A Unified Treatment of the Self-Force Problem

This paper introduces formalism, based on worldline effective field theory, which allows one to systematically calculate the gravitational self-force on a compact object immersed in an environment with non-vanishing stress energy. Using the closed time path integral we present a universal effective action that can be used to calculate the equations of motion of a compact object due to its interaction with the environment to any order in the relevant expansion parameters, the relative importance of which depends upon the choice of environment. We show that the leading order equations of motion can be universally written in terms of the retarded two-point function of the stress-energy tensor for the environment. The resulting action can be used to calculate both dissipative (dynamical friction) and conservative forces in a completely relativistic fashion. We demonstrate the utility of the result by calculating the dissipative force due to dust, an inviscid fluid, and a coherent field. Our results agree with those previously derived in the literature. We furthermore show that the famous ``Coulomb Log" found by Chandrasekhar should be interpreted as a renormalization group log due to a UV divergence that renormalizes the dissipative part of the in-in action. We then prove that this log is universal in the Newtonian limit in that its value is universal, for a generic class of environments and trajectories. This log is the leading log in an RG flow in the dissipative action that has yet to be explored.