Regular ultracompact objects with anti-de Sitter cores as polymerized vacuum solutions

Abstract

We present a systematic derivation of regular black hole solutions -- and their horizonless counterparts -- that achieve regularization via an anti-de Sitter core. These geometries emerge as polymerized vacuum solutions inspired by loop quantum gravity, constituting effective quantum gravity configurations that admit a Birkhoff-type theorem and are uniquely determined by their mass. Using an auxiliary relational dust clock, together with the absence of gravitational waves in spherical symmetry, we exploit the structural ultralocality of the system to decompose the dynamics into independent shell degrees of freedom. The dust field acts as a reference clock for deparameterization and does not source the vacuum geometries considered here. These assumptions tightly constrain the Lemaitre-Tolman-Bondi shell Hamiltonian to a factorized form and the static vacuum metric function to a universal expression. We examine the possibility of a bounce and analyze how its presence is encoded, or missed, in finite-order effective truncations of the full model. The procedure for deriving the explicit physical Hamiltonian is described for a generic case before specializing to a specific model of interest. Finally, we construct a four-dimensional covariant completion of the spatially covariant Lagrangian, showing that it belongs to the class of generalized extended mimetic gravity models.