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A light wave travels along the x-axis. The equation for the variation of an electric field with respect to location on the x-axis and time is as follows:

$ E = E_{max} \sin( k x -kc t )$ where $k = {2{\pi}/{\lambda}}$ where ${\lambda}=wavelength$

Therefore;

$ E = E_{max} \sin(2{\pi}/{\lambda})( x -c t )$

If we differentiate this with respect to time we get;

$ {\partial}E/{\partial}t = ({-2{\pi}{c}/{\lambda}})E_{max} \cos(2{\pi}/{\lambda})( x -c t )$

Let's set our position to x=0 and our time to t=0. We get;

$ {\partial}E/{\partial}t = ({-2{\pi}{c}/{\lambda}})E_{max} (1)$

If $E_{max}>{\lambda}$ then ${\partial}E/{\partial}t > c$

Is it even possible for a physical quantity to be changing at a pace that's a (>1) multiple of the speed of light?

Inquisitive
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2 Answers2

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The comparison $E_\mathrm{max} > \lambda$ does not make sense. The two sides have different units, so you cannot compare them.

The only way for the time-derivative of a quantity $Q$ to be comparable to the speed of light is for $Q$ to have units of length.

Now some things can very well be faster than $c$, for example the velocity of the point made by a laser reflecting off a distant wall as you change the laser direction.

Community
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@Chris White gave you the example of the dot made by a laser reflecting on a surface. If you move the laser fast enough, and your reflecting surface is far enough, then the dot appears to be moving faster than the speed of light. Appears is the keyword here.

I believe there is a MinutePhysics video that explains this nicely. The idea is that physics forbids any sort of information to travel faster than the speed of light, and what you see when waving your laser around is only the image of the photons. The photons themselves, once emitted, travel in a straight line at the speed of light. If you were able to perceive them separately, you would actually see a dotted line on the surface, each dot representing the impact of a photon.

Without going into the unknown territory of faster-than-light particles (tachyons), an example of a physical example would be inflation. Inflation is a solution to the horizon problem, and requires that space-time itself expanded faster than the speed of light for a fraction of a second after the Big Bang.

Demosthene
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