Environmental Effects in Post-Minkowskian Dynamics: Effective Field Theory, Feynman Rules, and Ward Identities for Compact Objects in Relativistic Fluids

Abstract: We develop a hydrodynamic effective field theory for the gravitational dynamics of compact objects moving through an inviscid fluid environment. The fluid is described by the effective theory of perfect fluids, whose Goldstone phonons are kept as explicit fields coupled to gravity. We establish a consistent power counting for the resulting Feynman rules and derive the propagators and interaction vertices through four points in $D$ dimensions, together with the generalized on-shell Ward identities they obey. These identities relate amplitudes with an external graviton to those with the graviton replaced by a phonon, and thus provide nontrivial checks on higher-order calculations. As an application, we recover the leading-order relativistic dynamical-friction force from the tree-level amplitude for single-phonon emission. This work is a first step towards a toolkit for incorporating environmental effects into the scattering-amplitude pipeline used in post-Minkowskian calculations.