Abstract
Loop quantum gravity introduces two characteristic modifications in the classical constraints of general relativity: the holonomy and inverse-triad corrections. In this paper, a systematic construction of anomaly-free effective constraints encoding such corrections is developed for spherically symmetric spacetimes. The starting point of the analysis is a generic Hamiltonian constraint where free functions of the triad and curvature components as well as non-minimal couplings between geometric and matter degrees of freedom are considered. Then, the requirement of anomaly freedom is imposed in order to obtain a modified Hamiltonian that forms a first-class algebra. In this way, we construct a family of consistent deformations of spherical general relativity, which generalizes previous results in the literature. The discussed derivation is implemented for vacuum as well as for two matter models: dust and scalar field. Nonetheless, only the deformed vacuum model admits free functions of the connection components. Therefore, under the present assumptions, we conclude that holonomy corrections are not allowed in the presence of these matter fields.
Highlights
One of the main aspects a quantum theory of gravity must face is that regarding the singularities of general relativity
We find two main branches: classical divergences when the volume of a region tends to zero are regularized and encoded in the so-called inverse-triad modifications, whereas holonomy corrections are directly related to the spacetime discreteness predicted by the theory
The canonical formalism of general relativity shows that this theory is a completely constrained dynamical system
Summary
One of the main aspects a quantum theory of gravity must face is that regarding the singularities of general relativity. The usual approach to obtain effective equations is to modify the Hamiltonian in such a way that expected quantum effects from loop quantum gravity are included Among these modifications, we find two main branches: classical divergences when the volume of a region tends to zero are regularized and encoded in the so-called inverse-triad modifications, whereas holonomy corrections are directly related to the spacetime discreteness predicted by the theory. Symmetric models are of particular relevance, as they would be a first step to describe black holes and gravitational collapse In this context, it was found that the Abelianization of the constraints could lead to a consistent quantization [30], even under the presence of a scalar matter field [31].
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