Maxwell's equations from spacetime geometry and the role of Weyl curvature

Abstract

This research article demonstrates how the field equations of electrodynamics can be shown to be a special case of Einstein field equations of General Relativity. By establishing a special conjecture between the electromagnetic four-potential and the metric of the spacetime, it is first shown how the relativistic wave equation of electrodynamics is a condition for the metric to be Ricci-flat. Moreover, the four-current is identified with a certain four-gradient, which allows one to conjecture that electric charge is related to the covariant divergence of the electromagnetic four-potential. These considerations allow one to understand the Einstein field equations as a nonlinear generalization of Maxwell's equations. Finally, it is argued that the four-current induces Weyl curvature on the spacetime.