Electromagnetism from 5D gravity: beyond the Maxwell equations

Abstract

Ever with the work of Kaluza, it has been known that 4D Einstein- and Maxwell-type equations emerge from the equations for 5D gravity, in Ricci-flat space-times having a space-like Killing vector. We revisit these equations and compare them with the Maxwell equations and the Ohm’s law. Although 5D gravity and traditional electromagnetic theory are mathematically related, a paradigm shift in traditional electromagnetic concepts is required for unification. First, both the Maxwell equations and the Ohm’s law are found among the field equations for 5D gravity, which turns Ohm’s law into a field equation. Second, the concept of 4-current density, bringing together electrical charge and current density, is no longer fundamental for the electromagnetic theory resulting from 5D gravity. Third, the 5D gravity equations suggest a generalization for the model of the perfect conductor, to describe a broader range of electromagnetic phenomena and bypass the concept of 4-current density. We thus propose an amendment to the Maxwell equations, based on the field equations for 5D gravity, and discuss a few applications. Namely, we discuss the propagation of electromagnetic waves through an imperfect conductor and topics in electrostatics and magnetostatics, touching ground with the Hall effect and the phenomenon of the persistent current.