Holonomy and inverse-triad corrections in spherical models coupled to matter

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.

Nonsingular spherically symmetric black-hole model with holonomy corrections

“…In vacuum it is well known how to choose this deformation function in order to produce an anomaly-free deformed theory [15,20]. But for dynamical spacetimes coupled to matter with local degrees of freedom, such simple procedure does not provide a closed algebra [15,16], and further modifications are needed.…”

“…There are recent proposals that address some of the mentioned problems [4][5][6][7][8][9][10][11][12] and predict the formation of either an inner horizon or a spacelike transition surface towards a white hole. However, every model so far violates the usual notion of covariance [13,14], particularly when introducing matter fields [15,16]. The use of self-dual variables has been suggested as a possible way out for these no-go results [17] but we will work with real Ashtekar-Barbero variables and, more precisely, with their spherical reduction [18,19].…”

Anomaly-free deformations of spherical general relativity coupled to matter

“…From this perspective, some interior region of the black hole would be Euclidean, and the usual notions of causality and time evolution would not apply. Unfortunately, strong no-go results point out that these models can not be extended to include matter fields with local degrees of freedom [37]. This could mean that the possibility of polymerizing the curvature components is a consequence of an excessive symmetry assumption.…”

An effective model for the quantum Schwarzschild black hole

“…For instance, the extension to the exterior static region, the asymptotic flatness, the slicingindependence and the confinement of quantum effects to large-curvature regions are open issues present in most of the models in the literature. Moreover, none of the mentioned studies addresses explicitly the covariance of the theory [17][18][19][20][21][22]: quantum effects may thus depend on the particular gauge choice and not yield conclusive physical predictions.…”

“…Although a careful choice of the functions allows to define an anomaly-free polymerized Hamiltonian in vacuum, the presence of matter with local degrees of freedom rules out that possibility [17,20,21].…”