The Carter constant for inclined orbits about a massive Kerr black hole: I. Circular orbits

DOI10.1088/0264-9381/27/22/225023zbMATH Open1223.83031arXiv1007.4189OpenAlexW3098808186MaRDI QIDQ3065960

M. Houde, P. G. Komorowski, Sree Ram Valluri

Publication date: 7 January 2011

Published in: Classical and Quantum Gravity (Search for Journal in Brave)

Abstract: In an extreme binary black hole system, an orbit will increase its angle of inclination (i) as it evolves in Kerr spacetime. We focus our attention on the behaviour of the Carter constant (Q) for near-polar orbits; and develop an analysis that is independent of and complements radiation reaction models. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at which Q is at its maximum value for given values of latus rectum (l) and eccentricity (e). The introduction of spin (S = |J|/M2) to the massive black hole causes this boundary, or abutment, to be moved towards greater orbital inclination; thus it no longer cleanly separates prograde and retrograde orbits. To characterise the abutment of a Kerr black hole (KBH), we first investigated the last stable orbit (LSO) of a test-particle about a KBH, and then extended this work to general orbits. To develop a better understanding of the evolution of Q we developed analytical formulae for Q in terms of l, e, and S to describe elliptical orbits at the abutment, polar orbits, and last stable orbits (LSO). By knowing the analytical form of dQ/dl at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the di/dl of hypothetical circular orbits that evolve along the abutment. From these values we have determined that di/dl = -(122.7S - 36S^3)l^-11/2 -(63/2 S + 35/4 S^3) l^-9/2 -15/2 S l^-7/2 -9/2 S l^-5/2. Thus the abutment becomes an important analytical and numerical laboratory for studying the evolution of Q and i in Kerr spacetime and for testing current and future radiation back-reaction models for near-polar retrograde orbits.

Full work available at URL: https://arxiv.org/abs/1007.4189

zbMATH Keywords

Kerr black hole Carter constant abutment near-polar orbits

Mathematics Subject Classification ID

Black holes (83C57) Gravitational energy and conservation laws; groups of motions (83C40) Equations of motion in general relativity and gravitational theory (83C10)