This paper studies the quantum behavior of the region between the two horizons of a Reissner–Nordström black hole. The authors use a simplified “minisuperspace” model, which keeps only a few time-dependent geometric variables, and apply affine quantization — a version of quantum rules designed for quantities that must stay positive, such as a radius squared. The main result is that the quantum constraint equation becomes solvable by separation of variables and that the affine method brings extra short-distance terms that alter the small-radius behaviour of the wave function.

What the researchers did: they started from the Reissner–Nordström metric for a charged, spherically symmetric black hole and focused on Region II, the interior slice between the outer and inner horizons where the radial coordinate behaves like time. They reduced the full field problem to a homogeneous minisuperspace model with a positive geometrical variable h (the square of the areal radius) and one other real degree of freedom. Instead of the usual canonical quantization, they applied affine quantization, which replaces the momentum p by the product d = p q for a positive coordinate q. This choice preserves positivity and yields well-defined quantum operators.

How it works at a high level and what they found: the quantum Hamiltonian constraint (the Wheeler–DeWitt equation, the quantum analogue of the classical constraint) separates into two sectors. One sector reduces to a harmonic-oscillator–type problem with Hermite-polynomial eigenmodes. The other, radial sector has Gaussian-like solutions influenced by an effective potential. Affine quantization produces additional ~ħ^2-dependent terms, including a characteristic repulsive contribution proportional to 1/h^2 (or 1/q^2 in general). That extra term modifies the wave function near small radius and helps produce normalizable semiclassical wave packets. The analysis also keeps explicit dependence on the electric charge Q, so the authors can trace how charge affects the quantum dynamics.