Quadrupolar tidal effects destroy the integrability of black hole geodesics: analytic proof and numerical evidence of chaos

Abstract

In general relativity, the motion of a test mass around a rotating black hole is described by Kerr geodesics. Owing to the symmetries of the Kerr spacetime, these geodesics possess four constants of motion, rendering the associated Hamiltonian system integrable. This integrability underlies much of the analytical framework used to model asymmetric-mass-ratio inspirals, key sources for future gravitational-wave detectors. Real compact bodies, however, are not test masses: their internal structure couples to the background curvature. In this work, we show that a non-spinning body endowed with a tidally induced quadrupole admits no deformation of the geodesic Carter constant that remains conserved, for generic tidal couplings and generic Kerr spin. Consequently, the leading-order tidal dynamics is generically non-integrable. The proof is analytic and relies on two key ingredients: a covariant Hamiltonian formulation of tidal dynamics on the same phase space as the geodesic problem, valid in arbitrary background spacetimes, and a novel relation between curvature tidal scalars and the geodesic Carter constant derived from the algebraic and Killing symmetries of Kerr spacetime. We complement this result with numerical diagnostics of the tidally perturbed dynamics, including Poincar\'e sections, Lyapunov exponents, and escape-time maps. These reveal chaotic structures in phase space, such as stochastic layers, sensitivity to initial conditions, and fractal basin boundaries, consistently with the analytic non-integrability result.