The electromagnetic field tensor

by Stéphane Haussler

Table of Contents:

• Deriving the Faraday Tensor from the 1865 Maxwell’s Equations
• Maxwell’s Equations via Differential Forms
• Potential Formulation via Differential Forms
• All Electromagnetic Field Tensors

Summary

My aim with these pages is to improve upon other sources by providing a clear and natural derivation of the Faraday tensor as an electromagnetic 2-form, along with the expression of Maxwell’s equations via geometric differential forms. I seek to clarify the connection of the Faraday tensor to rotations in Minkowski spacetime.

I could not find a truly satisfying derivation of Maxwell’s equations in differential form. Typically, the standard approach consists in assuming both the final equations in differential form, as well as the electromagnetic Faraday 2-form, and then prove the equivalence to Maxwell’s equations in the standard vector form. Here, I start from a mathematical point of view, expressing rotations in Minkowski spacetime as 2-forms. I then systematically compute the exterior derivative \(d\) of these rotations, thereby exploring how to express twists, or to be more precise torque, in Minkowski spacetime. This procedure naturally leads to the emergence of the Faraday tensor. By utilizing the exterior derivative \(d\) and the Hodge dual \(⋆\), we will derive the theory of electromagnetism into a single equation:

\[(d ⋆ - ⋆ d) \; F^{♭♭} = J^{♭♭♭}\]

The Faraday tensor \(F\) will be identified as a generic rotation in spacetime and expressed as a 2-form \(F^{♭♭}\). The 4-current \(J\) is expressed as the 3-form \(J^{♭♭♭}\), represents the amount of torque applied. Expanding in component form, we will obtain the following set of equations:

\[\begin{split}(d ⋆ - ⋆ d ) \left[ \begin{aligned} -& \E^x \; dt ∧ dx \\ -& \E^y \; dt ∧ dy \\ -& \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned}\right] = \begin{bmatrix} + μ_0 c ρ \; dx ∧ dy ∧ dz\\ - μ_0 J^x \; dt ∧ dy ∧ dz\\ - μ_0 J^y \; dt ∧ dz ∧ dx\\ - μ_0 J^z \; dt ∧ dx ∧ dy\\ \end{bmatrix}\end{split}\]

The wedge symbol \(∧\) denotes the exterior product, \(\tilde{E}^i\) the electric field components divided by the speed of light \(c\), and \(B^i\) the magnetic field components. \(μ_0\) is the vacuum permeability, and \(c ρ\) and \(J^i\) are the components of the 4-current.

Maxwell’s equations are rooted in experimental observations, except for Maxwell’s modification to Ampère’s circuital law, which originates from purely mathematical considerations. These equations mathematically express empirical data and are established experimental facts. We approach the formulation in geometric differential form via a systematic mathematical analysis of rotations. There, the reader will likely immediately recognize the presence of the equations governing electromagnetism. The conclusion will then consist of a straightforward identification.

To double-check the results, I recommend Michael Penn’s video on Maxwell’s equations via differential forms.

Rotations in differential form

Rotations in Euclidean space and Minkowski spacetime are systematically expressed and analyzed using differential forms.

2–forms (rotations)

We systematically analyse the exterior derivative applied to rotations. In layman’s terms, we examine how to apply a rotational force, or torque, to express a twist in 4-dimensional spacetime.

Deriving the Faraday Tensor from the 1865 Maxwell’s Equations

We revisit the original expression of Maxwell’s equations, moving away from the now-standard vector form proposed by Mr. Heaviside, and drawing significant inspiration from the work of Mr. Minkowski.

Maxwell’s Equations via Differential Forms

This is the cornerstone of our analysis, where everything falls into place in the most straightforward manner. From the expressions derived while studying rotations in differential forms, Maxwell’s equations naturally emerge, allowing us to identify the fundamental equations governing electromagnetism.

All Electromagnetic Field Tensors

We systematically calculate all representations of the Faraday tensor: the doubly covariant form, the doubly contravariant form, both mixed forms, as well as the Hodge duals of all these quantities. I hope this will clarify the expression of all Faraday tensors using metric signature \((+,-,-,-)\) (Please let me know if you find mistakes!).