Classical model of an elementary particle with a Bertotti-Robinson core and extremal black holes

Abstract

We discuss the question, whether the Reissner-Nordstr\"om (RN) metric can be glued to other solutions of Einstein-Maxwell equations in such a way that (i) the singularity at $r=0$ typical of the RN metric is removed, and (ii) matching is smooth. Such a construction could be viewed as a classical model of an elementary particle balanced by its own forces without support by an external agent. One choice is the Minkowski interior that goes back to the old Vilenkin and Fomin's idea who claimed that in this case the bare deltalike stresses at the horizon vanish if the RN metric is extremal. However, the relevant entity here is the integral of these stresses over the proper distance which is infinite in the extremal case. As a result of the competition of these two factors, the Lanczos tensor does not vanish and the extremal RN cannot be glued to the Minkowski metric smoothly, so the elementary-particle model as an empty ball inside fails. We examine the alternative possibility for the extremal RN metric---gluing to the Bertotti-Robinson (BR) metric. For a surface placed outside the horizon there always exist bare stresses but their amplitude goes to zero as the radius of the shell approaches that of the horizon. This limit realizes the Wheeler idea of ''mass without mass'' and ''charge without charge.'' We generalize the model to the extremal Kerr-Newman metric glued to the rotating analog of the BR metric.