Flux-balance laws for spinning bodies under the gravitational self-force

The motion of an extended, but still weakly gravitating body in general relativity can often be determined by a set of conserved quantities. Much like for geodesic motion, a sufficient number of conserved quantities allows the motion to be solved by quadrature. Under the gravitational self-force (relaxing the ``weakly gravitating'' assumption), the motion can then be described in terms of the evolution these ``conserved quantities.'' This evolution can be calculated using the (local) self-force on the body, but such an approach is computationally intensive. To avoid this, one often uses flux-balance laws: relationships between the average evolution (capturing the dissipative dynamics) and the values of the field far away from the body, which are far easier to compute. In the absence of spin, such a flux-balance law has been proven in Isoyama et al. [``Flux-balance formulae'' for extreme mass-ratio inspirals, Prog. Theor. Exp. Phys. 2019, 013E01 (2019).] for any of the conserved action variables appearing in a Hamiltonian formulation of geodesic motion in the Kerr spacetime. In this paper, we derive a corresponding flux-balance law, to linear order in spin, directly relating average rates of change to the flux of a conserved current through the horizon and out to infinity. In the absence of spin, this reproduces results consistent with those in Isoyama et al. [``Flux-balance formulae'' for extreme mass-ratio inspirals, Prog. Theor. Exp. Phys. 2019, 013E01 (2019).]. To linear order in spin, we construct flux-balance laws for four of the five constants of motion for spinning bodies in the Kerr spacetime, although not in a practical form. However, this result provides a promising path toward deriving the flux-balance law for the (generalized) Carter constant.