Theoretical Universe#
Maxwell Field Equations via Differential Forms
Within these pages, you will find the formulation of electromagnetism in terms of Élie Cartan’s differential forms, along with a systematic analysis of this formulation.
In 1865, James Clerck Maxwell presented his groundbreaking work, A Dynamical Theory of the Electromagnetic Field (pdf). This seminal paper united electrostatics and magnetism, leading to significant discoveries. Mr. Maxwell noted that his equations implied the propagation of waves at a constant speed, which exactly matched the measured speed of light, thereby identifying light as electromagnetic waves. Others observed that these equations implied an astonishing fact: the speed of light is constant, regardless of the relative motion of objects. This was experimentally confirmed by Albert Michelson and Edward Morley, paving the way for the theory of special relativity. Later, Albert Einstein drew inspiration from Mr. Maxwell’s work once more when developing the theory of general relativity, seeking a formulation that resembled the field equations for electromagnetism.
The mathematical work of Mr. Maxwell is deeply rooted in experiments, translating ideas, most notably from Mr. Michael Faraday, into mathematical expressions. The original equations were presented using differential expressions, which differ significantly from the standard modern vector formulation. Mr. Oliver Heaviside proposed both the concept of vectors, as well as the modern standard vector form of the Maxwell’s equations.
Further mathematical formulations include the tensor approach, quaternions, geometric algebra, and differential forms. There are more formulations I did not look at (yet), but have for now settled on the differential form formulation, as it feels to me the most elegant and complete approach. I could however not find (free) material to my exact liking. Therefore, I am doing it myself.
Some of the content of these articles, although certainly not new, may offer insights not widely known. I present the details of all calculations to give the reader a chance to follow and find mistakes. If you are aware of freely available resources I should cite, open an issue and I will include a reference. If you find errors, don’t hesitate to directly provide corrections by sending a merge request to the Theoretical Universe Github repository.
I point out the article Deriving the Faraday Tensor from the 1865 Maxwell’s Equations for its simplicity. This derivation is straightforward and self-contained, requiring only knowledge of vector calculus and matrix multiplication.
Subsequent articles assumes you, the reader, have a working understanding of tensor calculus, the concept of vector/covector duality, differential forms, musicality, the exterior product, as well as hodge duality. I gather all these tools into what I call the the Cartan-Hodge formalism. There you will find a personal overview of these concepts, which can be used as review material, though certainly not as primary learning material. I also introduce the free matrix representation, which I hope the prospective reader will find obvious and usefull for performing actual computations. For those looking to learn about differential forms, I loved the excellent video serie by Michael Penn.
The pivotal aspects the present study lie in three articles:
In the first article, differential forms are presented as a simple yet powerfull concept for representing rotations, and generalize to four-dimensional Minkowski spacetime. The connection to standard rotation matrices is laid plain, along with the relation to the Lie algebra of the Lorentz group \(\mathfrak{so}(1,3)\). The second article explores how to perform a twist in spacetime, essentially applying a derivation operator to a rotation. There, the exterior derivative of rotations in spacetime is systematically analyzed. The final article links the exterior derivative of rotations to Maxwell’s equations by identification, revealing that we are dealing with a twist, or to be more precise torque, in Minkowski spacetime. This provides a comprehensive and coherent demonstration of how Maxwell’s equations emerge from the framework of differential forms and electromagnetism is reduced to a single equation.
This serie was originally envisioned as a collection of short, standalone articles. However, it has expanded beyond my initial expectation and evolved into something more comprehensive.
This section reviews the mathematical objects used in this work. Basis vectors are identified as partial derivatives \(\mathbf{e_μ} = ∂_μ\), which in turns permits to express basis covectors as differentials \(\mathbf{e^μ} = dx^μ\). The musical sharp operator \(♯\) transforms covectors to vectors, and the musical flat operator \(♭\) transforms vectors to covectors. The concept of the Hodge dual and the star operator \(⋆\) are explained. Finally, I introduce the free matrix representation, which I hope readers will find obvious enough.
I encapsulate these concepts in what I name the Cartan-Hodge formalism. This comprehensive framework facilitates the systematic analysis of rotation representation in differential form and a thorough examination of differential operators within this context.
Rotations in differential form
This section contains a systematic analyis of the representation of rotations in Euclidean three-dimensional space, as well as its generalization to four-dimensional Minkowski spacetime. Rotations are expressed as linear combinations of basis rotations in each planes. The expressions as double covariant tensors, double contravariant tensors, and mixed tensors are derived. Symmetries are systematically analyzed. In this section, we get the first insight about the deep connection between the electromagnetism and rotations in spacetime.
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