background. We extend prior results to resummation over all partial waves, restoring contributions from transverse separation and correctly taking into account the particle masses in the pole structure. We consider leading order interactions mediated by scalar-scalar-graviton vertices and scalar electrodynamics. We perform our calculations in the black hole eikonal phase. The $2\to 2$ eikonal amplitude is measured by the transverse Green's function $(-\Delta_{\Omega}+a)G(\Omega,\Omega')=\delta^{(2)}(\Omega-\Omega')$. As a consistency check, we use our formalism in flat space to find an exact match with the known flat space eikonal $2\to 2$ amplitude in literature. We then extend the eikonal amplitude to arbitrarily many particles for the first time in both flat space and on the black hole background. We show that the black hole amplitude matches the black hole S-matrix as derived by 't Hooft. We conclude that this amplitude provides the most general elastic contribution one can achieve in the eikonal phase.