  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.
