Derivation of Maxwell's equations via the covariance requirements of the special theory of relativity, starting with Newton's laws

TL;DR

This paper derives Maxwell's equations from Newton's laws and special relativity principles, emphasizing covariance and symmetry, and clarifies the roles of charge, current, and magnetic monopoles within this framework.

Contribution

It presents a novel derivation of Maxwell's equations starting from Newton's laws, emphasizing covariance, symmetry, and the reinterpretation of Newton's third law.

Findings

Maxwell's equations are derived from Newton's laws and covariance requirements.

The Lorentz force law naturally emerges from the tensor formalism.

Magnetic monopoles are excluded by symmetry considerations under spatial inversion and time reversal.

Abstract

A connection between Maxwell's equations, Newton's laws, and the special theory of relativity is established with a derivation that begins with Newton's verbal enunciation of his first two laws. Derived equations are required to be covariant, and a simplicity criterion requires that the four-vector force on a charged particle be linearly related to the four-vector velocity. The connecting tensor has derivable symmetry properties and contains the electric and magnetic field vectors. The Lorentz force law emerges, and Maxwell's equations for free space emerge with the assumption that the tensor and its dual must both satisfy first order partial differential equations. The inhomogeneous extension yields a charge density and a current density as being the source of the field, and yields the law of conservation of charge. Newton's third law is reinterpreted as a reciprocity statement, which requires that the charge in the source term can be taken as the same physical entity as that of the test particle and that both can be assigned the same units. Requiring covariance under either spatial inversions or time reversals precludes magnetic charge being a source of electromagnetic fields that exert forces on electric charges.