Citation: Rastgoo, S.; Das, S. Probing the Interior of the Schwarzschild Black Hole Using Congruences: LQG vs. GUP. Universe 2022, 8, 349. https://doi.org/10.3390/ universe8070349 Academic Editor: Wlodzimierz Piechocki Received: 8 May 2022 Accepted: 22 June 2022 Published: 24 June 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). universe Review Probing the Interior of the Schwarzschild Black Hole Using Congruences: LQG vs. GUP Saeed Rastgoo 1, * and Saurya Das 2 1 Department of Physics and Astronomy, York University, 4700 Keele Street, Toronto, ON M3J 1P3, Canada 2 Theoretical Physics Group and Quantum Alberta, Department of Physics and Astronomy, University of Lethbridge, 4401 University Drive, Lethhbridge, AB T1K 3M4, Canada; saurya.das@uleth.ca * Correspondence: srastgoo@yorku.ca 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. Keywords: quantum gravity; quantum black hole; loop quantum gravity; generalized uncertainty principle; singularity resolution; Raychaudhuri equation 1. Introduction Black holes are one the most important objects in the Universe with regards to quan- tum gravity. The singularity in their interior is a prediction of general relativity (GR), which in turn is a prediction of its eventual breakdown. Furthermore, it is believed that this singularity resides in a small spatial region where quantum effects cannot be neglected. Thus, one has the natural expectation that a final theory of quantum gravity should be able to resolve this singularity. Various theories of quantum gravity or effective gravity have been utilized to study such objects. Among these are loop quantum gravity (LQG) [1], a nonperturbative canonical theory of quantization of the gravitational field, and the general- ized uncertainty principle (GUP), which is a rather phenomenological approach resulting from the assumption of noncommutativity of spacetime or existence of a minimum length. In LQG, there have been numerous works studying both the interior and the full spacetime of the Schwarzschild black hole [2–41]. In particular, the interior of such a black hole has been studied in various ways. One of the most common approaches uses the so called polymer quantization [42–46], which was originally inspired by loop quantum cosmology (LQC), dealing with a certain quantization of the isotropic Friedmann-Lemaitre- Robertson-Walker (FLRW) model [47,48]. Since the interior of the Schwarzschild black hole is isometric to the Kantowski-Sachs cosmological model, the idea in this polymer approach is to apply the same techniques of the polymer quantization of the Kantowski-Sachs model to the Schwarzschild interior [49,50]. Polymer quantization introduces a parameter in the theory called the polymer scale, that sets the minimal scale of the model close to which the quantum gravity effects become important. The approach in which such a parameter is taken to be constant is called the µ 0 scheme (which in this paper we refer to as the ˚ µ scheme), while approaches where it depends on the phase space variables are denoted by ¯ µ schemes. The various approaches were introduced to deal with some important issues resulting from quantization, namely, to have the correct classical limit (particularly in LQC), Universe 2022, 8, 349. https://doi.org/10.3390/universe8070349 https://www.mdpi.com/journal/universe