"We calculate the first order metric perturbation to a Schwarzschild background spacetime induced by a spinning secondary body in the Regge-Wheeler and Zerilli gauges. In particular we specialise to a secondary with spin (anti-)aligned to the total orbital angular momentum in a quasi-circular orbit. From the metric perturbation we can calculate gauge invariant self-force quantities such as Detweiler’s redshift invariant and compare with known PN results. In doing so we present the first strong field calculation of a conservative self-force quantity with a spinning secondary and emphasise:
1) The treatment of the additional spin term to the singular field by deriving additional tensor harmonic regularisation parameters.
2) Parametrising at fixed frequency and practically extracting the linear in spin contribution to the metric perturbation."
