Dynamics of information content in quantum black holes

We investigate the evolution of information content in black holes using a quantum probabilistic entropy within canonical quantum gravity. Employing the Wheeler-DeWitt framework with a Kantowski-Sachs metric for the black hole interior and a minimally coupled Klein-Gordon field, we derive and solve the Wheeler-DeWitt equation in the minisuperspace. The resulting wave function yields a quantum probabilistic entropy that quantifies the information content of the black hole. A challenge in studying its evolution is the absence of an explicit time parameter, reflecting the problem of time in quantum gravity. To address this, we use the Schwarzschild radius as an effective clock, utilizing its monotonic decrease during Hawking evaporation. The resulting entropy evolution exhibits a curve remarkably similar to the Page curve. Although quantum probabilistic entropy is distinct from von Neumann entropy, this behavior suggests that effective notions of time and information evolution can emerge naturally from non-perturbative quantum gravitational dynamics.