Black hole singularity resolution in Wheeler–DeWitt quantum gravity

Ever since Hawking highlighted the puzzle of the existence of a singularity in the classical spacetime of general relativity, implying breakdown of all laws of physics living in the classical spacetime, singularity resolution became a problem of significantly high importance. This undesirable feature, indicating loss of predictability, has intrigued physicists over several decades. It has been hoped that the classical singularity would be resolved in a quantum theory of gravity. However, no appropriate wave function in the vicinity of the black hole singularity has been obtained so far to reach a definite conclusion.

In this paper, we focus upon the interior of the Schwarzschild black hole, plagued by a non-removable classical singularity. We consider the interior spacetime of the black hole to be represented by the Kantowski–Sachs metric. Since in a quantum mechanical scenario, existence of spontaneous fluctuations of matter fields should not be ignored, we include a Klein–Gordon field in the system. Quantizing this simple gravity-matter model in the canonical scheme, we obtain the Wheeler–DeWitt equation and find an exact solution in the minisuperspace variables.

We find that there exist three classes of solutions belonging to three different subregions of the eigenvalue space. Two of these classes of solutions admit the DeWitt criterion, a necessary condition for singularity resolution, implied by vanishing of the wave function at the singularity. These solutions are well-behaved and finite in the vicinity of the singularity and they indicate the existence of regular black holes in quantum gravity. In these classes of solutions, we further find that the expectation value of the Kretschmann operator is well-behaved and regular near the singularity, confirming a definite resolution to the puzzle of classical black hole singularity. On the other hand, there exists a small subregion in the eigenvalue space where the solution does not satisfy both conditions, the DeWitt criterion and finiteness of the Kretschmann expectation value at the singularity. This last class of solutions does not represent regular quantum black holes.