Probing the Interior of the Schwarzschild Black Hole Using Congruences: LQG vs. GUP

Abstract

We review, as well as provide some new results regarding the study of the structure of spacetime and the singularity in the interior of the Schwarzschild black hole in both loop quantum gravity and generalized uncertainty principle approaches, using congruences and their associated expansion scalar and the Raychaudhuri equation. We reaffirm previous results that in loop quantum gravity, in all three major schemes of polymer quantization, the expansion scalar, Raychaudhuri equation and the Kretschmann scalar remain finite everywhere in the interior. In the context of the eneralized uncertainty principle, we show that only two of the four models we study lead to similar results. These two models have the property that their algebra is modified by configuration variables rather than the momenta.

Black hole interior quantization: a minimal uncertainty approach

“…In order to see the fate of the singularity in the quantum regime we compute the expectation value of K with respect to the quantum state (3.10). The classical K in terms of the Ashtekar-Barbero variables for our model is expressed as [41]…”

“…Applying semiclassical techniques to the interior of the Schwarzschild written in terms of Ashtekar-Barbero connection shows that the singularity can be resolved with such techniques [39,40]. However, as a comparative study of the interior of the Schwarzschild black hole shows, although some GUP approaches lead to similar qualitative results as LQG, not all of them actually resolve the singularity [41].…”