Unitary and finite self-energy of a single classical point charge and naked point singularity spacetimes

Abstract

We analyze linear Einstein--Maxwell perturbations of the superextremal Reissner--Nordstr\"om geometry in its static Kerr--Schild rest frame, viewing it as the nonlinear self-field of a single static point charge. In optical radial coordinates, and using the Kodama--Ishibashi gauge-invariant formalism, each radiative multipole is encoded by a single scalar master field on the half-line. The resulting master equation is of Regge--Wheeler type, with an inverse-square potential core at the optical apex (controlled by a Hardy inequality) and a short-range tail at infinity. The spatial-plus-potential part of the Einstein--Maxwell $T$-energy is closable and bounded below, which defines a positive quadratic form on the natural energy space. Its Friedrichs extension then gives the canonical self-adjoint realization of the master operator. The static Coulomb field and its nonlinear gravitational backreaction are treated as the exact background. All linear Einstein--Maxwell perturbations with finite spatial $T$-energy evolve unitarily on the energy space. The naked singularity at finite optical distance is ``silent'' in the technical sense that it carries no $T$-energy flux. We also construct the forward radiation field at future null infinity, obtaining a translation representation of the self-field dynamics in which the conserved $T$-energy coincides with the $L^2$ norm of the radiation profile in retarded time.