Abstract
The "flux-balance formulae" that determine the averaged evolution of energy, azimuthal angular momentum, and Carter constant in terms of the averaged asymptotic gravitational-wave fluxes for inspirals of small bodies into Kerr black holes were first derived about 15 years ago. However, this derivation is restricted to the case that the background Kerr geodesics are non-resonant (i.e., the radial and angular motions are always incommensurate), and excludes the resonant case that can be important for the radiative dynamics of extreme mass-ratio inspirals. We give here a new derivation of the flux formulae based on Hamiltonian dynamics of a self-forced particle motion, which is a valuable tool for analyzing self-force effects on generic (eccentric, inclined) bound orbits in the Kerr spacetime. This Hamiltonian derivation using action-angle variables is much simpler than the previous one, applies to resonant inspirals without any complication, and can be straightforwardly implemented by using analytical/numerical Teukolsky-based flux codes.
Highlights
Introduction and summaryConsider an extreme mass-ratio inspiral of a smaller non-spinning body of mass μ into a larger Kerr black hole of mass M ≫ μ, which is a main target of the space-based gravitational wave observatory of LISA [1,2,3]
It is known that every extreme mass-ratio inspiral that will be observable by LISA is expected to pass through at least one resonance [30], and neglecting this effect would lead to a loss of detectable gravitational-wave signals [31]; see Sec. 5.3 of Ref. [4] as well as Refs. [29, 32] for the further implications of this r-θ resonance for gravitational-wave astronomy. 2
We first need a set of canonical variables (Xα, Pα) such that the canonical momenta Pα recover the constants of motion for Kerr geodesics in the test-mass limit η → 0 [12]
Summary
Consider an extreme mass-ratio inspiral of a smaller non-spinning body of mass μ into a larger Kerr black hole of mass M ≫ μ, which is a main target of the space-based gravitational wave observatory of LISA [1,2,3]. A natural strategy would be to apply the “flux-balance formulae” that determine the time-averaged rates of change of {E, L, Q} under the self-force (“radiation-reaction force”) effects from the associated total gravitational radiation out to infinity and down to the horizon of the Kerr black hole: see Sec. 6 of Ref. It is known that every extreme mass-ratio inspiral that will be observable by LISA is expected to pass through at least one resonance [30], and neglecting this effect would lead to a loss of detectable gravitational-wave signals [31]; see Sec. 5.3 of Ref. The purpose of this paper is to eliminate the remaining non-resonant restrictions, and to establish the complete set of flux formulae for radiative inspirals of small bodies into Kerr black holes at O(η), including the case of resonant orbits
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