Regular and singular solutions for charged dust distributions in the Einstein--Maxwell theory

Solutions for the static spherically symmetric extremally charged dust in the Majumdar–Papapetrou system have been found. For a certain amount of the allocated mass/charge, the solutions have singularities of a type which could render them physically unacceptable, since the corresponding physically relevant quantities are singular as well. These solutions, with a number of zero-nodes in the metric tensor, are regularized through the $\delta$-shell formalism, thus redefining the mass/charge distributions. The bifurcating behaviour of regular solutions found before is no longer present in these singular solutions, but quantized-like behaviour in the total mass is observed. Spectrum of regularized solutions restores the equality of the Tolman–Whittaker and ADM mass, as well the equality of the net charge and ADM mass, which is the distinctive feature of Majumdar–Papapetrou systems.