Quantum induced shock dynamics in gravitational collapse: insights from effective models and numerical frameworks

We explore the formation and evolution of shock waves in spherically symmetric gravitational collapse within a Loop Quantum Gravity (LQG) inspired effective framework. In this setting, the classical singularities are replaced by quantum-induced shell-crossing singularities, which are resolved through weak solutions such as shock waves. By formulating the dynamics in a generalized Painlev\'e--Gullstrand coordinate system, we derive a first-order partial differential equation that governs the propagation of the shock surface, while enforcing metric continuity via thin-shell junction conditions. To handle the non-trivial square-root structures and source terms that arise in these equations, we develop a novel numerical scheme capable of simulating quantum-corrected spacetime dynamics. Our results show that for small mass black holes near the Planck scale, the shock surface remains timelike and is shielded behind both inner and outer horizons. In the long-time limit, the shock accumulates the entire mass of the collapsing star. In contrast, for larger black hole masses, the shock surface develops spacelike segments, indicating a transition in the effective dynamics driven by quantum effects. The framework also reveals discontinuities in curvature invariants across the shock surface, which can be traced back to stress-energy redistributions caused by quantum effects. Overall, the proposed computational framework provides a general tool for modeling quantum-corrected gravitational collapse and offers new insights into black hole formations, singularity resolution, and the interplay between quantum geometry effects and effective spacetime structures.