Schwinger instability, modular flow, and holographic entropy for near-extremal charged BTZ black hole

We investigate the quantum dynamics of a charged scalar field in the near-horizon region of a near-extremal charged BTZ black hole. A controlled expansion of the Einstein-Maxwell equations reveals an emergent warped AdS$_2 \times S^1$ throat geometry threaded by a constant electric field--an ideal setting for studying Schwinger pair production, Hawking radiation, and entropy flow. By solving the Klein-Gordon equation using both tunneling and field-theoretic methods, we compute the pair production rate and identify an effective Unruh-like temperature. In particular, we apply the WKB approximation for Hawking tunneling, justified by the infinite blueshift experienced by outgoing modes near the horizon. Instability arises when local acceleration exceeds the AdS curvature scale, linking near-horizon dynamics to thermal emission. Through the generalized uncertainty principle, which implies the existence of a minimal length, we argue that quantum gravity effects can drive the Hawking and Schwinger-like temperatures to zero. To connect quantum radiation to geometry, we analyze the flux of the modular Hamiltonian across the horizon and show that its variation matches the entropic contribution of the produced pairs. Using Tomita-Takesaki theory and the type II\_\infty von Neumann algebra of horizon observables, we derive a semiclassical gravitational constraint involving the second variation of the stress tensor, recovering the null-null component of the Einstein equations from entropy extremization. The intersection point of inward and outward RT geodesics marks both the peak of pair production and the vanishing of entropy variation, revealing a geometric alignment between entanglement, quantum matter, and backreaction. The near-horizon geometry does not merely support quantum effects--it organizes them.