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

TL;DR

This paper develops a systematic method to construct anomaly-free effective constraints with holonomy and inverse-triad corrections in spherically symmetric models of loop quantum gravity, analyzing their compatibility with matter fields.

Contribution

It introduces a general framework for anomaly-free deformations of spherical general relativity with matter, extending previous work and clarifying the conditions for holonomy corrections.

Findings

Holonomy corrections are not compatible with certain matter fields.

A family of consistent deformations of spherical GR is constructed.

The approach applies to vacuum, dust, and scalar field models.

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.