Effective stress–energy tensors, self-force and broken symmetry

Abstract

Deriving the motion of a compact mass or charge can be complicated by the presence of large self-fields. Simplifications are known to arise when these fields are split into two parts in what is known as a Detweiler–Whiting decomposition. One component satisfies vacuum field equations, while the other does not. The force and torque exerted by the (often ignored) inhomogeneous ‘S-type’ portion are analyzed here for extended scalar charges in curved spacetimes. This field has previously been shown to effectively renormalize a body's linear and angular momenta. We show here that if the geometry is sufficiently smooth, it actually shifts all multipole moments of the body's stress–energy tensor (and does nothing else). This greatly expands the validity of statements that the homogeneous R field determines the self-force and self-torque up to renormalization effects. The forces and torques exerted by the S field directly measure the degree to which a spacetime fails to admit Killing vectors inside the body. A number of mathematical results related to the use of generalized Killing fields are therefore derived, and may be of wider interest. As an example of their application, the effective shift in the quadrupole moment of a charge's stress–energy tensor is explicitly computed to lowest nontrivial order.