Pointlike electric charge in gravitational field theory

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 energy-momentum 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: $r_{\rm c}{=}\sqrt{r_{\rm e}r_o/2}$---with $r_{\rm e}{=}Q^2\!/4\pi\epsilon_omc^2$ being the 'classical' radius associated with the total charge $Q$ and where $r_o{=}2mG/c^2$ is the Schwarzschild radius belonging to the observable mass $m$ (for the electron one has $r_{\rm e}{\approx}10^{-15}$m and $r_o{\approx}\,10^{-57}$m). The resulting 'Einstein-Maxwell' gravitational electron radius can also be written as $r_{\rm c}{=}\ell_{\rm P}\sqrt{\alpha_{\rm e}}$, where $\ell_{\rm P}{=}\sqrt{\hbar G/c^3}{\approx}10^{-35}$m is the fundamental Planck length and $\alpha_{\rm e}{=}e^2\!/4\pi\epsilon_o\hbar c{\approx}1/137$ the fine-structure constant, which yields $r_{\rm c}^{\rm electron}{=}1.38063{\times}10^{-36}$m.