Black hole interior quantization: a minimal uncertainty approach

Abstract

In a previous work we studied the interior of the Schwarzschild black hole implementing an effective minimal length, by applying a modification to the Poisson brackets of the theory. In this work we perform a proper quantization of such a system. Specifically, we quantize the interior of the Schwarzschild black hole in two ways: once by using the standard quantum theory, and once by following a minimal uncertainty approach. Then, we compare the obtained results from the two approaches. We show that, as expected, the wave function in the standard approach diverges in the region where classical singularity is located and the expectation value of the Kretschmann scalar also blows up on this state in that region. On the other hand, by following a minimal uncertainty quantization approach, we obtain 5 new and important results as follows. 1) All the interior states remain well-defined and square-integrable. 2) The expectation value of the Kretschmann scalar on the states remains finite over the whole interior region, particularly where used to be the classical singularity, therefore signaling the resolution of the black hole singularity. 3) A new quantum number is found which plays a crucial role in determining the convergence of the norm of states, as well as the convergence and finiteness of the expectation value of the Kretschmann scalar. 4) A minimum for the radius of the (2-spheres in the) black holes is found 5) By demanding square-integrability of states in the whole interior region, an exact relation between the Barbero-Immirzi parameter and the minimal uncertainty scale is found.