Adiabatic Anisotropic Gravitational Collapse in Painlev\'e-Gullstrand Coordinates: A Geometric Analysis
We present a detailed geometric analysis of adiabatic, anisotropic gravitational collapse formulated in a single Painlev\'e-Gullstrand coordinate system that covers both the interior and exterior, thereby eliminating cross-chart matching artifacts. Building on the Oppenheimer-Snyder framework with a phenomenologically motivated energy-density profile, we enforce the Israel junction conditions and obtain closed-form surface evolution. Within this unified chart we derive exact solutions for the complete collapse process, characterize the causal structure, and track horizon formation and evolution. In particular, we identify and analyse a double apparent-horizon phase inside the matter and show that the event horizon stabilizes at the Schwarzschild radius. We further obtain critical parameter relations that govern the dynamics, including a threshold linking initial compactness to immediate horizon formation. The model is geometrically self-consistent within Einstein's equations but exhibits violations of the standard point-wise energy conditions, highlighting known limitations of idealized anisotropic matter models and delineating the boundary where classical descriptions become inadequate. Together, these results provide geometric insights, compact analytic benchmarks and a didactic, coordinate-uniform perspective on collapse and horizon dynamics.