By the way, as a quick note, I should probably mention that simultaneity, and all the other consequences of special relativity, are not a matter of one observer or another “getting it wrong”. When I say that event *B* comes before *A* according to one observer, but another observer says that *A* comes before *B*, it’s not that one of them has made a mistake. In the first observer’s frame, *B really does* come before *A*, and in the other observer’s frame, it* really doesn’t*. It’s weird.

I mention this because sooner or later one of you is going to read a bad relativity book, and it’s going to say something about one observer not accounting for the travel time of light, or something. But that has nothing to do with relativity. If I take a measurement of something, I have to be clever enough to subtract out any effect due to the signal taking time to reach me. Actually, if I’m really clever, I’ll get a bunch of low-ranking grad students and space them at intervals with a stopwatch and a ruler so they can just write down measurements and bring them back to me. It’s like when you see lightning and then later you hear the thunder, you might incorrectly believe that they didn’t come from the same source, because you didn’t account for their different travel times. But this isn’t relativity, this is just making a mistake.

Anyway, the weirdness this week is about time dilation. Simply put, time dilation says this:*In a moving frame, time runs slow.*

How much slower? That’s the subject of this week’s question. (By the way, if you happen to get stuck, you can always write me a comment; I’m happy to help.)

Let’s say Bob hops on a spaceship that goes rocketing by the Earth at near the speed of light. Just as he passes by, Alice peeks in the window. If right at that moment Bob is, I dunno, say playing a game of pool, then Alice sees it as though it were happening in slow motion. If Bob jumps the cueball, it seems to float slowly through the air; if he sinks a ball, it drifts gently downwards to the bottom of the pocket; and a blast break doesn’t look nearly as impressive when Bob’s traveling at 90% of the speed of light.

Of course, if you ask Bob about all this, he’ll say he’s just moving at normal speed. As far as he’s concerned, his spaceship is completely stationary, and it’s the

*Earth*that’s rocketing by in the

*other*direction. Hmm…

With any luck, at least a couple of you are scratching your heads at this point. I just told you that time runs slow in a moving frame, so Bob is moving slow according to Alice because Bob is moving with respect to her. But by the same argument,

*Alice*must be moving slow according to

*Bob*, because in Bob’s reference frame (the spaceship) it’s Alice who’s doing the moving.

So what’s the deal? Common sense says that if Bob looks slow to Alice, then Alice must look fast to Bob, right? If they both pull out stop watches and start them just as they pass by, whose watch reaches one minute first? Is it possible that one of them is wrong? If so, which one? I’ll try to unstick this one for you next week, but in the meantime, if you have any opinions or guesses, leave me a comment. (Christos, if you’re reading this blog, you’re not allowed to give away the answer.)

Problem on time dilation: https://mywebspace.wisc.edu/mweinberg/web/TimeDilation.pdf

## 4 comments:

So, this is probably sleep deprivation talking, but I'm fairly sure that you said that you can call any frame of reference stationary in the absence of acceleration. I'm going to guess that the rocket is not accelerating, but simply moving extraordinarily fast past the Earth. If we say the Earth is stationary, like Alice does, Bob looks fast. If we say the rocket is stationary, like Bob could, Alice looks fast. I'm guessing this means their points of view are interchangeable, and that their watches would reach one minute at the same time.

Do I win a cookie?

Wait, no! If each one thinks the other is slow, each one would think the other's clock is slow. For Alice, her watch would reach a minute first, and for Bob, his would.

Now I get a cookie!

Absolutely, I agree that both the Earth and the rocket must be in inertial frames (neither is accelerating), or they couldn't claim to be stationary. (In fact, any time I talk about a "frame", I'll always mean an inertial frame.)

As for Alice and Bob, you're definitely on the right track, but I'm not sure you've convinced me yet. For example, what do you mean by "for Alice" and "for Bob"? Is it possible that *both* their watches reach one minute first?

Please inform Rebecca that I said she can give you a cookie.

I guess I'll have to be satisfied with the cookie

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