Why can’t I go faster than the speed of light? Hints from electrodynamics

I should state from the very beginning that the title of this post is a bit of a tease.  I don’t know why nothing can go faster than the speed of light.  No one does.  It is a fundamental postulate of the theory of relativity: “The speed of light in a vaccum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.”  It is a weird assumption, and it leads to even weirder consequences.  It also happens to stand up to every conceivable test we have thought to put it through.  But it seems so strange and arbitrary; how did Einstein think of it?

It is a little-known fact that Einstein’s ground-breaking paper about relativity was called “On the Electrodynamics of Moving Bodies.”  It is even less well-known that the rule “nothing moves faster than the speed of light” is a consequence of the laws of electricity and magnetism, which had been known since the mid 1800’s.  Einstein was just the first scientist with enough imagination to realize how important this consequence was, and how generally it could be applied.

In this post, I’ll try to explain how simple rules for electric and magnetic fields demonstrate that all velocities must be less than the speed of light.  It can be done by imagining the consequences of a simple experiment, where a charged particle is watched by two different observers.  The demonstration done here doesn’t really explain why the speed of light is an absolute “speed limit”; it just shows that the question “how come nothing can go faster than the speed of light?” is equivalent to the question “how come the laws of electricity and magnetism look the way they do?”.  And even if this isn’t a real answer, there’s something profoundly satisfying about realizing how two hard questions are just different ways of looking at the same thing.

Imagine a very long, thin, rod that has been loaded up with positive charge.  The rod creates an electric field which pushes other positive charges away from it.  So, if you were watching as someone placed a small object with positive charge q next to the rod, you would see that it was repelled with some force .  You can solve directly for the strength of the force, if you want, using Gauss’s Law: , where Q is the amount of charge on the rod, L is its length, r is the distance between the rod and the object, and is a fundamental constant of nature called the “vacuum permittivity”.

If the rod is moving, however, it also creates a magnetic field.  This is also a well-known property of electricity and magnetism: moving charges create magnetic fields.  The shape of the magnetic field is circular, curling around the rod as it moves.  If some charged object is moving next to the rod, it will respond to the rod’s magnetic field.  If you assume that the rod and the object are moving parallel to each other with the same speed v, you can work out the strength of this “magnetic force” using Faraday’s Law: , where is another fundamental constant called the “vacuum permeability”.  The negative sign is there because the magnetic force works in the opposite direction as the electric force.  It pulls the object toward the rod.

Here we see the beginnings of a paradox.  If the rod is standing still next to a stationary charge, there is an electric force repelling the two.  If the rod and the charge are both moving, however, there are two forces: an electric force repelling them and a magnetic force pulling them together.  How is that possible?

To make the paradox more apparent, consider the picture below.  One observer, named A, watches as the rod and the charged object whisk by him at speed v.  Another observer, named B, moves alongside the rod and the object with the same speed v.  Because A sees a moving rod, he observes an electric and magnetic field emanating from it, and consequently sees the object being pulled by two competing forces and .  From B’s perspective, however, nothing is moving at all.  Neither the rod nor the charged object are getting any closer or farther from him.  Therefore, he sees that the rod creates an electric field pushing the object away, but no magnetic field.  For him, then, there is only one force .

A thought experiment with moving charges. Observer A sees electric field (black lines) and magnetic field (red lines) emanating from the rod. B sees only electric field.

The situation seems strange.  How can two different things be happening depending on how fast you’re moving when you look at it?  Maybe something subtle is happening, but one thing is clear: there is no way that A should see the object move upwards while B sees it move down.  If A sees that the total force on the object is upwards (), then he will also see the object move upward, closer to the rod.  But B sees only one force, pushing the object downward.  This is a major problem.  One universe cannot accomodate both outcomes.

So, it must mean that the magnetic force is always weaker than the electric force , regardless of the velocity.  Since we have the expressions for and , we can check when this is true:

.

The quantity meters per second is exactly the speed of light.  So why can’t these two charged objects move faster than the speed of light?  Because if they did, then there would be two completely different realities: one for everyone who wasn’t moving, where the object got pulled up, and another for everyone who moved with the object, where it was pushed down.

So, by imposing the “speed limit” of meters per second, we have avoided a universe-splitting catastrophe.  But you might notice that there is still a problem.  Even if v is less than the speed of light, B still sees a stronger downward force than A sees, which means that B will watch the object move downward more quickly.  How is that possible?  Einstein realized this problem too, and to solve it he came up with a drastic fix: time dilation.  The only way to resolve the paradox is if time is moving at different rates for the two observers.  A must observe B to be “living in slow motion”.  That way, A sees the object falling slowly next to a person living in slow motion, which leads him to conclude “B must see that object falling faster than I do”.  Which he does, although from his perspective it is for very different reasons.