New General Relativistic Formulations for an Electromagnetic Theory of Gravity, Applied to Self-Consistently Model an Elementary Charge

New general-relativistic formulations to model an elementary charge are presented, based on an electromagnetic theory of gravity, where the gravity is equivalently expressed as a gradient function of an effective permittivity distribution of the empty space. The metric tensor elements of general relativity are directly related to the effective permittivity function of the empty space, using which the energy density associated with the electric field surrounding the charge is properly defined. The empty space, represented by the equivalent permittivity function, would be fundamentally non-linear, in which case the definition of energy density, as conventionally applied for a linear medium, needs to be corrected. Further, the definition of energy density itself is modified allowing both positive and negative values, such that the total energy remains unchanged while the local values are much stronger, resulting in much stronger local gravitational effects. Solutions for the metric-tensor elements and the resulting energy/mass of the charge particle are studied, based on the Einstein-Maxwell equations with the different new formulations of the energy density and of the associated full stress-energy tensor. The solutions are verified with Schwarzschild and Reissner-Nordstrom metrics, as well as for calculation of light deflection by a massive body, as validation of the general new formulations for the specific reference cases of conventional modeling. Stable solutions for energy/mass are successfully derived for a spherically symmetric, surface distribution of an elementary charge, with specific modified definitions of energy density. A stable solution of the charge with the lowest possible energy/mass is associated with a ''static'' electron without spin. Significance of the new results and formulations, specifically established for the electron, are recognized in relation to the fine-structure constant of quantum electrodynamics, and towards further application of the theory to model other elementary particles and general electrodynamic problems.