The Mystery of the Missing Time

All right, faithful readers!  Time’s a wasting, and we have some very important business to take care of today. You and George have been independently measuring the time for the light flash to travel up and down Shirley’s shaft while Shirley is traveling 150,000 kilometers per second (half the speed of light).  I captured the geometry in the illustration in the last blog, and I show it again in the left side of the figure below.  From your stationary point of view on the ground, in one second Shirley’s movement creates a Line B that’s 150,000 kilometers long.  Her motion thus forces your version of the light pulse to have to climb up and down the hypotenuse of the two adjoining right-angle triangles.  From Pythagoras, when Line B is 150,000 kilometers long, it adds almost 36,000 extra kilometers to Line A.  Adding the same amount to Line A on the way back down (the hypotenuse of the mirror-image triangle), a total of just less than 72,000 kilometers of extra distance is created. 

At the speed that light travels, that extra distance should have added about 0.236 extra seconds to your base measurement of two seconds.  Instead, though, your average measurement was around 2.301 seconds, or about seven tenths of a second longer than it should be. 

Not a big deal, you say?  Don’t count on it!  To understand the significance of this seemingly trivial discrepancy, I’ll repeat the straightforward question I posed at the end of the previous blog.  During the extra 0.236 seconds that the light pulse takes to cover the almost 72,000 extra kilometers along the two diagonals, what is Shirley doing?  Answer: She’s continuing to move away from you at the same 150,000 kilometers per second! 

But wouldn’t this constant movement make Line B even longer? 

Indeed it would!  In the 0.118 seconds it takes the light pulse to cover the extra 36,000 or so kilometers up the first hypotenuse, Shirley moves about 17,000 kilometers further away, stretching Line B from 150,000 kilometers, the length it is in the triangles on the left side of the figure above, to 167,000 kilometers, the length in the triangles 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 her motion stretches Line B by the same amount while the light pulse is going in both directions).

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 details), 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 time above the “8” in the figure).  Still, even that little sliver of additional time is enough for Shirley to slide an extra 4,000 kilometers to your right, which is 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 that 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 distance along Line A increases the travel time for your light pulse.  That increase in travel time creates an additional temporal “window of opportunity” for Shirley to move even further away, and further lengthen Line B.  The bottom line: your spacecraft is now traveling fast enough to set up repeated cycles of space and time creation!

At the same time, though, these cycles are shrinking very rapidly. Within only about 18 cycles in total (each one smaller than the last), the amount of additional distance being added to Line A is driven to virtually zero, choking off any opportunity for Shirley to move further away.  The whole space/time creation thing comes to a screeching halt and, even from your perspective, the light pulse reaches the ceiling.  Then, the whole routine of space/time creation is repeated while the light pulse comes back down the shaft.  Eventually, even in your frame of reference, the pulse reaches the floor, gets reflected 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) to the return of the pulse to her floor yields 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 gives the 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 0.301 extra seconds (actually, it’s .306666 seconds; I’m rounding to keep things straightforward).  In other words, exactly the amount of additional time you measured on your stopwatch for the pulse to complete its journey, over and above the flat two seconds measured by George!

We’re really starting to get somewhere, now, readers! The notion that, from your frame of reference, time and space mutually construct each other over successive cycles is kind of amazing.  And make no mistake: these cycles are perfectly real.  Your stopwatch doesn’t lie about the extra time that the event has taken.  As for space, 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 onboard odometer would read exactly 300,000 kilometers.  In your frame of reference, though, by the time 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 really think about this, it raises all sorts of puzzles, which I’ll explore in future blogs. To avoid straining your brain right now, you might be forgiven for saying to me: “What’s a few tens of thousands of kilometers, and less than a tenth 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?  At this point, all I can say is: Hold on to your hats!  In the next blog, we’re going to pull out all the stops and crank up Shirley’s speed to 299,999 kilometers per second, just one kilometer less than the speed of light itself. 

299,999 kilometers per second is virtually twice as fast as Shirley was going in today’s blog.  That extra speed is going to do nothing to George’s measurements, of course.  It’ll be the same old boring two seconds for him.  However, anybody care to speculate what time you’re going to record on your stopwatch?