Giesel K, Li BF, Singh P, Weigl S (2022)
Publication Language: English
Publication Type: Journal article, Original article
Publication year: 2022
Book Volume: 105
Pages Range: 066023
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 Lemaître-Tolman-Bondi spacetimes and report the noncommutativity of gauge fixing and polymerization and inherent limitations of some choices made in the literature with a consistent effective dynamics.
Giesel, K., Li, B.-F., Singh, P., & Weigl, S. (2022). Consistent gauge-fixing conditions in polymerized gravitational systems. Physical Review D - Particles, Fields, Gravitation and Cosmology, 105, 066023. https://dx.doi.org/10.1103/PhysRevD.105.066023
Giesel, Kristina, et al. "Consistent gauge-fixing conditions in polymerized gravitational systems." Physical Review D - Particles, Fields, Gravitation and Cosmology 105 (2022): 066023.