Rotating Black Holes in the U-Cell Model: The Kerr Metric from the Doran Substrate Flow

The Schwarzschild metric arises in the U-Cell Model (UCM) from a purely radial substrate flow. We extend this to rotating black holes by showing that the Kerr metric is reproduced exactly by the Gullstrand–Painlevé framework with the Doran flow — a substrate velocity field with both a radial inflow vʳ = -c√ (rS r/Σ) and an azimuthal swirl v^φ = - (ac/ϱ²) √ (rS r/Σ). The GP metric with this flow is algebraically identical to the Kerr metric in Doran coordinates (exact, not an approximation). The Doran flow is introduced as a kinematically motivated ansatz whose uniqueness follows from the Carter–Robinson theorem: conditional on the EFE holding for non-irrotational substrate flows, stationarity, axisymmetry, asymptotic flatness, and vacuum uniquely fix the Kerr metric (Carter 1971, Robinson 1975), and the Doran flow is its natural Gullstrand–Painlevé gauge representation. Frame dragging is substrate swirl: the azimuthal flow reproduces the Lense–Thirring precession ΩLT = GJ/ (c²r³) confirmed by Gravity Probe B. Stability for |a| < M is inherited from the Klainerman–Szeftel theorem (2021–2023). Three open problems are stated: extreme Kerr (|a|=M), the exact Helmholtz decomposition for a≠0, and the Ricci tensor for rotational flow.