The Carter Constant for Inclined Orbits About a Massive Kerr Black Hole: near-circular, near-polar orbits

TL;DR

This paper introduces the abutment as a tool to test and improve the evolution equations of the Carter constant for inclined orbits around Kerr black holes, demonstrating its utility with elliptical orbit calculations.

Contribution

It presents the abutment as a new independent testing ground for Carter constant evolution equations in Kerr spacetime.

Findings

Derived dQ/dt from published dl/dt and de/dt formulas matched known results to second order in eccentricity.

Validated the abutment as a useful tool for testing Carter constant evolution equations.

Suggested potential for improving accuracy of evolution equations at higher orders.

Abstract

In an extreme mass-ratio binary black hole system, a non-equatorial orbit will list (i.e. increase its angle of inclination, {\iota}) as it evolves in Kerr spacetime. The abutment, a set of evolving, near-polar, retrograde orbits, for which the instantaneous Carter constant (Q) is at its maximum value (Q_{X}) for given values of latus rectum (l) and eccentricity (e), has been introduced as a laboratory in which the consistency of dQ/dt with corresponding evolution equations for dl/dt and de/dt might be tested independently of a specific radiation back-reaction model. To demonstrate the use of the abutment as such a laboratory, a derivation of dQ/dt, based only on published formulae for dl/dt and de/dt, was performed for elliptical orbits on the abutment. The resulting expression for dQ/dt matched the published result to the second order in e. We believe the abutment is a potentially useful tool for improving the accuracy of evolution equations to higher orders of e and l^{1}.