[Paper Review] Maxwell's Equations, Hodge Theory, and Gravitation

D. H. Sattinger|arXiv (Cornell University)|May 20, 2013

Relativity and Gravitational Theory8 references3 citations

TL;DR

This paper demonstrates that Maxwell’s equations arise naturally from Hodge theory and the fundamental laws of Gauss and Ampère, unifying electromagnetic and gravitational fields as geometric phenomena on Minkowski spacetime. By reinterpreting the electric and magnetic constants with reversed signs for gravity, the theory extends Maxwell’s formalism to attractive forces, showing that weak-field gravity is inherently spin-1 and distinct from Einstein’s spin-2 linearized gravity.

ABSTRACT

A mathematical proof is given that Maxwell's equations are an {\it artifact} of Hodge theory together with the laws of Gauss and Ampère, 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.

Motivation & Objective

• To establish a geometric foundation for Maxwell’s equations using Hodge theory, independent of specific physical mechanisms.
• To extend Maxwell’s formalism to gravitational fields by reversing the signs of the permittivity and permeability constants to reflect attraction.
• To clarify the distinction between Maxwell’s spin-1 field theory and Einstein’s linearized spin-2 gravity in weak-field regimes.
• To show that the energy-momentum tensor for both electromagnetic and gravitational fields has positive energy density, resolving historical objections to field energy sign.

Proposed method

• Apply the Hodge decomposition theorem to differential forms on 3D Euclidean space to decompose vector fields into exact and co-exact components.
• Use Hodge duality to express Gauss’s and Ampère’s laws in terms of differential forms, linking them to the electromagnetic and gravitational potentials.
• Formulate Maxwell’s equations in 4D Minkowski spacetime using the Faraday 2-form $F$ and the Maxwell-Ampère 2-form $G$, related by Hodge duality.
• Derive the Lagrangian for Maxwell’s equations from the Hodge inner product of $F$ and $G$, ensuring gauge invariance and geometric consistency.
• Reinterpret the electromagnetic constants $\epsilon$ and $\mu$ as geometric parameters with dimensions of inverse velocity squared, consistent with special relativity.
• Apply the same formalism to gravity by flipping the signs of $\epsilon$ and $\mu$, yielding a consistent spin-1 theory of weak gravity.

Experimental results

Research questions

• RQ1 Can Maxwell’s equations be derived as a geometric consequence of Hodge theory and fundamental laws of Gauss and Ampère, independent of physical mechanisms?
• RQ2 How can Maxwell’s equations be extended to describe attractive forces such as gravity, given their original formulation for repulsive electromagnetic fields?
• RQ3 What is the relationship between Maxwell’s spin-1 field theory and Einstein’s linearized spin-2 theory of gravity in weak-field limits?
• RQ4 Why does the energy-momentum tensor for both electromagnetic and gravitational fields have positive energy density, resolving historical concerns about negative field energy?
• RQ5 To what extent does Maxwell’s field theory serve as the leading-order approximation in the Post-Newtonian expansion of General Relativity?

Key findings

• Maxwell’s equations are shown to be a geometric consequence of Hodge theory and the axioms of Gauss and Ampère, valid for any force field—repulsive or attractive—generated by a material density and current.
• By reversing the signs of $\epsilon$ and $\mu$, the same formalism applies to gravitational fields on Minkowski spacetime, yielding a consistent spin-1 theory of weak gravity.
• The product $\mu\epsilon$ is proven to have dimensions of inverse velocity squared, and Einstein’s postulate of constant light speed ensures $\mu\epsilon = c^{-2}$, validating the theory for all force fields.
• The energy-momentum tensor $T^{00}$ for both electromagnetic and gravitational fields is positive, resolving the classical objection that gravitational field energy is negative.
• In the weak-field limit, Maxwell’s field theory dominates over Einstein’s linearized gravity, with the latter providing only a small relativistic correction, as seen in Mercury’s perihelion precession.
• The Post-Newtonian approximation shows that Maxwell’s equations constitute the leading-order term in the expansion, consistent with the observed dominance of electromagnetic-like fields in weak gravitational regimes.