Lyapunov timescales and black hole binaries

TL;DR

This paper investigates the Lyapunov timescales of unstable orbits in black hole binaries, analyzing their chaotic behavior and observational relevance through gravitational wave phase decoherence.

Contribution

It provides a detailed analysis of Lyapunov exponents in black hole binary orbits, highlighting their limitations and the conditions under which chaos can be observed.

Findings

Lyapunov timescales can be comparable to gravitational wave dissipation timescales.

Lyapunov exponents depend on coordinate choices and orbit specifics.

Transient behaviors can obscure the detection of chaos.

Abstract

Black holes binaries support unstable orbits at very close separations. In the simplest case of geodesics around a Schwarzschild black hole the orbits, though unstable, are regular. Under perturbation the unstable orbits can become the locus of chaos. All unstable orbits, whether regular or chaotic, can be quantified by their Lyapunov exponents. The exponents are observationally relevant since the phase of gravitational waves can decohere in a Lyapunov time. If the timescale for dissipation due to gravitational waves is shorter than the Lyapunov time, chaos will be damped and essentially unobservable. We find the timescales can be comparable. We emphasize that the Lyapunov exponents must only be used cautiously for several reasons: they are relative and depend on the coordinate system used, they vary from orbit to orbit, and finally they can be deceptively diluted by transient behaviour for orbits which pass in and out of unstable regions.