We study the relativistic orbit of binary black holes in systems with small mass ratio. The trajectory of the smaller object (another black hole or a neutron star), represented as a particle, is determined by the geodesic equation on the perturbed massive black hole spacetime. Here we study perturbations around a Schwarzschild black hole using Moncrief's gauge invariant formalism. We decompose the perturbations into ℓ multipoles to show that all ℓ-metric coefficients are C0 at the location of the particle. Summing over ℓ, to reconstruct the full metric, gives a formally divergent result. We succeed in bringing this sum to a Riemann's ζ-function regularization scheme and numerically compute the first-order geodesics.