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Full general relativity requires that chaos indicators should be invariant invarious spacetime coordinate systems for a given relativistic dynamicalproblem. On the basis of this point, we calculate the invariant Lyapunovexponents (LEs) for one of the spinning compact binaries in the conservativesecond post-Newtonian (2PN) Lagrangian formulation without the dissipativeeffects of gravitational radiation, using the two-nearby-orbits method withprojection operations and with coordinate time as an independent variable. Itis 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 positiveLEs 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 slightlydifferent treatments of the LEs. It takes much more CPU time to obtain thestabilizing limit values as reliable values of LEs for the former than to getthe slopes (equal to LEs) of the fit lines for the latter. Due to coalescenceof some of the black holes, the LEs from the former are not an adaptiveindicator of chaos for comparable mass compact binaries. In this case, theinvariant fast Lyapunov indicator (FLI) of two-nearby orbits, as a verysensitive tool to distinguish chaos from order, is worth recommending. As aresult, we do again find chaos in the 2PN approximation through differentratios of FLIs varying with time. Chaos cannot indeed be ruled out in realbinaries.

Revisit on "Ruling out chaos in compact binary systems"