Spherically Symmetric Loop Quantum Gravity: Connections to Two-Dimensional Models and Applications to Gravitational Collapse

We review the spherically symmetric sector of General Relativity and its midisuperspace quantization using Loop Quantum Gravity techniques. We exhibit anomaly-free deformations of the classical first class constraint algebra. These consistent deformations incorporate corrections presumably arising from a loop quantization and accord with the intuition suggesting that not just dynamics but also the very concept of spacetime manifolds changes in quantum gravity. Our deformations serve as the basis for a phenomenological approach to investigate geometrical and physical effects of possible corrections to classical equations. In the first part of this work we couple the symmetry reduced classical action to Yang-Mills theory in two dimensions and discuss its relation to dilaton gravity and the more general class of Poisson sigma models. We show that quantum corrections for inverse triad components give a consistent deformation without anomalies. The relation to Poisson sigma models provides a covariant action principle of the quantum corrected theory with effective couplings. We also use our results to provide loop quantizations of spherically symmetric models in arbitrary D space-time dimensions. In the second part, we turn to Lemaitre-Tolman-Bondi models of spherical dust collapse and study implications of inverse triad quantum corrections, particularly for potential singularity resolution. We consider the whole class of LTB models, including nonmarginal ones, and as opposed to the previous strategy in the literature where LTB conditions are implemented first and anomaly-freedom is used to derive consistent equations of motion, we apply our procedure to derive anomaly-free models which first implements anomaly-freedom in spherical symmetry and then the LTB conditions. While the two methods give slightly different equations of motion, which may be expected given the ubiquitous sprawl of quantization ambiguities, conclusions are the same in both cases: Bouncing solutions for effective geometries, as a mechanism for singularity resolution, seem to appear less easily in inhomogeneous situations as compared to quantizations of homogeneous models, and even the existence of homogeneous solutions as special cases in inhomogeneous models may be precluded by quantum effects.