Gravitational collapse in loop quantum gravity

In this paper we study the gravitational collapse in loop quantum gravity. We consider the space-time region inside the Schwarzschild black hole event horizon and we divide this region in two parts, the first one where the matter (dust matter) is localized and the other (outside) where the metric is Kantowski-Sachs type. We calculate the state solving Hamiltonian constraint and we obtain a set of three difference equations that give a regular and natural evolution beyond the classical singularity point in “r = 0” localized. Introduction Quantum gravity, the theory that wants reconcile general relativity and quantum mechanics, is one of major problem in theoretical physics today. The lesson of general relativity is that also the spacetime is dynamical, then it is not possible to study the other interaction on a fixed background. The background itself is a dynamical field. One of more diffuse theory of quantum gravity is the theory called of “loop quantum gravity” [1]. This is one of the non perturbative and background independent approach to quantum gravity (another non perturbative approach to quantum gravity is called ”asymptotic safety quantum gravity” [2]). In this paper we will apply ideas suggested by full loop quantum gravity to a minisuperspace model where we will impose symmetries on the full metric to obtain a reduced model. It is possible to implement the Dirac quantization program following the fundamental ideas of loop quantum gravity. Some interesting results in this theory are related to the problem of space-like singularity. In fact it was shown in [3], [4], [5] and [6] that it is possible to solve the cosmological singularity problem and the black hole singularity problem by using the ideas developed in full loop quantum gravity theory. In [7] the problem of black hole singularity has been analyzed also under the contest of ”asymptotic safety quantum gravity”; in that paper authors showed that non perturbative quantum gravity effects give a much less singular Schwarzschild metric. In this paper we would like to study the gravitational collapse of a dust sphere inside the event horizon [8]. We consider the phase of the gravitational collapse when all (dust) matter has crossed the Schwarzschild radius. In this phase the symmetric reduced metric is homogeneous and we can solve completely the quantum Einstein equations for the minisuperspace model. In particular we have two regions (Region 1 and Region 2), the first one is where the matter is localized (Region 1) with a FriedmannRobertson-Walker type metric and the other one outside the matter (Region 2) where the metric is of Kantowski-Sachs type [9]. We summarize the model in ADM variables following the paper of Nambu and Sasaki [8]. After this we will pass to Ashtekar variables. In Ashtekar variables we quantize the system following ideas suggested from full loop quantum gravity [1]. The technology used in this paper was developed in the preview papers [5], [10], [11], [6] and [13].