Singularity resolution by holonomy corrections: Spherical charged black holes in cosmological backgrounds

We study spherical charged black holes in the presence of a cosmological constant with corrections motivated by the theory of loop quantum gravity. The effective theory is constructed at the Hamiltonian level by introducing certain correction terms under the condition that the modified constraints form a closed algebra. The corresponding metric tensor is then carefully constructed ensuring that the covariance of the theory is respected, that is, in such a way that different gauge choices on phase space simply correspond to different charts of the same spacetime solution. The resulting geometry is characterized by four parameters: the three usual ones that appear in the general relativistic limit (describing the mass, the charge, and the cosmological constant), as well as a polymerization parameter, which encodes the quantum-gravity corrections. Contrary to general relativity, where this family of solutions is generically singular, in this effective model the presence of the singularity depends on the values of the parameters. The specific ranges of values that define the family of singularity-free spacetimes are explicitly found, and their global structure is analyzed. In particular, the mass and the cosmological constant need to be nonnegative to provide a nonsingular geometry, while there can only be a bounded, relatively small, amount of charge. These conditions are suited for any known spherical astrophysical black hole in the de Sitter cosmological background, and thus this model provides a globally regular description for them.