Quantum corrected thermodynamics and horizon quantization of the Reissner--Nordstr\"om black hole

Abstract

In this letter, we develop a unified semiclassical framework for the thermodynamics and quantization of the Reissner--Nordstr\"om (RN) black hole (BH) based on the Misner--Sharp--Hernandez (MSH) mass. Treating the quasi-local horizon energies as the relevant thermodynamic variables, we formulate a horizon-by-horizon first law and Smarr relation. Using a reduced phase-space quantization, we obtain a discrete MSH mass spectrum for both horizons, which reproduces the minimal entropy spacing. Quantum transitions between adjacent levels yield Planck-scale corrections to the Hawking temperatures and a universal logarithmic contribution to the entropy, consistent with independent approaches to quantum gravity. We encode these corrections into a quantum-deformed RN geometry via a simple multiplicative factor that preserves the classical horizon positions while reproducing the corrected surface gravities. The associated effective stress tensor behaves as a conserved vacuum-polarization source with characteristic $r^{-4}$ falloff and a small trace, providing a compact representation of semiclassical backreaction. The deformation slightly lowers both horizon temperatures, weakens the inner-horizon instability, and induces tiny shifts in photon-sphere and shadow observables for macroscopic BHs.