Status of Birkhoff's theorem in polymerized semiclassical regime of Loop Quantum Gravity

TL;DR

This paper investigates the status of Birkhoff's theorem within the semiclassical regime of Loop Quantum Gravity, analyzing solutions to polymerized Einstein equations for dust collapse and their implications for spherical symmetry.

Contribution

It provides a detailed analysis of polymerized Einstein equations in LQG, exploring conditions under which Birkhoff's theorem holds semiclassically and identifying unique and potential time-dependent solutions.

Findings

Existence of a unique, parameter-dependent class of static solutions.

Potential for more complex, time-dependent solutions not fully ruled out.

Comparison with classical spherical collapse models.

Abstract

The collapse of a spherically symmetric ball of dust has been intensively studied in Loop Quantum Gravity (LQG). From a quantum theory, it is possible to recover a semiclassical regime through a polymerization procedure. In this setting, general solutions to the polymerized Einstein field equations (PEFE) will be discussed both for the interior and the exterior of the dust cloud. Exterior solutions are particularly interesting since they may lead to a semiclassical version of the Birkhoff's theorem. It is seen that if time independence of the vacuum is imposed, there exists a unique class of solutions depending on two parameters. Nevertheless, the possibility of more intricate time dependent solutions is not ruled out completely. Ultimately, these results will be compared to a model of spherical collapse obtained independently from the Einstein equations.