Time Dilates When You’re Chasing Bunnies

We’re just starting out on a walk, and no sooner do I open the gate from the back yard than the dog takes off at a run, hitting the end of the leash and nearly pulling my arm out of the socket.

“Whoa, there,” I say. “Take it easy.”

“Come on,” she says, “We need to go fast! Let’s go, let’s go, letsgo!”

“What’s the hurry? It’s a nice day, there’s no rush.”

“We need to go fast. If we go fast, I’ll be younger than that annoying dog out back.”

“You know, I’m all in favor of exercise, but I think you’re overstating the benefits.”

“No, silly, it’s not about exercise. It’s physics. When I go fast, time slows down. It’s Simple Relativity.”

“The name is Special Relativity, actually.”

“Whatever. The point is, light bounces around in triangles, so time slows down, and I get younger. So let’s gooooo!!!”

“OK, you’re garbling the explanation a little. The reason time slows down is that the laws of physics are invariant under the Lorentz transformation between moving frames. The business with light clocks is just an illustration of the basic effect, a gedankenexperiment that Einstein used to explain things.”

“You just love busting out the big German words, don’t you?”

“You have to like a language where you can just keep gluing words together. Germans must play Scrabble on a board the size of a parking lot.” We’ve reached the corner of the street, and I stop to let a car pass. When we start going again, she races right to the end of the leash, and resumes pulling. “Anyway, the point is, it’s a particular thought experiment that demonstrates the general principle.”

“Right. Light, triangles, slow time. Let’s go. If you insist on explaining this, do it while we walk.”

“You’ve got to admit, it’s an ingenious idea. You can imagine making a ‘clock’ by bouncing a pulse of light back and forth between two mirrors, and recording a ‘tick’ every time it hit one of the two.”

“That’d be one fast clock.”

“Sure. If you put the two mirrors one meter apart, it’d tick roughly once every six nanoseconds, which is pretty fast. It would keep very regular time, though. And, anyway, any kind of clock you care to use would have to give you the same results.”

“Why is that?”

“Well, otherwise, you would be able to tell the difference between moving and standing still by comparing two different types of clocks, and that’s not allowed.”

“Oh, right. So what’s the deal with the triangles again?”

“Well, the point of the gedankenexperiment is that an observer at rest with respect to the clock will see the light bouncing straight up and down, say, covering a distance of twice the mirror separation between ‘ticks.’ A moving observer, though, will see the light taking a sort of zig-zag path.”

“Like that kid over there with the ball?” I look up ahead, and there’s a kid standing in a driveway bouncing a basketball. “He sees it going straight up and down, but if we go by really fast, it looks like it’s going sideways, too.” She redoubles her pulling on the leash, and almost yanks me off balance.

“Slow down, will you?” I haul on the leash, and she lets up a little. “Anyway, yes, that’s the basic idea. The ball looks like a series of parabolic arcs, where light would travel in straight lines, but in either case, the path seen by a moving observer is different. And that makes it take longer.”

“This is the bit with the triangles?”

“That’s right. In the light clock experiment, the moving observer sees the light moving along the hypotenuse of a right triangle, with one of the other two sides being the separation between the mirrors, and the third side being half the distance moved by the mirror between ticks. The path taken by the light is longer than it is according to the observer standing next to the clock. Which means the time between ticks is longer.”

“OK,” she says, “That’s the part I don’t quite get. Why does it take longer?”

“Well, the path is longer, but the speed of light is the same. The speed of light is determined by the laws of physics, and the laws of physics are the same for any observer. Otherwise, you’d be able to tell the difference between moving and standing still.”

“It always comes back to that, doesn’t it?”

“That’s why they call it relativity.”

But can’t you tell that you’re moving when you see the zig-zag path?”

“You know that you and the clock are moving relative to one another, but not which of you is moving in an absolute sense. You could be moving while they stand still, or you could be standing still while they move. It looks the same, either way.”

“Oh. That’s pretty weird.”

“You get used to it. Anyway, the point is, as you said, that moving clocks run slower than stationary ones. The faster the motion, the bigger the difference.”

“Right, so we need to go fast!”

“The thing is, ‘fast’ in this context means ‘at a speed comparable to the speed of light.’ Even at our current exceedingly brisk walking pace, we’re not going more than a few meters per second. Which means you would need to walk for a billion years to gain one second on that annoying dog out back.”

She stops dead. “A billion years?”

“A billion years.”

“That’s a long time.” She thinks for a minute. “I don’t think I can walk that long.”

“No, you can’t. So we might as well walk at a slower pace, all right?”

“I guess.” We start off down the street again, at a much more comfortable walking pace. “Of course, there’s another way we could do this, you know.”

“Really? What’s that?”

“I could go a lot faster if you would let me off this leash…”

1. #1 Dave Gill
May 19, 2008

“Germans must play Scrabble on a board the size of a parking lot.”

Love that…..

2. #2 Scott Spiegelberg
May 19, 2008

Are you sure you want to say “that’s not allowed”? That phrase implies a rule-maker who will penalize you if you do determine if you are moving or standing still. “that can’t happen” or “that’s impossible” seem to fit better, avoiding the anthropomorphizing of physical laws. One anthropomorphism per story is the limit.

3. #3 Odysseus
May 19, 2008

Just checked my Scrabble board, and it’s just 15×15. However, it would be easy to fill actually any given size with German words. And a good player can easily make use of two (or, in lucky cases, even three) triple-word squares using only one word, thus achieving ridiculously high scores. Yes, German definitely *is* the ultimate Scrabble language.

4. #4 Susan B.
May 19, 2008

One thing I always get confused about–in your example, time goes slower for the person moving relative to the two mirrors, so they would end up younger than a person standing by the mirrors. But by the same argument, since the mirrors are moving relative to the other person, shouldn’t time dilate for them as well, so that a person standing next to the mirrors ends up younger than the other person relative to whom they’re moving? This seems like a paradox to me, although I know there is an explanation.

5. #5 Dan Miller
May 19, 2008

I don’t know if it’s a harder concept or what (I am but a poor political science major), but I found this entry notably tougher to understand than previous items in the genre. It might have something to do with the Lorenz transformation reference.

May 19, 2008

Susan: One thing I always get confused about–in your example, time goes slower for the person moving relative to the two mirrors, so they would end up younger than a person standing by the mirrors. But by the same argument, since the mirrors are moving relative to the other person, shouldn’t time dilate for them as well, so that a person standing next to the mirrors ends up younger than the other person relative to whom they’re moving? This seems like a paradox to me, although I know there is an explanation.

That’s exactly right. A stationary observer looking at a moving clock thinks that the moving clock is running slow, while an observer moving with the clock thinks that the stationary observer’s clock is running slow. Weirder still, they’re both right.

The resolution of the paradox is that in order to compare “ages,” the two observers need to end up back in the same frame, and the act of moving between frames allows you to distinguish between the two. One of the observers accelerated and decelerated to switch frames (in the usual paradox, anyway), while the other did not, and that difference allows you to determine who is actually older.

It’s not that acceleration causes the difference in ages, but the fact that one accelerated and the other did not means that the situations are no longer perfectly symmetric, and that resolves the paradox.

I may try to write that up for a later post, if the dog is interested.

7. #7 Eric Lund
May 19, 2008

And a good player can easily make use of two (or, in lucky cases, even three) triple-word squares using only one word

If you play by standard rules (seven tiles in your rack at any time), you can, with luck, use two triple word scores at once, but (barring enormous amounts of luck) never three. It takes an 8-letter word to use two triple word squares: all seven from your rack plus one already on the board. To use three triple word scores you need 8 tiles already on the board, with an opponent who doesn’t grab the chance to use one (or two) of the triple word scores while you’re setting this up. Of course, you could play compound words one component at a time, but you don’t get to count the two triple word scores you’ve already counted when you cover the third.

The post is a nice simple explanation of the concept, though.

8. #8 CCPhysicist
May 19, 2008

Emmy might want to look at the Physics FAQ
with particular emphasis on the Spacetime Diagram explanation and the Too Many Explanations explanation (after reading the Introduction, of course).

The spacetime diagram clearly shows that one path is “longer” than the other (which results in a shorter elapsed time because of the weird metric of spacetime), while the last one includes diagrams that allow you to see the symmetry (you are slow, no you are slow) that is there in the Doppler shift explanation part for the first half of the trip … before the turnaround point.

9. #9 Roman Werpachowski
May 19, 2008

“Are you sure you want to say “that’s not allowed”? That phrase implies a rule-maker who will penalize you if you do determine if you are moving or standing still. “that can’t happen” or “that’s impossible” seem to fit better, avoiding the anthropomorphizing of physical laws. One anthropomorphism per story is the limit.”

But saying “that can’t happen” is, on the other hand, a very objectivist position.

10. #10 Scott Spiegelberg
May 19, 2008

Since Chad had already explained why “that can’t happen” in a previous post/chapter (I’m sensing a sequel in the making), it is objective. At least from my frame of observation.

11. #11 Lynn
May 20, 2008

I *LOVE* relativity!!!! It’s one of the few topics in physics that I understand and most people don’t!!! Hehe I may have to read this blog more often!!!

12. #12 Brandon
May 20, 2008

Okay, here’s another question. Two people get into spaceships and fly in opposite directions so that each is moving 0.5c from the other. After ten years or so, they each e-mail each other a picture of themselves. Many, many years later, they will receive the picture the other guy e-mailed. Whose picture will have more wrinkles? After the initial take-off, neither one is accelerating, so both guys claim they were standing still while the other guy was moving at 0.5c.