Creating Time and Creating Space #7

by Whabby

All right, owls. Time’s a wasting (LOL), and we have very important business today. Last week, you and naked George (who, by popular demand, is now not only naked but erect as well) decided to measure the time for the light flash to travel up and down Shirley’s shaft while Shirley was traveling 150,000 kilometers per second (half the speed of light). This situation is captured in the illustration on the left side of the figure below. According to our calculations, from your stationary point of view on the ground, Shirley’s movement creates a Line B that’s 150,000 kilometers long. This forces the light pulse to have to climb up and down the sloping sides of the two adjoining right-angle triangles in the figure. From Pythagoras, the 150,000 kilometer Line B added about 36,000 extra kilometers to Line A, compared to the journey taken by the light pulse from the perspective of naked, erect George. Adding the same amount to Line A on the way back down (the hypotenuse of the mirror-image triangle), and 72,000 odd kilometers of extra distance would be added.

At the speed light travels, that extra distance should have added about 0.22 extra seconds to your base measurement of two seconds. Instead, though, your average measurement was 2.32 seconds, about a tenth of a second more than, at first glance, it should be.

Not a big deal, you say? Don’t count on it! To understand the significance of this seemingly tiny discrepancy, I need to pose a straightforward question to you. During the extra 0.22 seconds that the light pulse was taking to cover the extra 72,000 odd kilometers along the two diagonals (36,000 extra kilometers along each Line A), what was Shirley doing? Answer: She was continuing to move away at the same 150,000 kilometers per second! But, wouldn’t this constant movement stretch Line B, forcing it into getting longer? Indeed it would! Specifically, in the 0.118 seconds it takes the light pulse to cover the extra 36,000 kilometers along one diagonal (one Line A), Shirley moves about 17,000 kilometers further away. This stretches Line B from 150,000 kilometers, the length it is on the left side of the figure above, to 167,000 kilometers, the length it is on the right side of the figure. Of course, Shirley is moving at the same speed while the light flash is going up to the ceiling as she is when the flash is going down, so Line B stretches by the same amount on both of the mirror-image triangles.

From Pythagoras’ theorem, any time you add length to Line B, you’re adding length to Line A. Doing the math (I’ll spare you the actual numbers this time), lengthening Line B by 17,000 kilometers adds roughly 8000 kilometers to Line A; this is the source of the number 8 at top of Line A on the “uphill” triangle. But, hold on a minute! Isn’t your version of the light pulse going to have to take even more time to cover that additional 8000 kilometers? Sure. Since light travels so fast, its not much time, about 0.028 seconds to be precise (the number above the 8 in the figure). Still, even that little sliver of additional time is enough for Shirley to move an extra 4,000 kilometers to your right, which is of course also added directly to the length of Line B. I haven’t shown this in the figure, but 4000 extra kilometers along Line B lengthens the all-important Line A by a little under 2000 kilometers, which takes the light pulse an additional 0.0066 seconds to cover, which allows Shirley to move even a little further away… And so on.

See the pattern? Each additional increment in travel time for your light pulse creates an additional “window of opportunity” for Shirley to move even further away, and further lengthen Line B. From the Pythagoras Theorem, any increase in Line B has to donate additional length to Line A, which, in turn, takes additional time for the light pulse to cover. The bottom line: your spacecraft is now traveling fast enough to set up repeated cycles of space/time creation!

Of course, as we’ve also seen, these cycles of time/space creation shrink very rapidly. Within only about 18 of them (each one smaller than the last), the amount of additional distance along Line A is driven to virtually zero, choking off the opportunity for Shirley to move any further away before the light pulse reaches your fellow experimentalist, naked erect George. The whole space/time creation thing comes to a screeching halt and, even from your perspective, the light pulse reaches its destination, gets reflected by Shirley’s floor back to your eyes, and you hit your stopwatch.

Adding up all the additions to the length of Line B donated by Shirley’s movement from the point of departure (which corresponds to the time when she passed directly over you, and you and George both started your stopwatches), we get a final length of about 173,000 kilometers for Line B, which adds a total of about 46,000 kilometers to Line A. Multiplying 46,000 kilometers by two (to take into account the same additional distance added onto the light pulse’s journey after it was reflected off Shirley’s mirrored ceiling) we get a total increase in the distance your version of the light pulse has to travel, compared to George’s version, of 92,000 kilometers.

How much extra time does it take light to cover that much extra distance? You guessed it… approximately 0.32 extra seconds (actually, it’s .306666 seconds; I’m rounding to keep things straightforward).

Exactly the amount of additional time you measured on your stopwatch, compared to the two seconds measured by the steady, unfailing, naked, erect George!

We’re really starting to get somewhere, now, owls! The notion that, from your frame of reference, time and space mutually construct each other over successive cycles is pretty amazing. And make no mistake: these cycles of mutual creation are real. Suppose there was an odometer onboard Shirley that recorded the distance she covered from the point she passed over top of you to the point when the light pulse returned to the floor, two seconds later (as measured by George onboard). Since Shirley is traveling 150,000 kilometers per second, the odometer would read exactly 300,000 kilometers. In your frame of reference, though, when the light pulse hits the floor, Shirley has traveled almost 50,000 kilometers further. That’s a lot of extra distance out into space!

When you think about this, it raises all sorts of puzzles. To avoid thinking about them, you might be forgiven for saying to me: “What’s a few tens of thousands of kilometers, and a couple of extra tenths of a second, among friends”? True, there’s still not a dramatic discrepancy between your time measurements and those of George. Does that mean time dilation’s not that interesting, after all? Hold on, owls! Next week, we’re going to pull out all the stops, and crank up Shirley’s speed all the way to 299,999 kilometers per second, just one kilometer less than the speed of light itself.

This is going to do nothing to George’s measurements, of course (I mean his measurements of time, not his erection). It’ll be the same old boring two seconds for him. But. Anybody care to speculate on what time you’re going to record on your stopwatch?