Thursday, October 18, 2007

Length contraction problems

Well, since no one left a comment, I’m assuming that no one had any thoughts the length problem (or, surprisingly, the Veyron). But as you might anticipate, the explanation is similar to the one for time dilation. So, if A says that B’s measuring stick is short, and B says that A’s measuring stick is short, who is right? Answer: they both are. It makes more sense if we think about how length is actually measured.

Let’s say, once again, that A is driving the Maserati, and we’d like to measure its length on the ground. If A happens to be parked, this is easy: we just lay our ruler on the ground, check the positions of the front and back bumpers, and subtract the two. No problem.

For that matter, it’s not hard to check the length of the car if it happens to be moving, either. Same procedure, but this time obviously we have to be sure to check the two positions at the same time. If you don’t, then clearly the car will move while you’re measuring and you’ll get the wrong answer.

Hmmm. Just maybe, this could be starting to sound a little familiar to you. The thing that bails us out of the paradox, once again, is the first consequence of relativity: two events that are simultaneous in one frame are not going to be simultaneous in another. We say we were careful to measure both points at the same time, but no matter how careful we are, the dude in the car is never going to agree. He’s going to complain that we read the front end of the Maserati first, and then read the back end after the car had already moved forward, so of course we got a number that was too small.

Well, A has always been something of a complainer, and we’re thinking he’s really just full of it, but it’s important to remember that neither of us is “actually” right. In his reference frame, it really is true that we measured the two points at different times. In our reference frame, we really did measure at the same time, and his car really is shorter. For that matter, someone moving in a third frame would tell us we were all wrong: they would claim that we mismeasured, and his car is shorter (exactly how short would depend on how fast they were going, but they’d get a different length than us).

Is there really a paradox hiding somewhere in length contraction? Once again, it sure feels like it, but I’ve been looking for years and I’ve never actually found one. Still, no one’s ever accused me of being too clever, so maybe you’ll have better luck. I encourage you to roll this one around in your heads for a while, see if you can come up with something that looks like a contradiction. To help get you started, this week’s problems involve a couple of entertaining “paradoxes” about length contraction. No worries; you won’t need to figure them out to answer the questions, but if you have any guesses on how to fix the paradox, leave me a comment. (Seriously. I get lonely out here in these French villages.)

Here are this week's problems:

and here are a couple of pictures to help you visualize the ladder paradox:


Liz said...

I'm confused. If something shrinks when it is moving, does it "grow" back to normal size when it stops moving?

Liz said...

Also, who is Ehrenfest?

marc2718 said...

That's exactly right; moving objects are shortened, but only until they stop moving (and as they slow down, they'll get closer and closer to their normal size). Unfortunately, the details are a little complicated because in slowing down, the object undergoes acceleration, so you can't use special relativity any more. But it does end up back at its normal size when it's completely stopped.

Paul Ehrenfest was a goofy looking guy who worked on physics from about the turn of the century until World War II. He mostly worked on statisical mechanics (a different branch of physics), but every now and again he'd publish papers with a sort of "puzzle" theme, like his relativity paradox.

Magalindan... said...

Okay, so that means that when something is accelerating, special relativity doesn't apply?

That makes sense because it should be at constant velocity and a non-accelerating frame.

What other kinds of "puzzles" did Ehrenfest publish? Or were they all the same kind of type of paradox?

Katie said...

That must be either a pretty big ladder, or a really tiny garage. If that ladder doesn't fit, where do they park their cars?

You know, they should try driving their car really fast and then slamming on the brakes inside their tiny garage in order to "shrink" the car and make it fit. That would be pretty entertaining.

Magalindan... said...

[Btw, I'm still Liz, though this name is more specific to me...]

marc2718 said...

Hi Magalindan,

Yeah, for acceleration you need general relativity. This is actually the key to "solving" Ehrenfest's paradox. (I'm about to post the solutions; check them out for details.)

Most of the other puzzles Ehrenfest posed were in stat mech and quantum mechanics. In particular he worked quite a bit with things called adiabatic invariants. His problems all had this sort of flavor: physics relies on the fact that fundamental processes are reversible, but in stat mech lots of things are irreversible, so how is it possible that stat mech is right?

I actually think he was on to something there. Never cared for stat mech myself.

marc2718 said...

Hi Katie,

Yeah, I guess it's just a huge ladder. 10 m is pretty reasonable for a garage; figure a typical Mercedes-Benz S-Class is a bit over 5 m. I don't see why they don't just buy a smaller ladder. Or one that folds up.

Of course, I can't believe they actually think their plan is going to work. If you send a ladder rocketing into a garage at near light speed, the ladder, the garage, and anything nearby is probably just going to blow up. But hey, it's not our problem.

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