Growth of orbital resonances around a black hole surrounded by matter

TL;DR

This paper investigates how orbital resonances evolve around a black hole influenced by surrounding matter, revealing non-integrable dynamics and quantifying phase shifts during inspirals.

Contribution

It introduces a model for perturbed black hole spacetime, demonstrating non-integrability and linking resonance widths to perturbation parameters.

Findings

System exhibits non-integrable geodesic motion.

Resonance widths are calculated using rotation numbers.

Phase shifts during inspirals are estimated from resonance properties.

Abstract

This work studies the dynamics of geodesic motion within a curved spacetime around a Schwarzschild black hole, perturbed by a gravitational field of a far axisymmetric distribution of mass enclosing the system. This spacetime can serve as a versatile model for a diverse range of astrophysical scenarios and, in particular, for extreme mass ratio inspirals as in our work. We show that the system is non-integrable by employing Poincar\'e surface of section and rotation numbers. By utilising the rotation numbers, the widths of resonances are calculated, which are then used in establishing the relation between the underlying perturbation parameter driving the system from integrability and the quadrupole parameter characterising the perturbed metric. This relation allows us to estimate the phase shift caused by the resonance during an inspiral.