Quantum Dynamics of the Schwarzschild Interior in Ashtekar-Barbero Variables with Minimal Length Effects

Abstract

We study the quantum dynamics of the Schwarzschild interior in the Ashtekar-Barbero formulation, focusing on the fate of the classical singularity and the annihilation-to-nothing scenario. Using minisuperspace Wheeler-DeWitt quantization, we first analyze the standard Schrödinger representation and show that the annihilation-to-nothing behavior appears only for a specific choice of factor ordering and is not generic. We then introduce a generalized uncertainty principle (GUP), which induces minimal-length effects through a deformation of the canonical algebra. Solving the modified Wheeler-DeWitt equation and constructing Gaussian wave packets localized at the horizon, we find that the annihilation-to-nothing behavior is suppressed once the GUP corrections are included. Our results indicate that minimal-length effects qualitatively alter the quantum interior dynamics and challenge the robustness of this scenario as a mechanism for singularity resolution.

Figures (6)

• Figure 1: Probability density of the wave function for $C=\sigma=r_s=1$. The integral over the wave number $k$ is taken over the interval $[-8,8]$. Adapted from Ref. Bouhmadi-Lopez:2019kkt.
• Figure 2: Probability density of the wave function for $C=\sigma=\gamma'=1$, $a=1$. The integral over the wave number $k$ is taken over the interval $[-8,8]$.
• Figure 3: Probability density of the wave function for $C=\sigma=\gamma'=1$, $a=2$. The integral over the wave number $k$ is taken over the interval $[-8,8]$.
• Figure 4: Probability density of the wave function for $a=\frac{5}{6},\gamma=\frac{r_s}{2}=2\beta_b=\beta_c=1$.
• Figure 5: Probability density of the wave function for $a=1,\gamma=\frac{r_s}{2}=2\beta_b=\beta_c=1$.