Chaos and order in models of black hole pairs

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 nonspinning 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.

Figure 1

Basin boundaries in the 2PN-Hamiltonian system with spin-orbit couplings. The pair has mass ratio $m_2/m_1=1/3$. The heavier black hole is maximally spinning ($S_1=m_1^2$) with an initial angle with respect to the $\widehat{z}$-axis of 95° while the lighter companion is not spinning ($S_2=0$). The initial center of mass separation in ADM coordinates is $r_i/M=5$. The orbital initial conditions vary along the $x$-axis from $0.02\le p_r\le 0.05$ and along the $y$-axis from $0.1195\le p_{\varphi }\le 0.1200$. $200\times 200$ orbits are shown. Initial conditions that are color-coded white correspond to stable orbits and those color-coded black correspond to merging pairs. The basin boundary is clearly smooth and not fractal. Therefore, there is no evidence of chaos.

Figure 2

Left: An orbit taken from the stable basin of Fig. 1. The initial conditions for the orbit are $p_r=0.03$ and $p_{\varphi }=0.20645$. Right: A projection of the phase space motion onto the $(r,p_r)$ plane. The topology of the projection confirms that the motion clearly lies on a torus and therefore is not chaotic.

Figure 3

The pair has mass ratio $m_2/m_1=1/3$. The initial center of mass separation in harmonic coordinates is $5M$. Using the notation $\mathbf{x}=(x^{(1)},x^{(2)},x^{(3)})$, the orbital initial conditions vary along the $x$-axis from $0\le {\dot{x}}^{(1)}\le 0.035$ and along the $y$-axis from $0.425\le {\dot{x}}^{(2)}\le 0.443125$. $200\times 200$ orbits are shown. Initial conditions that are color-coded white correspond to stable orbits and those color-coded black correspond to merging pairs. The basin boundary is fractal indicating at least a thin region of chaos. Left: A reproduction of Fig. 4 from Ref. 3. The heavier black hole is maximally spinning ($S_1=m_1^2$) with an initial angle with respect to the $\widehat{z}$-axis of $95°$ while the lighter companion is not spinning ($S_2=0$). Right: The heavier black hole is not spinning ($S_1=0$) while the lighter object is maximally spinning ($S_2=m_2^2$) with an initial angle with respect to the $\widehat{z}$-axis of 95°.

Figure 4

Left: An orbit taken from the stable basin of Fig. 3. The initial conditions for the orbit are ${\dot{x}}^{(1)}=0.00105$ and ${\dot{x}}^{(2)}=0.43074$. Right: A detail of the projection of the motion in phase space onto the radial $(x,\dot{x})$ plane, where again $x=\sqrt{\mathbf{x}·\mathbf{x}}$.