Pointlike Electric Charge in Gravitational Field Theory

Hans Dekker

doi:10.24297/jap.v14i2.7596

Abstract

The existence of charged elementary ’point particles’ still is a basically unsolved puzzle in theoretical physics.The present work takes a fresh look at the problem by including gravity—without resorting to string theory. Using Einstein’s equations for the gravitational fields in a general static isotropic metric with the full energymomentum tensor (for the charged material mass and the electromagnetic fields) as the source term, a novel exact solution with a well-defined characteristic radius emerges where mass and charge accumulate:—with being the ’classical’ radius associated with the total charge and where is the Schwarzschild radius belonging to the observable mass (for the electron one has m and m). The resulting ’Einstein-Maxwell’ gravitational electron radius can also be written as , where m is the fundamental Planck length and the fine-structure constant, which yields m

Highlights

Modern theoretical physics is essentially based on the existence of a finite set of elementary ’point particles’—leptons and quarks—and their electromagnetic, gravitational, and weak or strong interactions
Why should it be possible to accumulate a finite amount of electric charge in an infinitely small volume? What internal force does the work against the repulsive self-interaction? without such a force the charged particle should immediately explode
The problems with a point charge were already recognized in classical physics

Summary

Introduction

Modern theoretical physics is essentially based on the existence of a finite set of elementary ’point particles’—leptons and quarks—and their electromagnetic, gravitational, and weak or strong interactions (see, e.g., Refs. [1,2,3,4,5]). In Poincaré’s ’electron model’ [7] the electric force on the charged sphere was counteracted by an elastic force of unspecified, non-electromagnetic and non-gravitational nature in order to define a total energy-momentum tensor satisfying the condition characteristic of a closed system In non-gravitational quantum theory the electron can be treated successfully as a structureless point particle—at least, if the problem of its infinite self-energy is being ’swept under the carpet’. The present work takes a fresh look at the problem by including gravity—as yet without resorting to string theory, ignoring the weak interaction and without a priori assuming the mass-charge density to be rigorously zero outside some

The gravitational field equations

The static isotropic case

The equilibrium condition

Results

Final remarks