Sam Auyeung

Sam Auyeung

A Simple But Powerful Thought Experiment for Special Relativity

4 minute read

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Imagine two parallel straight copper rods set a centimeter apart that extend infinitely out in both directions and suppose there are positive unit charges spaced out evenly on the two rods. We’ll also suppose that the charges on one rod are lined up with charges on the other. You’re an observer some distance away and you take note that the rods want to repel each other because everything is positively charged; call this amount $\alpha$. We’ll suppose the rods are magical and can never be more than one centimeter apart. So they aren’t actually repelled.

Now, you start moving parallel to the direction that the rods run, eventually reaching a constant velocity. A moving charge is exactly what current is so from your point of view, you observe there to be current though your friend who is staying put with the rods, does not observe current. Because of this, you might form the following the conclusion.

1. Physical phenomena depend on your frame of reference.

Of course, you don’t need to imagine such a fancy situation to come up with this postulate. As you’re driving in a car with your friend, you know that people on the side of the road see your friend moving past them. But to you, your friend isn’t moving at all; they’re sitting right next to you the whole time. Since you and the people on the side of the road don’t share the same inertial frame of reference, this is why you observe different phenomena. (1) gets at the first postulate of special relativity though the correct version is sort of an inverse to this statement: 1’: Physical laws are unchanged in all inertial frames of reference; i.e. reference frames without acceleration.

So what’s the point of this parallel rods thought experiment? Maxwell’s equations tell us that that when there is a current (moving charge), it induces a magnetic field perpendicular to the motion of travel. Since the charges are all positive and appear to be moving at constant velocity as you move along, from your frame of reference, there should be a magnetic field. Moreover, because of the setup, the magnetic field should attract the rods towards each other; call this particular force $-\beta$. We said before that the rods can never be more than one centimeter apart because of magic but suppose this magic allows the rods to move closer together. You take a close look and conclude that the rods are not moving closer. What can you conclude from this? With some thought, you recall that light can be viewed as electromagnetic waves and after some math, you realize that the reason the rods are not coming closer together is ultimately because Maxwell’s Equations assume that the speed of light is constant. So you make the following postulate.

2. There is an exception to (1’): the speed of light will always be measured as the same in all frames of reference.

But now, this is very strange. You empirically verified that the rods aren’t being attracted towards each other as you move despite the fact that you observe an attracting magnetic field. In fact, you measure that the force is still repelling in the exact same amount $\alpha$ as when you consider the positive charges and you’re not moving with respect to the rods. So there has to be something that cancels out $-\beta$. What could it be? There aren’t any other entities around to introduce a third force. Hence, you conclude, against classical intuition, that the repelling force has increased from $\alpha$ to $\alpha+\beta$.

And how might that happen? One way is for the rods to get shorter and thus, the charge density on the rods increase. With an increase in density of the charge, the repelling force increases. Indeed, this is the correct solution and is called length contraction in special relativity. So from knowing Maxwell’s equations, it is possible to derive some consequences of special relativity.

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