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

Abstract

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.

Prospects for fundamental physics with LISA

“…While in GR the geodesic motion of a test particle around a Kerr black hole is fully integrable, in some theories the orbital motion might even be chaotic [108], which would also impact on the structure of the QNMs [109]. The presence of chaos in EMRIs is expected to be encoded in chaotic plateaus in the temporal evolution of the fundamental frequencies of the motion [110][111][112][113][114].…”

Eikonal quasinormal modes of black holes beyond general relativity. II. Generalized scalar-tensor perturbations

“…These orbits/modes appear provided the spacetime is separable with a Carter-like constant; in the case of Kerr black holes, they correspond to the well-known 'nonequatorial' |m| < ℓ modes. Similar QNMs could characterize the black holes of dynamical Chern-Simons gravity where geodesic motion has been conjectured to be endowed with a higher-than-two rank Killing tensor [45].…”

Chaotic photon orbits and shadows of a non-Kerr object described by the Hartle-Thorne spacetime

“…This number characterises the frequency structure of the phase space for each trajectory [27]. In this paper, we use a variation of the definition (15), that will indicate to us if an orbit is indeed sticky.…”

“…In Fig. (27), we present the shadow of the object along with the shadow of a Kerr black hole with the same spin. The resolution is 90000 pixels.…”