Gauge-Controlled Breakdown of Kerr Integrability in Weakly Non-Kerr Spacetimes

The Kerr geometry is distinguished not only by stationarity and axisymmetry, but also by a hidden tensorial structure responsible for the Carter constant and the complete integrability of geodesic motion. This talk presents a perturbative and gauge-controlled framework for studying how these manifest and how hidden symmetries break down in weakly non-Kerr spacetimes. Instead of starting from a particular phenomenological deformation, the analysis is formulated directly at the level of symmetry restoration: whether the stationary and axial Killing vectors, the Killing-Yano tensor, and the associated rank-two Killing tensor can be consistently continued under a small metric perturbation.

The central result is a hierarchy of symmetry obstruction. Manifest symmetry breaking controls the drift of the energy and axial angular momentum, while hidden-sector obstruction diagnoses the possible failure of the Carter-type invariant even when stationarity and axisymmetry remain intact. This distinction is essential for relativistic orbital dynamics, because a perturbation may preserve the usual conserved charges but still destroy the separability structure that organizes Kerr geodesics.

I will discuss how these obstruction tensors can be projected onto relativistic orbits and used to classify weak environmental deviations from Kerr, including stationary–axisymmetric perturbations that isolate the hidden sector. The resulting framework provides a geometrically transparent route toward understanding frequency-map drift, resonant structures, and long-time phase accumulation in extreme-mass-ratio inspirals. It therefore connects black-hole perturbation theory, hidden symmetry, and future precision tests of the Kerr paradigm via gravitational-wave observations.