Regular black holes and their relationship to polymerized models and mimetic gravity

We present further applications of the formalism introduced by the authors in arXiv:2308.10949, which allows embedding of a broad class of generalized LTB models into effective spherically symmetric spacetimes. We focus on regular black hole models, where a broad class of models can be considered, including for example LQG-inspired models as well as the model with a regular center, e.g. of Bardeen and Hayward. For a certain class of regular black hole models, we can formulate a Birkhoff-like theorem in LTB coordinates. We further show that depending on the properties of the polymerization functions characterizing such regular black hole models in this formalism, the uniqueness of the effective spherically symmetric vacuum solutions might not be given in general in Schwarzschild-like coordinates. Furthermore, we introduce a reconstruction algorithm that allows for a subclass of this models to construct from a given metric in Schwarzschild-like coordinates the corresponding effective spherically symmetric model, its dynamics as an 1+1-dimensional field theory as well as a corresponding covariant Lagrangian of extended mimetic gravity in four dimensions. Such a reconstruction allows us to obtain Lagrangians of extended mimetic gravity models for black holes with a regular center, e.g. the Bardeen and Hayward metric as well as for effective LQG inspired models. Moreover, the reconstruction enables us to extend regular black hole models to general inhomogeneous dust collapse models. For the latter, within this formalism, we can investigate and look at the physical properties of the models such as the existence of weak shell-crossing singularities from a novel perspective.