Embedding generalized Lemaître-Tolman-Bondi models in polymerized spherically symmetric spacetimes

We generalize the existing works on the way (generalized) Lema\^{\i}tre-Tolman-Bondi (LTB) models can be embedded into polymerized spherically symmetric models in several aspects. We reexamine such an embedding at the classical level and show that a suitable LTB condition can only be treated as a gauge fixing in the nonmarginally bound case, while in the marginally bound case, it must be considered as an additional first class constraint. A novel aspect of our formalism, based on the effective equations of motion, is to derive compatible dynamics LTB conditions for polymerized models by using holonomy and inverse triad corrections simultaneously, whereas in earlier work, these were only considered separately. Further, our formalism allows one to derive compatible LTB conditions for a vast of class of polymerized models available in the current literature. Within this broader class of polymerizations, there are effective models contained for which the classical LTB condition is a compatible one. Our results show that there exist a class of effective models for which the dynamics decouples completely along the radial direction. It turns out that this subsector is strongly linked to the property that in the temporally gauge fixed model, the algebra of the geometric contribution to the Hamiltonian constraint and the spatial diffeomorphism constraint is closed. We finally apply the formalism to existing models from the literature and compare our results to the existing ones.