Detweiler's redshift invariant for extended bodies orbiting a Schwarzschild black hole

Understand

We compute the first-order self-force contribution to Detweiler's redshift invariant for extended bodies endowed with both dipolar and quadrupolar structure (with spin-induced quadrupole moment) moving along circular orbits on a Schwarzschild background.

• Our analysis includes effects which are second order in spin, generalizing previous results for purely spinning particles.
• The perturbing body is assumed to move on the equatorial plane, the associated spin vector being orthogonal to it.
• The metric perturbations are obtained by using a standard gravitational self-force approach in a radiation gauge.