Nonlinear field dynamics and spacetime self-organization in Kerr geometry

We develop a unified, field-theoretic framework for coherent dynamics in the near-horizon environment of rotating (Kerr) black holes that bridges general-relativistic geometry, nonequilibrium phase transitions, and multimessenger diagnostics. Starting from a covariant effective action with a Landau–Ginzburg potential for a complex order parameter A coupled to a slow reservoir X (curvature/density proxy), we identify a critical ridge X ₂ that organizes metastability and defines the natural control parameter =X-X ₂. Linearization in a corotating Kerr patch yields a dispersion relation in which Doppler shifting and Lense–Thirring precession enter explicitly, while coupling to X reshapes the unstable band and links superradiant/corotation amplification to pattern-forming thresholds. Near onset, a multiple-scale reduction leads to a complex Ginzburg–Landau (TDGL/CGL) envelope governing nonlinear self-organization. The resulting states exhibit quantized defects and rotating spiral morphologies whose core precession and arm-pass motion set a pattern clock. Energy and angular momentum cycle quasi-periodically between A and X, producing windowed anti-correlations with an entropy-like coherence measure and establishing scale links via core radius and interface width. GW-like and EM-like proxies display a common dominant peak with sidebands and finite coherence, with frequencies and bandwidths that track the Kerr spin a_ and the unified coupling strength. Together, these results provide a minimal, predictive backbone for interpreting horizon-scale morphology, energy transport, and partially coherent GW/EM spectra around rotating black holes.