Chaos and Order in Models of Black Hole Pairs

TL;DR

This paper investigates chaos in black hole binary models, highlighting discrepancies among different approximations and emphasizing the need for more reliable methods to understand chaos in highly nonlinear regimes.

Contribution

It compares various approximations of black hole pairs, revealing inconsistent predictions of chaos and stressing the importance of improved models for nonlinear dynamics.

Findings

Chaos appears in some approximations but not others.

Current models are inconsistent in predicting chaos for spinning black hole binaries.

Better approximations are needed for conclusive results in nonlinear regimes.

Abstract

Chaos in the orbits of black hole pairs has by now been confirmed by several independent groups. While the chaotic behavior of binary black hole orbits is no longer argued, it remains difficult to quantify the importance of chaos to the evolutionary dynamics of a pair of comparable mass black holes. None of our existing approximations are robust enough to offer convincing quantitative conclusions in the most highly nonlinear regime. It is intriguing to note that in three different approximations to a black hole pair built of a spinning black hole and a non-spinning companion, two approximations exhibit chaos and one approximation does not. The fully relativistic scenario of a spinning test-mass around a Schwarzschild black hole shows chaos, as does the Post-Newtonian Lagrangian approximation. However, the approximately equivalent Post-Newtonian Hamiltonian approximation does not show chaos when only one body spins. It is well known in dynamical systems theory that one system can be regular while an approximately related system is chaotic, so there is no formal conflict. However,the physical question remains, Is there chaos for comparable mass binaries when only one object spins? We are unable to answer this question given the poor convergence of the Post-Newtonian approximation to the fully relativistic system. A resolution awaits better approximations that can be trusted in the highly nonlinear regime.