Maxwell's Equations, Hodge Theory, and Gravitation

Abstract

A mathematical proof is given that Maxwell's equations are an {\it artifact} of Hodge theory together with the laws of Gauss and Amp\`ere, taken as axioms. They are thus geometric in nature, independent of any specific physical mechanisms, and valid for any force field -- attractive or repulsive -- generated by a material density and current. In particular, with appropriate sign changes to reflect the attractive nature of the field, they apply to gravitational fields on Minkowski space-time as well. The linearization of the Einstein Field Equations on Minkowksi space-time leads also to a linear theory of gravity; but this theory is spin 2, while Maxwell's field theory is spin 1. Hence the two theories are distinct. The relationship of Maxwell's field theory to Einstein's geometric theory is explained for weak fields.