Here’s a classroom activity intended to demonstrate the issue of simultaneity in measuring a stick’s length. Students need a calibrated metre stick (I’m trying to get into the habit of spelling it that way), another stick approximately 1/3 m long although the precise length is unimportant, two coins of the same denomination or two small pea-size balls of sticky clay or something similar, and a smooth tabletop. All references to “the stick” are to the second, smaller stick.
First, you students must establish an operational definition of what it means to measure a stick’s length.
1) Place the metre stick along the table’s edge, place the other stick on the table anywhere along, and parallel to, the metre stick and then write down an operational definition for “measuring the stick’s length.” Use this operational definition to measure the stick’s length and record it with an appropriate unit. Write your operational definition on a whiteboard, and label is as Definition I.
Answers varied more widely than I ever expected in today’s class, but eventually everyone should settle on something like “record the numbers from the metre stick marking the ends of the other stick and simply subtract these two numbers so as to get a positive number and this number is the stick’s length.” There are other ways of saying it of course, but all should be equivalent to this simple articulation.
Now, there is a hidden assumption in the above operational definition that we now need students to be lead to see.
2) Hold the stick with both hands such that the fingertips hold one coin (or clay ball) under each tip. Dropping a coin (or clay ball) onto the metre stick constitutes “taking a reading” or “recording the number” and of course we’re ignoring the duration of the actual drop. Move the stick slowly from left to right, making sure both ends stay within the ends of the metre stick. At a given moment, drop the left coin (or clay ball). Then after one or two seconds, drop the right coin (or clay ball). Using the operational definition articulated above, measure the stick’s length and record it with an appropriate unit.
If you ask students how the two measurements compare, they should find that the second one is greater than the first one. This may not seem significant to them, but ask them to consider which one is “real.” Most students at this point will say that the first measurement is the “real” one because obviously the stick couldn’t have simply lengthened because it’s the same length it always was.
3) Repeat step 2, but this time drop the right coin (or clay ball) first. Then after one or two seconds, drop the left coin (or clay ball). Using the operational definition articulated above, measure the stick’s length and record it with an appropriate unit.
If you ask students how this third measurements compares to the original one, they should find that it’s less. Again, students will say that this isn’t the stick’s “real” length because the stick obviously hasn’t shrunk. In a sense, they’re more correct than they realize. Students have stumbled onto the fact that there is something significant, as opposed to “real,” about the measurement from step 1.
Note the wording of the next step.
4) Repeat step 3, but this time drop the left and right coins (or clay balls) in whatever way is necessary to replicate the measurement from step 1. The stick must be moving!
Eventually, students will realize they must drop the coins (or clay balls) simultaneously or at least as simultaneously as possible given the situation. This is the assumption missing from the initial operational definition, which they now must refine.
5) Revise your operational definition in step 1 to explicitly include this new finding. Record your revised operational definition on a whiteboard, label it as Definition II.
Students should now see that simultaneity play a role in measuring what we have called the stick’s “real” length. Now, they must also be explicitly lead, by facilitating a class discussion, that all of these measurements are as “real” as they can be, and are as “real” as the walls of their classroom.
Now for a very important question.
6) Which definition will give a result that could give different results in different situations, depending on how fast the stick is moving and how great the duration between dropping the coins (or clay balls)?
Students should, perhaps after discussion, unanimously agree that it’s Definition I.
7) Which definition will give a result that will always be the same regardless of how great the duration between dropping the coins (or clay balls)?
Students should, perhaps after discussion, unanimously agree that it’s Definition II. Interestingly, one of my students tried hard several times to home in on this very question, but he tried to articulate it as an issue of whether or not there was something material between the two points marked on the metre stick. In other words, he was asking of “length” depended on something material, like a wooden stick, existing in the space between the two endpoints. His reasoning seemed to be that the presence of a material stick somehow made the measurement more “real” than otherwise. Finally, he articulated his question as in step 7.
Now another important connection.
8) What does step 7 imply about the stick’s motion?
Students should realize that step 7 is equivalent to saying the stick is stationary relative to the observer. Don’t go any further until this sinks in, and it may indeed take a while.
9) Now that students have an operational definition of “measuring a stick’s length” that always gives the same result, we can, in the spirit of Arons, give this idea a name: proper length. Because all observers will agree on the stick’s proper length, we call it an invariant.
My sense is that what students have been conditioned to call “the length” of a stick is what we call the proper length. They all seemed to collectively say “AHA!” at this notion.
There’s an even more important lesson here, and that is once we have a mutually agreed to operational definition of something (in this case, measuring a stick’s length), we must agree to use that definition even if it gives results that don’t match our intuition. I think this is the deepest lesson in this activity. If our definition is good, then we can’t abandon it suddenly when we’re faced with apparent inconsistencies. Instead, we may need to retune our intuition and accept the inconsistencies. In a way, this is what special relativity is all about. Our classical conceptions of space and time were so deeply ingrained that abandoning, or even modifying them, seemed out of the question.
This activity could be far less structured. You could define “measuring a stick’s length” as “subtracting the two numbers onto which the coins (or clay balls) fall” and then ask students to perform measurements that give results greater than and less than what they get if the stick is stationary. Then ask them to discuss the implications of these results.
Okay, now for the big finish.
10) What really happened to the stick as its moving length was measured?