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

Daxx W. Delucchi

TL;DR

The work derives a canonical, unitary evolution for linear Einstein--Maxwell perturbations on the fixed superextremal Reissner--Nordström self-field by recasting the exterior as an optical half-line and deploying gauge-invariant Kodama--Ishibashi master fields. It shows the spatial Schrödinger-type operators on the optical line admit Friedrichs extensions $H_ ext{±}^ ext{F}$, with a silent apex enforced by a sharp Hardy control, yielding no bound states or zero-energy resonances, and a purely absolutely continuous spectrum on $[0, ablafty)$. A Doob ground-state transform provides a transparent energy representation, while the forward radiation field $ ext{R}_+$ furnishes a unitary isomorphism between the Einstein--Maxwell energy space and $L^2( abla ext{R}_u)$, ensuring all finite-$T$-energy perturbations radiate to $ uture ext{null infinity}$ and are unitarily equivalent to translations in retarded time. Collectively, these results establish a robust, unitary, radiation-dominated dynamics for the self-field of a naked point charge and illuminate how energy conservation manifests as an isometry with the radiation profile, despite the absence of a horizon. The findings also suggest a structural framework for potential quantum interpretations built on the radiation map and ground-state factorization.

Abstract

We analyze linear Einstein--Maxwell perturbations of the superextremal Reissner--Nordström 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.

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

TL;DR

The work derives a canonical, unitary evolution for linear Einstein--Maxwell perturbations on the fixed superextremal Reissner--Nordström self-field by recasting the exterior as an optical half-line and deploying gauge-invariant Kodama--Ishibashi master fields. It shows the spatial Schrödinger-type operators on the optical line admit Friedrichs extensions , with a silent apex enforced by a sharp Hardy control, yielding no bound states or zero-energy resonances, and a purely absolutely continuous spectrum on . A Doob ground-state transform provides a transparent energy representation, while the forward radiation field furnishes a unitary isomorphism between the Einstein--Maxwell energy space and , ensuring all finite--energy perturbations radiate to and are unitarily equivalent to translations in retarded time. Collectively, these results establish a robust, unitary, radiation-dominated dynamics for the self-field of a naked point charge and illuminate how energy conservation manifests as an isometry with the radiation profile, despite the absence of a horizon. The findings also suggest a structural framework for potential quantum interpretations built on the radiation map and ground-state factorization.

Abstract

We analyze linear Einstein--Maxwell perturbations of the superextremal Reissner--Nordström 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 -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 -energy evolve unitarily on the energy space. The naked singularity at finite optical distance is ``silent'' in the technical sense that it carries no -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 -energy coincides with the norm of the radiation profile in retarded time.
Paper Structure (23 sections, 6 theorems, 341 equations)

This paper contains 23 sections, 6 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. Rest frame and optical coordinate
  3. Master field reduction
  4. Potential asymptotics and Hardy closure
  5. Form-boundedness at the apex
  6. Short-range tails as infinitesimal form perturbations
  7. Global estimate on optical half-line
  8. The Kato--Lions--Milgram--Nelson and representation theorems
  9. Friedrichs extension and silent apex
  10. Doob ground-state representation and absence of bound states on the optical half-line
  11. Construction of the ground state near the apex
  12. Global continuation and large-x behaviour
  13. Doob transform and positivity of the T-energy forms
  14. Extension to H01 and factorization
  15. Absence of eigenvalues and static Einstein--Maxwell perturbations
  16. ...and 8 more sections

Key Result

Theorem 5.1

Let $V_\pm$ be a master potential for a fixed multipole $\ell\ge2$ in the superextremal Reissner--Nordström background, and let $q_\pm$ be the quadratic form eq:qpm-def on $C_0^\infty(0,\infty)$. Assume that $V_\pm$ satisfies the core and tail conditions eq:V-core and eq:short-range. Then:

Theorems & Definitions (6)

  • Theorem 5.1
  • Theorem 6.1
  • Theorem 7.1
  • Lemma 7.2
  • Theorem 7.3: Sá Barreto SaBarreto2003
  • Theorem 7.4