Tuesday, September 18, 2007

The twin "paradox"

Okay, so I promised last week that I’d try to clear up the confusion about Alice and Bob. Alice says time is running normally on Earth, but is slowed down on Bob’s spaceship. Meanwhile, Bob is claiming that time is running normally on his spaceship, and it’s on Earth that time is slowed down. Which one is right? You’re not going to like it, but it turns out they’re both right.

What gives? Well, let’s be a little clearer about what we’re saying. To know that time is moving slow, you’d have to check the time difference between two separate events. Suppose Alice and Bob agree that right when they pass each other, they’ll both hold up big digital stopwatches, and reset them to 0 just as they go by. Then they’ll simply watch each other through high powered binoculars and compare how long it takes for each of their stopwatches to get to 1 minute. Presumably, someone’s watch is going to reach 1 minute first.

Let’s tell the story from Alice’s point of view. Right as Bob zooms by, she sees him set his watch to 0, and she does the same. Of course, time is moving slow on his spaceship, so for every second that ticks by on her watch, only half a second goes by on his. Finally, her watch reaches one minute exactly, and looking through the binoculars, she observes triumphantly that his watch only shows 30 seconds.*** (See "Optional note" below.) Now, how to prove to Bob that her watch got to 1 minute first? Simple! She’ll just take a picture of her stopwatch and sent it to him in a light signal. Except she quickly realizes that there’s a problem. The light races away from her at the speed c, but Bob’s spaceship is going pretty close to that speed anyway, so the light is only catching up very slowly. In fact, the ship is going so fast that the light signal ends up taking 3 whole minutes to get to him! By the time her picture gets to him, his watch (which is running at half speed), shows 2 minutes have elapsed, and he incorrectly thinks it’s her watch that’s running at half speed!

That’s not how Bob would tell the story at all. He agrees that as the Earth went shooting by they both set their watches, but since it’s the Earth that’s moving, Alice’s watch is only running at half speed. He’s looking through his binoculars, and he observes that right when his watch reads 2 minutes, hers only reads 1. In fact, just to prove it, she takes a picture and sends it to him! Of course, he’s not moving at all, so the light reaches him very quickly (since, of course, it’s traveling at the speed c while he’s completely stationary), and confirms what he already knew: that it’s Alice’s watch that’s running at half speed.

Okay, so that didn’t work out quite the way Alice planned. Note that she’s perfectly willing to admit that when her picture reached Bob, his watch showed 2 minutes, she just says it’s because the picture took so long to catch up to him. The culprit is the second postulate of relativity, which says that light travels at the speed c according to all observers. They both agreed on how fast the signal was moving; what they couldn’t agree on was how fast Bob was moving.

But Alice is quite clever, and she’s got another idea. Forget about sending messages; that’s too complicated. Instead, she gets her good friend Cassie to stand one light-minute away with a stopwatch of her own. Then when Bob goes by Alice, all three of them will start their watches at the same time, and when Bob’s spaceship gets to Cassie she can just hold up the stopwatch and show him it displays a longer time than his, proving his watch is running slow. What happens this time?

Well, no doubt about it, as Bob zooms by Cassie, her stopwatch shows 1 minute, and Bob’s only shows 30 seconds. Does that mean that Bob’s watch really is the one that’s running slow? Not at all! I claimed that Alice and Cassie started their watches at the same time. Put another way, I claimed they started their watches simultaneously. But of course, we already know that two events that are simultaneous in one frame are not in another! So Alice and Cassie may say they’ve started their watches at the same time, but according to Bob, Cassie made a rookie mistake and started her watch too early, which is the only reason her watch shows more time than his does.

Okay, well, if you’re anything like me, just these two examples won’t convince you that there’s not a legitimate time paradox here. I’ve just covered two possible ways you might show there’s a contradiction, and in both cases there’s a slippery way out. But maybe you’re not satisfied; it sure feels like there’s a problem somewhere. If you’re not yet convinced, I’d be interested to hear your take on the so-called “twin paradox”, which goes like this:

On her 21st birthday, an astronaut takes off in a spaceship at near the speed of light. After 5 years have elapsed on her watch, she turns around and heads back at the same speed to rejoin her twin brother, who stayed at home. The traveling twin has aged 10 years (5 years out, 5 years back), so she arrives at home just in time to celebrate her 31st birthday. However, as viewed from Earth, her clock has been running slow, so her twin brother will now be much older than she is!

But what happens when you try to tell this story from the point of view of the traveling twin? She sees the Earth fly off at near the speed of light, turn around after 5 years, and return. From her point of view, it would seem, she’s at rest, whereas her brother is in motion, and hence it is he who should be younger at the reunion. So which one is really younger when they meet up, and why? For the answer you’ll have to wait until next week, but in the meantime you can entertain yourselves with these questions.


*** Optional note: I’m being very slightly dishonest here, because I didn’t mention that Alice has to account for the travel time of light. What Alice would actually see, looking through the binoculars, is that Bob’s clock would read less than 30 seconds, and she’d have to figure out that at that moment his clock actually read 30 seconds, but the light hadn’t reached her yet. I left it out because I didn’t want the story to be any more confusing than it had to be. If this is too much, just forget about it; it doesn’t make any difference to the story. As long as you’re reading, though, let me remind you that Bob’s clock really is running slow (in the reference frame of the Earth), and it’s not just a trick of observation.


Anonymous said...

Wait. The astronaut can't be moving at the speed of light all the time. He has to turn around. Does that mean the twin on Earth ages while the astronaut is accelerating to c and slowing to rest from c? I know the paradox is resolved somehow with the way the astronaut isn't always moving at the speed of light, but I don't know how.

Anonymous said...

Also, I'm still pretty fuzzy on Alice and Bob. If Alice's watch hits a minute first, like she says it does, why does proving it matter? Why should the travel time of light come into it at all?

marc2718 said...

Well, let me start with Alice and Bob. In Alice's rest frame, her watch reaches 1 minute first, but in Bob's rest frame, his watch does. So, supposing that she wants to show that Bob's watch is "actually" slow, she'd need to prove it somehow (or Bob would just come back from his trip and claim that it was her watch running slow). The point of the story is that it turns out there's *no way* she can prove it, because each of them is right, within their rest frame.

As for the astronaut, I definitely think you're on to something, but I don't think we need to worry about her speeding up or slowing down. If it bothers you, let's assume she was already up to speed when she zooms by the Earth and they just synch their watches like Alice and Bob. Then let's say when the astronaut has traveled for 5 years, she just hops on a rocket already going the other direction, so there's no real speeding up or slowing down. (By the way, remember the astronaut can't go at the speed of light, only near to it.)

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