Speaker
Description
We investigate the quantum behavior of the Kretschmann curvature invariant $\mathcal{K}$ in the deep quantum regime near black hole singularities, where the classical expression diverges as $r^{-6}$ as $r \to 0$. Our goal is to understand whether this singular behavior persists under a quantum mechanical treatment. Due to the involvement of both first- and second-order time derivatives of metric variables in $\mathcal{K}$, formulating a corresponding quantum operator $\hat{\mathcal{K}}$ poses significant challenges. To address this, we consider the Wheeler-DeWitt equation within a Kantowski–Sachs minisuperspace framework in the presence of quantum vacuum fluctuations modeled by a Klein–Gordon field. The separable structure of the equation allows for a Bohmian quantum mechanical interpretation, enabling the derivation of quantum potentials and a procedure to represent second-order time derivatives as quantum operators. With the resulting Kretschmann operator $\hat{\mathcal{K}}$, we compute its expectation value $\langle \hat{\mathcal{K}} \rangle$ using regular wave function solutions (Class I and II) of the Wheeler–DeWitt equation. Our analysis reveals that $\langle \hat{\mathcal{K}} \rangle$ remains finite in certain quantum subregions, provided the wave function eigenvalues are subject to stricter constraints than those originally defining the solution classes. These findings suggest a potential resolution of curvature singularities within a quantum gravitational framework.