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

TL;DR

The paper derives flux-balance laws for extended, weakly gravitating bodies in general relativity to linear order in spin, connecting averaged evolutions of generalized conserved quantities to asymptotic gravitational-field fluxes at the horizon and at null infinity. By developing a pseudo-Hamiltonian, phase-space framework with a Hamilton propagator and a conserved symplectic current, the authors relate local self-force effects to global fluxes, and show that in Kerr spacetime true flux-balance laws exist for non-spinning bodies and for certain spinning generalizations (notably $m^2$, $E$, $L_z$, and $K$). A conservative Hamiltonian via the symmetric part of the metric perturbation is introduced to disentangle dissipative and conservative effects, while the spin introduces complexities: the Rüdiger constant $Y$ and spin-constraint terms do not yield a straightforward flux-balance law at this order. The results open a path toward explicit expressions for spin-corrected Carter-type quantities and highlight both the utility and current limitations of flux-balance methods for spinning self-forced bodies in Kerr, with implications for modeling extreme mass-ratio inspirals and testing GR.

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.

Figures (1)

• Figure 1: The various surfaces surrounding the worldline $\gamma$ on which the flux-balance laws are integrated: the inner worldtube $\mathcal{W}(\tau, \Delta \tau; r)$, the outer boundary $\mathcal{B}(\tau, \Delta \tau)$, and the two pairs of endcaps, $\mathcal{W}_\pm (r)$ and $\mathcal{B}_\pm$. For clarity, we only label (and shade) two of the endcaps, $\mathcal{W}_+ (r)$ (at the top) and $\mathcal{B}_-$ (at the bottom).