Many people have heard of Maxwell’s famous four equations:

∇·E = 0

∇·B = 0

∇ x E = -∂B/∂t

∇ x B = μ_{0}J + μ_{0}ε_{0}(∂E/∂t)

But did you know that they can actually all be captured in one simple expression? When we extend classical electrodynamics into Minkowski 4-space (the geometry of special relativity), the time derivatives that appear in the derivation of Maxwell’s equations can be mixed into the same mathematics that takes care of the spatial expressions. By generalizing the Laplacian (∇^{2}) to 4-space, we obtain the d’Alembertian (□^{2}, sometimes written without the exponent). This operator allows us to rewrite Maxwell’s equations in tensor notation as one simple formula, sometimes called the Riemann–Sommerfeld equation:

□^{2}A^{μ} = -μ_{0}J^{μ}

Where A^{μ} is the four-potential (φ, cA) (with φ and A the familiar scalar and vector potentials) and J^{μ} the four-current (ρc, J) (with ρ and J the familiar charge and current densities), thereby using the power of differential geometry and tensor notation to capture all of classical electrodynamics in one swift stroke.

### Background

Maxwell’s Equations describe how electric and magnetic fields relate to each other and how they relate to the presence of charged particles and currents. The way we’ve presented them above is in *differential form*, which describes how electric fields (**E**) and magnetic fields (**B**) change in space and time; they can also be presented in *integral form*, which describes how the total measure of these fields scales with quantities like current density (**J**).

Although these equations were originally formulated in terms of fields, physicists would later find that the equations could be much cleaner when expressed in terms of potentials. A similar revolution happened in classical mechanics; although Newton’s forces were a good start, Lagrangian dynamics showed us that understanding the potential energy of the system made solving for equations of motion much simpler than trying to add up dozens of force vectors. In electrodynamics, we have two kinds of potentials: the scalar potential φ (or, sometimes, V), which most people know by the term “voltage”; and the vector potential, **A**, which is related to magnetic fields and may be less familiar.

Tensors notation (for our purposes) is little more than a convenient way to represent vectors and matrices. When you read the Riemann-Sommerfeld equation, then, you’re really reading 4 equations at once. What are those equations? Repeat the equation four times but each time replace μ with a different value; so, one equation might be □^{2}A^{1} = -μ_{0}J^{1}. The 1’s in this equation aren’t exponents, but rather tensor indices; just like you might see vector components written as v_{x}, v_{y}, and v_{z} representing different components of **v**, A^{0}, A^{1}, A^{2}, and A^{3} represent the four components of the (vector) tensor **A**.

What are the funny upside-down triangles? The symbol is called a nabla, and is often read as “del”. In some ways, it’s an abuse of notation, but it gives us a convenient way to write the gradient, divergence, curl, and Laplacian operators. The exercise for how this notation works is left to the reader, but note the definitions: ∇U = grad(U), the gradient; ∇·U = div(U), the divergence; ∇ x U = curl(U), the curl; and ∇^{2}U = ∇·(∇U) = div(grad(U)), the Laplacian. Each of these functions has a very important place in vector analysis. Since we represent electric and magnetic fields mathematically as vector fields, these operations give us information about the geometry of those fields. The gradient function tells us the direction that a field changes most at each point; the divergence tells us how much the field is dispersing (a field where everything pointed away from one center point would have a high divergence); the curl tells us how much a field is swirling, so to speak; and the Laplacian tells us how much a field changes in strength as it disperses.

And finally, we have our partial derivatives (∂B/∂t, for example) that give us rates of change (like how a magnetic field B changes with respect to the time t), and our constants, ε_{0} and μ_{0}. These constants have the interesting relation that, for the speed of light c, ε_{0}μ_{0}=1/c^{2}.

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