# The Unapologetic Mathematician

## The Meaning of the Speed of Light

Let’s pick up where we left off last time converting Maxwell’s equations into differential forms:

\displaystyle\begin{aligned}*d*\epsilon&=\mu_0c^2\rho\\d\beta&=0\\d\epsilon&=-\frac{\partial\beta}{\partial t}\\{}*d*\beta&=\mu_0\iota+\frac{1}{c^2}\frac{\partial\epsilon}{\partial t}\end{aligned}

Now let’s notice that while the electric field has units of force per unit charge, the magnetic field has units of force per unit charge per unit velocity. Further, from our polarized plane-wave solutions to Maxwell’s equations, we see that for these waves the magnitude of the electric field is $c$ — a velocity — times the magnitude of the magnetic field. So let’s try collecting together factors of $c\beta$:

\displaystyle\begin{aligned}*d*\epsilon&=\mu_0c^2\rho\\d(c\beta)&=0\\d\epsilon&=-\frac{1}{c}\frac{\partial(c\beta)}{\partial t}\\{}*d*(c\beta)&=\mu_0c\iota+\frac{1}{c}\frac{\partial\epsilon}{\partial t}\end{aligned}

Now each of the time derivatives comes along with a factor of $\frac{1}{c}$. We can absorb this by introducing a new variable $\tau=ct$, which is measured in units of distance rather than time. Then we can write:

\displaystyle\begin{aligned}*d*\epsilon&=\mu_0c^2\rho\\d(c\beta)&=0\\d\epsilon&=-\frac{\partial(c\beta)}{\partial\tau}\\{}*d*(c\beta)&=\mu_0c\iota+\frac{\partial\epsilon}{\partial\tau}\end{aligned}

The easy thing here is to just write $t$ instead of $\tau$, but this hides a deep insight: the speed of light $c$ is acting like a conversion factor from units of time to units of distance. That is, we don’t just say that light moves at a speed of $c=299\,792\,457\frac{\mathrm{m}}{\mathrm{s}}$, we say that one second of time is 299,792,457 meters of distance. This is an incredibly identity that allows us to treat time and space on an equal footing, and it is borne out in many more or less direct experiments. I don’t want to get into all the consequences of this fact — the name for them as a collection is “special relativity” — but I do want to use it.

This lets us go back and write $\beta$ instead of $c\beta$, since the factor of $c$ here is just an artifact of using some coordinate system that treats time and distance separately; we see that the electric and magnetic fields in a propagating electromagnetic plane-wave are “really” the same size, and the factor of $c$ is just an artifact of our coordinate system. We can also just write $t$ instead of $c t$ for the same reason. Finally, we can collect $c\rho$ together to put it on the exact same footing as $\iota$.

\displaystyle\begin{aligned}*d*\epsilon&=\mu_0c\rho\\d\beta&=0\\d\epsilon&=-\frac{\partial\beta}{\partial t}\\{}*d*\beta&=\mu_0c\iota+\frac{\partial\epsilon}{\partial t}\end{aligned}

The meanings of these terms are getting further and further from familiarity. The $1$-form $\epsilon$ is still made of the same components as the electric field; the $2$-form $\beta$ is $c$ times the Hodge star of the $1$-form whose components are those of the magnetic field; the function $\rho$ is $c$ times the charge density; and the vector field $\iota$ is the current density.

February 24, 2012 -

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