Quantum gravitational corrections to a Kerr black hole using Topos theory

We examine non-perturbative quantum gravitational corrections to a Kerr black hole using Topos theory. Black hole thermodynamics is modeled using sheaves, and these corrections are represented as additional morphisms on these sheaves. These corrections are thus expressed as intrinsic structural modifications modeled by these new morphisms, allowing both the original and quantum gravitationally corrected structures to exist within the same Topos. We also construct a functor between thermodynamics and information theory by using the Parikh–Wilczek formalism to represent the probability density of emitted particles as sheaves. Then, using the Kullback–Leibler divergence as another functor, we measure deviations of the quantum gravitationally corrected sheaves from the original sheaves. To obtain information about the modifications produced by quantum gravity, we construct an additional functor representing Fisher information. Topos theory provides a formalism that allowing us to study properties of the system in classical spacetime as well as cases where the classical spacetime breaks down near the Planck scale. This formulation reveals a novel information paradox in the ultraviolet regime of quantum gravity, where not only does information about particle states become inaccessible, but even information about the quantum gravitational modifications themselves becomes computationally unobtainable.