On consistent gauge fixing conditions in polymerized gravitational systems

TL;DR

This paper investigates whether gauge fixing conditions in polymerized gravitational systems remain consistent under effective dynamics, revealing that such consistency is rare and highlighting potential pitfalls in common approaches.

Contribution

The study derives conditions for the commutativity of gauge fixing and polymerization, showing it occurs only in special cases and identifying inconsistencies in previous literature.

Findings

Gauge fixing and polymerization do not generally commute.

Inconsistencies arise from common choices in the literature.

Examples include Schwarzschild interior and LTB spacetimes.

Abstract

For classical gravitational systems the lapse function and the shift vector are usually determined by imposing appropriate gauge fixing conditions and then demanding their preservation with respect to the dynamics generated by a canonical Hamiltonian. Effective descriptions encoding quantum geometric effects motivated by loop quantum gravity for symmetry reduced models are often captured by polymerization of connection (or related) variables in gauge fixing conditions as well as constraints. Usually, one chooses the same form of polymerization in both cases. A pertinent question is if the dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the polymerized canonical Hamiltonian is provided by the lapse function and the shift vector obtained from the polymerization of their classical counterparts. If this is the case, then we say that gauge fixing and polymerization commute. In this manuscript we investigate these issues and obtain consistency conditions for the commutativity of gauge fixing and polymerization. Our analysis shows that such a commutativity occurs in rather special situations and reveals pitfalls in making seemingly well motivated choices which turn out to be inconsistent with the effective dynamics. We illustrate these findings via examples of symmetry reduced models in the loop quantization of the Schwarzschild interior and Lema\^{\i}tre-Tolman-Bondi (LTB) spacetimes and report the non-commutativity of gauge fixing and polymerization, and inherent limitations of some choices made in the literature with a consistent effective dynamics.