The Complete Einstein Equation of the Electron

Abstract

We apply the full retarded Green operator to the harmonic-gauge linearized Einstein equation of a localized electron. The resulting causal field is compared with a classical truncation containing an instantaneous near-zone field and a finite outgoing radiation sector. Their difference defines a historical correction and gives a coupling-corrected Einstein equation. The electron carries both electric-charge and mass-charge four-currents along the same worldline. A calibrated charge projection of the \(0\mu\) metric response produces the electromagnetic four-potential, while harmonic gauge becomes Lorenz gauge. From this potential we derive the Maxwell tensor, all four Maxwell equations, retarded potentials, boundary conditions, and local \(U(1)\) symmetry. The mass-charge projection remains distinct and describes gravitoelectric and gravitomagnetic response. Both channels act simultaneously for an electron, whereas a neutral particle retains only the mass-charge channel. **Keywords** Complete Einstein equation; electron worldline; retarded Green function; linearized gravity; electromagnetic four-potential; Maxwell equations; Lorenz gauge; \(U(1)\) symmetry; gravitoelectromagnetism; historical correction.