Singularity avoidance in black hole interiors by quantum gravity effects

TL;DR

This paper addresses whether quantum gravity can resolve classical singularities inside static black holes by quantizing the Kantowski-Sachs interior with the Wheeler-DeWitt equation. Quantum effects are parameterized by the effective gravitational constant $κ$, enabling a controlled transition from semiclassical to fully quantum regimes. Numerical solutions show that in the semiclassical limit the wave packet follows the classical interior trajectory toward the singularity, but at strong quantum gravity the packet deviates and the computed time to singularity formation grows, signaling singularity avoidance. Using an internal clock based on minisuperspace variables, the authors quantify this delay and report exponential growth of the singularity time with $κ$ near small $κ$, indicating robust behavior across $\Lambda$ and horizon topologies. The results motivate further studies in more general black hole interiors, including charged or rotating cases, and in broader quantum cosmology contexts.

Abstract

The quantum nature of the Schwarzschild black hole interior is investigated through the Wheeler-DeWitt (WDW) equation. The interior of a static, spherically symmetric black hole is described by the Kantowski-Sachs (KS) metric, which represents a homogeneous but anisotropic cosmology. We derive the Hamiltonian for the gravitational system corresponding to the black hole interior and obtain the associated WDW equation. By varying the gravitational constant as a parameter controlling quantum effects, we examine how the solutions of the WDW equation change with respect to this parameter. In the parameter regime where quantum effects are negligible, we find that the wave packet solutions closely follow the classical trajectory of the black hole interior. On the other hand, as quantum effects are enhanced, the wave packet deviates from the classical trajectory and exhibits behavior suggestive of singularity avoidance. To quantify this behavior, we introduce an appropriate "clock" inside the black hole and compute the time to singularity formation with respect to this clock. The results show that stronger quantum effects lead to a longer formation time, suggesting a tendency toward the avoidance of singularity formation due to quantum gravity effects.

Figures (24)

• Figure 1: Schematic picture of the classical solution in the $(U,V)$ plane. The trajectory starts at the horizon $(-r_h, 0)$ and ends at the singularity $(0, \mu)$.
• Figure 2: $\Lambda=-3$ and $\kappa=0.01$
• Figure 3: $\Lambda=-3$ and $\kappa=0.1$
• Figure 4: $\Lambda=-3$ and $\kappa=1$
• Figure 5: $\Lambda=0$ and $\kappa=0.01$