Extended Maxwell equations derived from spacetime geometry

Abstract This paper derives a unified description of electromagnetic and gravitational interactions in the classical regime, addressing the fundamental question of why like charges repel while like masses attract. We formulate a flat spacetime geometry tailored to special relativity and introduce intrinsic operations of spacetime product and spacetime curl, naturally yielding a multi-branched formalism. Based on three axiomatic postulates—relativistic properties of spacetime, energy conservation, and standard charge condition—we rigorously derive four extended Maxwell equations and their corresponding force formulas by employing our formalism, without ad hoc parameters or phenomenological fitting. These simultaneously recover electromagnetic and gravitational interactions, along with their mirror-symmetric counterpart, thereby naturally resolving the aforementioned fundamental question. A thought experiment demonstrates consistency with the principle of relativity. We propose a unified charge system to facilitate the unified description. The framework remains strictly classical, does not incorporate spin and radiative-reaction effects, and applies to gravity in quasi-charge conditions where the motion of matter can be treated as a conserved Lorentz four-vector current. This axiomatic derivation provides a geometrically grounded unified description of long-range interactions and a transparent foundation for theoretical extensions.