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

TL;DR

This paper derives flux-balance laws for spinning bodies in Kerr spacetime under gravitational self-force, extending previous results to include linear spin effects and facilitating easier computation of orbital evolution.

Contribution

It introduces flux-balance laws for spinning bodies in Kerr spacetime to linear order in spin, generalizing prior non-spinning results and aiding gravitational self-force calculations.

Findings

Derived flux-balance law for spin-affected constants of motion.

Reproduced non-spinning flux-balance results as a special case.

Outlined a path to flux-balance law for the Carter constant.

Abstract

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., 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., 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 towards deriving the flux-balance law for the (generalized) Carter constant.