To the Moon, Alice! And beyond!

In the third (and final) version of our little thought experiment, we’re going to add a few “dilithium crystals" to Shirley’s engines so that we can crank her up to 299,999 kilometers per second, just one kilometer below the speed of light.  By now, you’re familiar with the drill.  At the exact instant Shirley passes over you, George flicks the searchlight on and off.  Both you and George measure the time it takes for the light pulse to travel up and down Shirley’s shaft.  You record your times, and repeat the exercise over and over, until you both get a stable average.  Luckily, your stopwatches are smart, so they help by automatically subtracting your brain processing time, and in addition, your stopwatch automatically subtracts the time for the pulse to travel from Shirley’s transparent floor back to your eyes.

When Shirley was going “only” 150,000 kilometers a second, your stopwatch registered about three tenths of a second more for the light pulse to complete its journey than George’s did.  Now, with Shirley moving almost twice as fast, a perfectly reasonable conjecture would be that your stopwatch should register a delay that’s about twice as big, or something in the neighborhood of 0.6 seconds.  Consequently, when you and George start the first measurement, you go in half expecting to see the light pulse still arriving only a little later than he does.

But I’m afraid that doesn’t happen. You don’t see the flash after 2.6 seconds.  You don’t see it after twenty-six seconds.  You don’t even see it after two hundred and fifty seconds!  As the time keeps mounting, you begin to panic, worried that you’ve somehow missed the flash and spoiled the experiment.  Just as you’re ready to call the whole thing quits, lo and behold, the flash finally arrives!  Breathlessly, you look down at your stopwatch, only to see that seven hundred and seventy-five seconds have passed.  Almost 13 minutes! 

13 minutes?  For the first time since you started this whole crazy time measurement gig, you scarcely believe the outcome of your own experiment.  How can a two second event be taking 13 whole minutes?  Your sense of disbelief persists as you and George perform the measurement again and again, only to confirm: two seconds for him; hundreds of seconds for you.

To make sense of this result, we have to go back to – where else – Pythagoras. In the one second it takes the light flash to go from the searchlight to Shirley’s mirrored ceiling, in George’s world, Shirley has been racing away along the line stretching off to your right at 299,999 kilometers per second.  Geometrically, Shirley’s movement has constructed a Line B that is 299,999 kilometers long.  Eager to find out how much distance 299,999 kilometers adds to the length of the all-important Line A (the line that traces out the path the light pulse takes from your perspective), you frantically crunch through the familiar Pythagorean calculations.  Once again, I’ll spare you the numerical details: Line A turns out to be 424,263 kilometers long.  That’s fully 124,263 kilometers longer than Line C, which is equivalent to the path that the pulse takes for George.  The resulting triangle is shown in the figure right below.

After one second of George’s time, his light pulse has covered 300,000 kilometers, enough distance to reach the ceiling.  After one second of your time, the light pulse has also covered 300,000 kilometers (light travels at the same speed for everybody), but since it has to travel up the diagonal of the right-angle triangle, it has 124,263 extra kilometers still to cover. 

That’s a lot of extra distance: About 10 times the diameter of the Earth, in fact.  Even traveling at 300,000 kilometers per second, 124,263 kilometers is going to take the pulse almost half a second - 0 .414 seconds to be exact - to traverse.  In 0.414 seconds, though, Shirley, zipping along at that blinding 299,999 kilometers per second, moves an additional 124,262 kilometers down the line (0.414 times 299,999), extending Line B by the same amount.  By the Pythagorean theorem, that, in turn, lengthens Line A by 95,000 kilometers or so. 

Just as it was when Shirley was going 150,000 km per second, successive cycles of space/time creation have begun.  The extra time needed for your light pulse to cover that additional 95,000 kilometers along Line A is about 0.317 seconds.  But in 0.317 seconds, Shirley moves 80,000 kilometers further along.  These numbers are pretty big. Moreover, exactly the same-sized cycles will be created by Shirley’s movement during the one second of George’s time that the light pulse is traveling back down Shirley’s shaft, stretching Line A for the mirror-image triangle by identical amounts (I’ve illustrated that in the figure above, too).   So, it’s already clear that the time you’re going to measure for the light flash to reach Shirley’s floor will be larger - quite a bit larger – than when Shirley was traveling at “just” half the speed of light.

Still, just as we saw at that slower speed, the numbers in each cycle are shrinking rapidly.   You would be forgiven for thinking that the cycles would, again, collapse fairly quickly to insignificance.  However, you’d be wrong!   As the cycles start to pile on, an important aspect of the geometry of the situation begins to exert an ever-increasing influence.  To understand that aspect, I’m afraid we need to revisit the details of the Pythagorean theorem (yeah, I know.  Back to math.  Sorry about that).

Recall that according to the theorem (shown again above), to get the length of Line A, you first square the lengths of Line B and Line C.  Then, you add the squared values together.  Finally, you take the square root of the sum.  Remember when Shirley was moving only one kilometer per second, and I talked about the relative contributions that Line B and Line C make to the overall length of Line A?  Remember how I took pains to emphasize that squaring the lengths of Lines B and C, and then adding them together, has the effect of inflating any initial difference between them when it comes to how much they donate to the length of Line A?  Simply stated, when the lengths of Lines B and C are very discrepant, with one relatively long and the other relatively short, squaring the numbers before adding them together ensures that the length of Line A is almost completely controlled by the length of the longer side. 

 The figure to the right once again illustrates the geometry of the situation back when Shirley was moving only one kilometer per second (slow relative to the speed of light) making Line B vastly shorter than Line C.   With Line C so much more in control of the length of Line A than Line B, the math “forced” Line A to be virtually the same length as Line C.  Remember, too, that I took some time out to illustrate the case of a triangle where Lines B and C are of equal lengths?  In that case, their squared values are equal too, and the fact that you simply sum their squared values means that they start donating equal amounts of their own lengths to the length of Line A.  

That’s the situation we’re in right now.  With Shirley moving at 299,999 kilometers per second, after just one second Line B and Line C are virtually the same length (Line B is just one kilometer shorter).  Consequently, Line C is supplying just a tiny trifle more than 71% of its length to Line A, and Line B just a tiny trifle less.

In the very next cycle of time/space creation, though, Line B increases by another 124,262 kilometers, making it considerably longer than Line C.  As the space/time creation cycles pile on, Line B grows ever longer, while Line C (of course) stays the same.  Consequently, it’s Line B that begins to control a larger and larger proportion of the overall length of Line A.   Critically, this means that Line B also starts to donate a larger and larger proportion of the portion of its length that was just added in the latest space/time creation cycle.  

The top triangle of the three triangles in the figure below illustrates how the situation has evolved by the 20^th cycle of space/time creation, close to the number of cycles that completely “closed things out” when Shirley traveled at only 150,000 kilometers per second.  Shirley has moved almost 1.5 million kilometers down the line, making Line B almost five times as long as Line C.   In the very next (21^st) cycle, illustrated in the middle triangle of the figure, Shirley moves an additional 30,941 kilometers down the line.  Crucially, with the big (and growing) imbalance between the lengths of Line B and Line C, the vast majority of that additional length is donated directly to Line A.  Specifically, Line A lengthens by about 30,303 kilometers, only a fraction less than the increase in Line B itself.

It takes just slightly more than a tenth of a second for the light flash to cover that extra 30,000 or so kilometers.  In that amount of time, Shirley moves another 29,999 or so kilometers away, and again, virtually all of that additional length is donated to Line A. As the cycles of space/time creation continue to mount, and Line B grows ever longer compared to Line C, the proportion of the additional length of Line B that’s donated to A grows every larger too, until they’re virtually identical.  The result is illustrated in the bottom triangle of the figure above by the relative spacing between the large number of light pulses along the diagonals (those spacings are obviously not completely accurate; they are meant to just illustrate the general point).  Eventually, things almost settle into perfect equilibrium, where on each cycle, almost as much new distance is added to Line A as was added on the immediately previous cycle.  As a result, on each cycle almost as much new space is created for the light pulse to have to cover in the future as it just covered in the past.   Even though it’s traveling at 300,000 kilometers per second, the light pulse makes very little headway in its journey toward the top of the triangle (Shirley’s mirrored ceiling), or (after one second of George’s time) in its journey back down to the floor. 

Luckily for your patience, though, the amount of Line B that gets donated to Line A never - quite - equalizes.  On every successive cycle, the temporal window of opportunity for Shirley to move further down the line grows just a little bit smaller, and the old girl covers just slightly less additional distance compared to the cycle before. 

Eventually, the cycle IS choked off, and the light pulse DOES reach the ceiling (floor).   As shown by the pile-up of light pulses toward the end, though, it takes many, many, many extra cycles for that to happen.  The final Line B distance says it all: by the time the pulse reaches the ceiling, Line B has stretched to over 116 million kilometers, and by the time the pulse reaches the floor, Shirley has covered twice that distance, for a total of 232 million kilometers (in your frame of reference), an impressively large fraction of the distance to Jupiter! 

Compared to that colossal distance, the length of line C, which remains fixed at 300,000 kilometers, pales into insignificance.  This is why the shape of the triangle is so squashed, and why Line A is now almost identical in length to Line B (of course, the true triangle is a great deal more squashed than this figure can do justice).

Let’s pause here to take stock.  We have exactly the opposite geometry of the situation we found when the spacecraft was traveling at only one kilometer per second, and it was Line B whose length was insignificant compared to Line C.  See, now, why time dilation is magnified by such a colossal extent when the spacecraft gets very close to the speed of light?  Line B has an opportunity to get so long compared to C that virtually all of the length of Line B is donated to A; thus, as Line B grows, so grows Line A, creating ever more distance for your version of the light flash to have to cover.

And, of course, light always takes more time to cover more distance.

With this description, we’ve virtually finished the quest to understand the “time” in time dilation!  The key to the entire phenomenon lies in how quickly the spacecraft is traveling, and how quickly the light pulse can reach the floor of the spacecraft.  If the pulse can do so in a relatively small number of extra cycles of space/time creation, Line C will always win the competition with Line B for who donates the bigger proportion of their length to Line A, and things won’t get out of hand.  If, however, Shirley is moving fast enough, Line B wins the “who donates the most to Line A” sweepstakes, and the time/space creation cycles acquire a life of their own.

Believe it or not, there are only a few loose ends to tie up, now.  One of those ends is really nifty, however, so I’m going to leave you today with a teaser to it.  We’ve seen that, in your frame of reference, when Shirley travels at very close to light speed, she covers over 230 million kilometers before the light pulse returns to her floor.  That distance puts her way out in outer space: in the context of our solar system, 230 million kilometers away marks a location well inside the asteroid belt that lies between Mars and Jupiter.

Meanwhile, what distance would the odometer onboard Shirley herself read when the pulse reaches the floor?  599,998 kilometers, of course: Her speed, 299,999 kilometers per second, multiplied by the two seconds of elapsed time that it takes for George to see the flash.  But that is only about one and a half times the distance to the Moon, and is many millions of kilometers short of Mars, let alone the asteroid belt.  So. At the exact point in time when the light pulse returns to the spacecraft floor, time dilation appears to have created an enormous discrepancy between where Shirley is located in George’s world, compared to where she is in your world.

This discrepancy raises a major conundrum.  Since, from your perspective, Shirley travels all the way into the asteroid belt, it is entirely possible (though unlikely) that an asteroid lies somewhere along her path, and Shirley actually hits the asteroid in a high-speed collision that immediately pulverizes her, and poor George, into dust.  If that happened, you’d obviously never see the light pulse, because there’d be no glass floor around to reflect it back to you.  You wouldn’t care much about that, because you’d be mourning the loss of poor dedicated George.  Meanwhile, from George’s perspective, Shirley doesn’t travel nearly far enough to even enter the asteroid belt.  The high-speed collision never takes place, the light pulse reaches the floor with no problem, and George lives to record the time of that event.

Can exactly the same event have two such different histories? How can George both live and die?  How can Shirley get pulverized and not get pulverized? I invite you to speculate on how to resolve this paradox in today’s comment section!  Or, if you like, just sit tight and wait for me to provide the (in my humble opinion) quite mind-boggling answer!