The Unapologetic Mathematician

Mathematics for the interested outsider

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.

About these ads

February 24, 2012 - Posted by | Electromagnetism, Mathematical Physics

4 Comments »

  1. [...] The Meaning of the Speed of Light (unapologetic.wordpress.com) [...]

    Pingback by Magnetic fields, again | cartesian product | February 26, 2012 | Reply

  2. [...] The Meaning of the Speed of Light [...]

    Pingback by The Faraday Field « The Unapologetic Mathematician | March 6, 2012 | Reply

  3. [...] If we have vectors and — with time here measured in the same units as space by using the speed of light as a conversion factor — then we calculate the metric [...]

    Pingback by Minkowski Space « The Unapologetic Mathematician | March 7, 2012 | Reply

  4. [...] off, I’m going to use a coordinate system where the speed of light is 1. That is, if my unit of time is seconds, my unit of distance is light-seconds. Mostly this helps [...]

    Pingback by The Higgs Mechanism part 2: Examples of Lagrangian Field Equations « The Unapologetic Mathematician | July 17, 2012 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 392 other followers

%d bloggers like this: