This week, we encountered what, in my opinion, is the most fundamental aspect of special relativity: the loss of absolute simultaneity. The Michelson-Morley experiment established that light’s speed must be invariant. An immediate consequence of this is that two events that are simultaneous in one inertial frame cannot be simultaneous in any other frame. Last week I did a demo with a stick on a rolling cart (like a large skateboard) and an LED in the center. Students seemed to readily accept that in the frame in which the stick on the cart is at rest, mirrors at either end of the stick catch light from the center simultaneously but this doesn’t happen in a frame in which the stick and cart are moving parallel to the stick’s length. The trailing mirror will always catch its light before the leading mirror catches its light. Students seemed to need no special convincing to accept this. I don’t know whether this is significant, but I noticed it nonetheless.
So I recalled that demonstration and then had students use the manipulatives I’d given them. I asked them to write out an operational definition for “measuring a stick’s length” by comparison against a calibrated measuring stick. The idea is to lead them to see that if the stick is at rest, you’re really taking simultaneous position readings of both ends of the stick. We normally overlook this because if the stick is at rest, we don’t HAVE to take the readings simultaneously and thus we rarely do. Next we frame this within the context of the demo mentioned above. Instead of mirrors on the ends of the stick catching light, the simultaneous events are taking readings of the positions of the ends of the stick. Thus, students already anticipate that these events cannot be simultaneous in any frame in which the stick is moving in a direction parallel to its length. The next step is for students to model taking the position readings while the stick is moving. There are two options, each corresponding to the position of one end being recorded before the other. In terms of the stick’s direction of motion, the leading end can have its position measured first or the trailing end can have its position measured first. One way gives a greater numerical value for the stick’s length (compared to its proper length in the frame where it’s at rest) and the other way gives a lesser numerical value for the stick’s length. Students need to figure out which one is consistent with predictions from the simultaneity experiment. Thus, they arrive at the conclusion that the measurement process the previously operationally defined gives a shorter length for the moving stick than for the stationary stick. They have discovered length contraction.
Next, we do the more traditional light clock derivation and arrive at the consequence that the duration of a clock’s tick-tock in a frame where it is moving is always greater than the duration of that same clock’s tick-tock in a frame where it is stationary. They have discovered time dilation.
I really want to write all this up as a formal activity when time permits.
Now, I point out two very important, yet frequently overlooked, facts.
- Nothing happens to a stick while it’s moving to make its length change.
- Nothing happens to a clock while it’s moving to make the duration of its tick-tock change.
It’s nonsense to say that “moving sticks contract along the direction of motion.” It’s similarly nonsense to say that “moving clocks run slow.” Length contraction and time dilation are merely consequences of carrying out the operational definition of measurement in relatively moving frames of reference. It’s just that simple. If I can get students to understand this, then they understand special relativity in a deep way that is not usually imposed in introductory courses, and in a way that prevents the spouting of nonsense about weird things happening in motion.
Students spent the remaining time this week working questions from the Arons handout.
Next week, we will wrap up special relativity with an introduction to the Lorentz transformation as a way to describe events from different inertial frames.
I welcome questions, comments, feedback, and constructive criticism.