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We investigate the possibility of sustained orbital resonances in extreme mass ratio inspirais. Using a near-identity averaging transformation, we reduce the equations of motion for a particle moving in Kerr spacetime with self-force corrections in the neighborhood of a resonant geodesic to a one-dimensional equation for a particle moving in an effective potential. From this effective equation we obtain the necessary and sufficient conditions that the self-force needs to satisfy to allow inspiralling orbits to be captured in sustained resonance. Along the way we also obtain the full nonlinear expression for the jump in the adiabatic constants of motion incurred as an inspirai transiently evolves through a strong resonance to first- order in the mass ratio. Finally, we find that if the resonance is strong enough to allow capture in sustained resonance, only a small fraction (order of the square root of mass-ratio) of all inspirais will indeed be captured. This makes observation of sustained resonances in extreme mass ratio inspirais--if they exist-- very unlikely for space-based observatories like eLisa. [ABSTRACT FROM AUTHOR]

This paper presents the first calculation of the gravitational self-force on a small compact object on an eccentric equatorial orbit around a Kerr black hole to first order in the mass ratio. That is the pointwise correction to the object's equations of motion (both conservative and dissipative) due to its own gravitational field, which is treated as a linear perturbation to the background Kerr spacetime generated by the much larger spinning black hole. The calculation builds on recent advances on constructing the local metric and self-force from solutions of the Teukolsky equation, which led to the calculation of the Detweiler-Barack-Sago redshift invariant on eccentric equatorial orbits around a Kerr black hole in a previous paper. After deriving the necessary expression to obtain the self-force from the Weyl scalar Ψ4, we perform several consistency checks of the method and numerical implementation, including a check of the balance law relating the orbital average of the self-force to the average flux of energy and angular momentum out of the system. Particular attention is paid to the pointwise convergence properties of the sum over frequency modes in our method, identifying a systematic inherent loss of precision that any frequency domain calculation of the self-force on eccentric orbits must overcome. [ABSTRACT FROM AUTHOR]