The exact dynamical Chern-Simons metric for a spinning black hole possesses a fourth constant of motion: A dynamical-systems-based conjecture

The recent gravitational wave observations by the LIGO/Virgo collaboration have allowed the first tests of general relativity in the extreme gravity regime, when comparable-mass black holes and neutron stars collide. Future space-based detectors, such as the Laser Interferometer Space Antenna, will allow tests of Einstein's theory with gravitational waves emitted when a small black hole falls into a supermassive one in an extreme mass-ratio inspiral. One particular test that is tailor-made for such inspirals is the search for chaos in extreme gravity. We here study whether chaos is present in the motion of test particles around spinning black holes of parity-violating modified gravity, focusing in particular on dynamical Chern-Simons gravity. We develop a resummation strategy that restores all spin terms in the general relativity limit, while retaining up to fifth-order-in-spin terms in the dynamical Chern-Simons corrections to the Kerr metric. We then calculate Poincaré surfaces of section and rotation numbers of a wide family of geodesics of this resummed metric. We find evidence for geodesic chaos, portrayed by thin chaotic layers surrounded by deformed invariant tori. This chaotic layers shrink in size as terms of higher-order in spin are included in the dynamical Chern-Simons corrections to the Kerr metric. Our numerical findings suggest that the geodesics of the as-of-yet unknown exact solution for spinning black holes in this theory may be integrable, and that there may thus exist a fourth integral of motion associated with this exact solution. The studies presented here begin to lay the foundations for chaotic tests of general relativity with the observation of extreme mass ratio inspirals with the Laser Interferometer Space Antenna.