  Full general relativity requires that chaos indicators should be invariant in
various spacetime coordinate systems for a given relativistic dynamical
problem. On the basis of this point, we calculate the invariant Lyapunov
exponents (LEs) for one of the spinning compact binaries in the conservative
second post-Newtonian (2PN) Lagrangian formulation without the dissipative
effects of gravitational radiation, using the two-nearby-orbits method with
projection operations and with coordinate time as an independent variable. It
is found that the actual source leading to zero LEs in one paper [J. D.
Schnittman and F. A. Rasio, Phys. Rev. Lett. 87, 121101 (2001)] but to positive
LEs in the other [N. J. Cornish and J. Levin, Phys. Rev. Lett. 89, 179001
(2002)] does not mainly depend on rescaling, but is due to two slightly
different treatments of the LEs. It takes much more CPU time to obtain the
stabilizing limit values as reliable values of LEs for the former than to get
the slopes (equal to LEs) of the fit lines for the latter. Due to coalescence
of some of the black holes, the LEs from the former are not an adaptive
indicator of chaos for comparable mass compact binaries. In this case, the
invariant fast Lyapunov indicator (FLI) of two-nearby orbits, as a very
sensitive tool to distinguish chaos from order, is worth recommending. As a
result, we do again find chaos in the 2PN approximation through different
ratios of FLIs varying with time. Chaos cannot indeed be ruled out in real
binaries.
