The differential cross section for scattering of a Dirac particle in a black hole background is found. The result is the gravitational analog of the Mott formula for scattering in a Coulomb background. The equivalence principle is neatly embodied in the cross section, which depends only on the incident velocity, and not the particle mass. The low angle limit agrees with classical calculations based on the geodesic equation. The calculation employs a well-defined iterative scheme which can be extended to higher orders. Repeating the calculation in different gauges shows that our result for the cross section is gauge-invariant and highlights the issues involved in setting up a sensible iterative scheme.
C.J.L. Doran and A.N. Lasenby, Perturbation Theory Calculation of the Black Hole Elastic Scattering Cross Section, Phys. Rev. D, 66(2), 024006 (2002)