Nonlinearly Self-Interacting Extended Bodies Move as Test Bodies in Effective External Fields

In electromagnetism, linearized general relativity, and other contexts, previous work has shown that the laws of motion which govern compact, self-interacting bodies can be obtained by applying “Detweiler-Whiting prescriptions” to the laws of motion that govern test bodies. These prescriptions replace any field that appears in a test-body law of motion with a certain effective field which is a quasilocal functional of the physical variables—a functional that can be interpreted as a regularization procedure in a point-particle limit. We generalize these results, presenting a formalism that allows Detweiler-Whiting prescriptions to be be directly derived for extended bodies, even in nonlinear field theories. If a generating functional with particular properties can be constructed, we find effective linear and angular momenta which evolve via Mathisson-Papapetrou-Dixon equations involving appropriate effective fields. These equations implicitly incorporate all self-force, self-torque, and extended-body effects. Although our main focus is on bodies coupled to nonlinear scalar fields, we also remark on the gravitational case.