Maxwell’s Equations via Differential Forms

The Connection to Rotations

by Stéphane Haussler

The present page presents a derivation of the geometric differential formulation of electromagnetism, emphasizing the direct connection between Maxwell’s equations and rotations in differential form. The mathematical expressions are based on previous investigations, where I utilize the Cartan-Hodge formalism to calculate the exterior derivative of rotations in spacetime. Through a straightforward identification process, we will see that the equations governing electromagnetism are, in fact, spacetime rotations with the exterior derivative applied. In layman’s terms, we can think of electromagnetism as a twist in spacetime.

My approach should feel more intuitive than the conventional method where the electromagnetic 2-form as well as the Maxwell’s equations in geometric differential form are given upfront, and equivalence to the vector formulation proven backward. For an example of the standard proof, see a nonetheless great video by Michael Penn. The mathematical expressions obtained in this section correspond to the well-recognized differential geometric formulation of Maxwell’s equations, where \(\mathbf{F}\) is the field 2-form and \(\mathbf{J}\) the current 3-form.

\[\begin{split}d\:\mathbf{F} = 0 \\ d⋆ \mathbf{F} = \mathbf{J}\end{split}\]

Unless explicitly mentioned otherwise, I use The Cartan-Hodge formalism, which permits to be explicit about the nature of the mathematical objects involved by using musical notation. The electromagnetic field 2-form \(\mathbf{F}\) is written \(F^{♭♭}\), the current 3-form \(J^{♭♭♭}\) and the Hodge dual current 1-form \(J^♭\).

The Equations of Mr. Maxwell

In the article Deriving the Faraday Tensor from the 1865 Maxwell’s Equations, I reordered and re-expressed the original Maxwell’s equations with modern notation. We use this form as the basis for formulating the electromagnetic field equations using geometric differential forms.

Inhomogenous Maxwell equations

Gauss’s Law and Ampère’s Circuital Law are gathered:

(1)\[\begin{split}\begin{alignedat}{5} & + ∂_x \E^x & + ∂_y \E^y & + ∂_z \E^z & = & + μ_0 c ρ \\ + ∂_t \E^x & & - ∂_y B^z & + ∂_z B^y & = & - μ_0 J^x \\ + ∂_t \E^y & + ∂_x B^z & & - ∂_z B^x & = & - μ_0 J^y \\ + ∂_t \E^z & - ∂_x B^y & + ∂_y B^x & & = & - μ_0 J^z \\ \end{alignedat}\end{split}\]

Homogenous Maxwell equations

Similarly, Gauss’s Law for Magnetism and Faraday’s Law of Induction are also be formulated in this manner:

(2)\[\begin{split}\begin{alignedat}{4} & + ∂_x B^x & + ∂_y B^y & + ∂_z B^z & = 0 \\ + ∂_t B^x & & + ∂_y \E^z & - ∂_z \E^y & = 0 \\ + ∂_t B^y & - ∂_x \E^z & & + ∂_z \E^x & = 0 \\ + ∂_t B^z & + ∂_x \E^y & - ∂_y \E^x & & = 0 \\ \end{alignedat}\end{split}\]

The Exterior Derivative of Rotations

In the article Rotations in Minkowski space, I investigate spacetime rotations in differential form and demonstrate that rotations can be expressed as:

\[\begin{split}R^{♭♭} = \left[ \begin{aligned} - &a \; dt ∧ dx \\ - &b \; dt ∧ dy \\ - &c \; dt ∧ dz \\ &d \; dy ∧ dz \\ &e \; dz ∧ dx \\ &f \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

In the subsequent article 2–forms (rotations), I systematically calculate the exterior derivative of arbitrary rotations and their Hodge dual, obtaining the following expressions:

Exterior derivative of the Hodge dual of rotations in differential form

(3)\[\begin{split}d ⋆ R^{♭♭} = \left[ \begin{alignedat}{5} (& & + ∂_x a & + ∂_y b & + ∂_z c \:&) \; dx ∧ dy ∧ dz \\ (& + ∂_t a & & - ∂_y f & + ∂_z e \:&) \; dt ∧ dy ∧ dz \\ (& + ∂_t b & + ∂_x f & & - ∂_z d \:&) \; dt ∧ dz ∧ dx \\ (& + ∂_t c & - ∂_x e & + ∂_y d & \:&) \; dt ∧ dx ∧ dy \\ \end{alignedat} \right]\end{split}\]

Hodge dual of the exterior derivative of rotations in differential form

(4)\[\begin{split}⋆\:d\:R^{♭♭} = \left[ \begin{alignedat}{5} (& \; & - ∂_x \; d & - ∂_y \; e & - ∂_z \; f \:&) \; dt \\ (& - ∂_t \; d & \; & - ∂_y \; c & + ∂_z \; b \:&) \; dx \\ (& - ∂_t \; e & + ∂_x \; c & \; & - ∂_z \; a \:&) \; dy \\ (& - ∂_t \; f & - ∂_x \; b & + ∂_y \; a & \; \:&) \; dz \\ \end{alignedat} \right]\end{split}\]

Identifying the Equations of Mr. Maxwell

From equations (1) and (3), identifiying the components of the electric field \(\tilde{E}^i=E^i/c\) and magnetic field \(B^i\) is trivial:

\[\begin{split}\begin{matrix} \E^x = a & B^x = d \\ \E^y = b & B^y = e \\ \E^z = c & B^z = f \\ \end{matrix}\end{split}\]

We could have equally used equations (2) and (3) for the identification. There, the sign of (2) can be flipped as needed. The doubly covariant Faraday tensor \(F^{♭♭}\) is identified as an arbitrary rotation \(R^{♭♭}\) in Minkowski spacetime:

The doubly covariant Faraday 2-form as a rotation in spacetime

\[\begin{split}R^{♭♭} = F^{♭♭} = \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right]\end{split}\]

Maxwell equations are therefore obtained by applying the exterior derivative to that rotation with \(d F^{♭♭}\) and its Hodge dual \(d ⋆ F^{♭♭}\).

Inhomogenous Maxwell equations via differential forms

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

Homogenous Maxwell equations via differential forms

\[\begin{split}⋆\:d \left[ \begin{aligned} - & \E^x \; dt ∧ dx \\ - & \E^y \; dt ∧ dy \\ - & \E^z \; dt ∧ dz \\ & B^x \; dy ∧ dz \\ & B^y \; dz ∧ dx \\ & B^z \; dx ∧ dy \\ \end{aligned} \right] = 0\end{split}\]

Thus and as advertised in the introduction, we fall back to the well-known expression of Maxwell equations in differential form where \(\mathbf{F}\) is the field 2-form and \(\mathbf{J}\) is the current 3-form.

\[\begin{split}d\:\mathbf{F} &= 0 \\ d⋆ \mathbf{F} &= \mathbf{J} \\\end{split}\]

A Single Equation

With the explicit component form of the Cartan-Hodge formalism, it may now be obvious that since \(⋆\:d\:F\) is a 1-form and \(d⋆F\) a 3-form, we can unambiguously merge inhomogeneous and homogeneous equations [note2].

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

Maxwell’s Equations in Differential Form

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

With a shorthand \(F^{♭♭}\) for the electromagnetic field 2-form and \(J^{♭♭♭}\) for the current 3-form, we conclude with the compact form [note3]:

Maxwell equations

\[(d ⋆ + ⋆ d) \; F^{♭♭} = J^{♭♭♭}\]

Maxwell’s equations are interpreted as a twist in spacetime, with a strength proportional to the 4-current.

Proof using the standard notation

Using the standard notation, we get to the same result:

\[⋆ \: d \: \mathbf{F} + d⋆\mathbf{F} = \mathbf{J}\]

Since \(d\:\mathbf{F}\) is a 3-form, its Hodge dual \(⋆\:d\:\mathbf{F}\) is a 1-form. We can thus rewrite the homogenous equation as acting on 1-forms, and the inhomogeneous equations as acting on 3-forms:

\[\begin{split}\begin{alignedat}{2} ⋆ \: d \: \mathbf{F} &= 0 & \qquad\text{1-forms} \\ d⋆\mathbf{F} &= \mathbf{J} & \qquad\text{3-forms} \\ \end{alignedat}\end{split}\]

Expanding the right hand side of both equations for the argument, we have:

\[\begin{split}⋆ \: d \: \mathbf{F} =& \: 0\:dt + 0\:dx + 0\:dy + 0\:dz \\ d⋆\mathbf{F} =& \: \mu_0 ρ dx∧dy∧dz - μ_0 J^x dt∧dy∧dz - μ_0 J^y dt∧dz∧dx - μ_0 J^z dt∧dx∧dy \\\end{split}\]

Therefore, the differential geometric formulation can be unambiguously reduced to a single equation involving both 3-forms and 1-forms:

\[⋆ \: d \: \mathbf{F} + d⋆\mathbf{F} = \mathbf{J}\]

Notes

[note2]

An equation containing 3-forms and 2-forms indeed cannot be reduced. For example, the following equation: \(a \; dx ∧ dy + b \; dx ∧ dy ∧ dz = c \; dx ∧ dy\) cannot be simplified. Surface 2-forms and volume 3-forms are distinct objects, but they can be represented in the same equation using the \(+\) symbol, even though they cannot actually be added together. Similar examples include combining the real and imaginary parts of complex numbers, or adding bivectors and trivectors in Clifford algebra. With the exemplary equation above, we thus necessarily have \(a = c\) as well as \(b = 0\). This is how we can write the Maxwell equations via differential forms into a single equation.

[note3]

Flipping the sign of \(⋆ d\) is equally valid.