Quantum-Gravitational Backreaction in BTZ via Curved Momentum Space

We study how quantum features of spacetime, specifically the curvature of momentum space, can backreact on classical gravity in a semiclassical (2+1)D setting with a negative cosmological constant. Starting from a first-order action, we derive an effective configuration-space action, revealing mass-dependent geodesics and a mild violation of the equivalence principle. Coupling this modified matter source to Einstein gravity, we obtain a corrected BTZ black hole, with ADM mass, Hawking temperature, and entropy acquiring explicit corrections and upper bounds dictated by the momentum-space geometry. We also compute the return time of a massless particle along null geodesics, illustrating observable signatures of underlying quantum-spacetime structures. Our results provide a concrete semiclassical framework showing how Planck-scale kinematic modifications leave imprints on classical geometry.