Relational quantum dynamics of the black hole interior: singularity resolution and quantum bounce
We study the interior of the Schwarzschild black hole which is isometric to the Kantowski-Sachs cosmological model, using a fully relational and gauge-invariant quantization framework. The physical Hilbert space is constructed via refined algebraic quantization, and quantum dynamics is recovered through the Page-Wootters formalism with a covariant POVM clock built from one of the two configuration variables, whose Hamiltonian is proportional to the momentum of the said variable. Gauge-invariant relational observables for the area of 2-spheres, the Kretschmann scalar, and the expansion scalar of null geodesic are constructed via group averaging (G-twirl) and evaluated on physical states. We find that the Kretschmann and expansion scalars remain finite throughout the black hole, while the area of 2-spheres is bounded below by a minimum value proportional to the uncertainty in the system variable, which is the other configuration variable distinct from the clock variable. In particular, the expansion scalar vanishes and changes sign at the quantum bounce, establishing a black-hole-to-white-hole transition. These results hold for any general clock whose operator forms a canonical pair with the clock Hamiltonian, and require no specific quantization scheme other than the Schrodinger representation. The singularity resolution emerges directly from relationality, the Heisenberg uncertainty principle, and the structure of the physical Hilbert space.