Renormalized Dynamical Friction in Black-Hole Spacetimes

We derive the renormalized local equation of motion for the gravitational self-force on a compact object moving through matter in a black-hole spacetime. We integrate out the environment and metric perturbations on a Schwinger–Keldysh contour, obtaining a causal effective action given by a sum of curved-spacetime Feynman diagrams organized by powers of the mass ratio and environmental density. At leading order, an environmental wake gravitationally backreacts on the worldline. The corresponding diagram contains a logarithmic ultraviolet divergence that renormalizes the leading dissipative fluid–particle coupling. We isolate the divergence with a local large-momentum expansion, in which the curved-spacetime retarded Green’s function reduces to its flat tangent-space form. We translate the resulting singularity into a Barack–Ori mode-sum scheme, deriving an analytic puncture and regularization parameters for an ideal-fluid environment. Mode subtraction fixes the divergent part but not the finite Wilson coefficient. For collisionless dust, we determine this coefficient by matching to an exact near-zone calculation for a black hole moving through the medium. This gives a complete renormalization prescription suitable for numerical self-force calculations.