To the Moon, Alice? Or beyond? #8

by Whabby

In the third (and final) version of our thought experiment, we’re going to add a few “dilithium” crystals to Shirley’s engines so she can crank her speed up to exactly 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, as you lie motionless on the ground, George will flick his searchlight on and off. Both you and George will measure the time for the light pulse to travel up and down Shirley’s shaft. You’ll 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.

Last week, 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. Today, with Shirley moving almost twice as fast (and as some owls guessed last week), a perfectly reasonable conjecture would be that the delay is going to be about twice as big, or just over 0.6 seconds. Consequently, when you and George start the first measurement, you go in still expecting to see the light pulse only a little later than he does.

But, 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 start 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, you finally do see it! Breathlessly, you look down at the time as measured on your stopwatch, and see 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 have serious trouble believing the outcome of your own experiment. How can a two second event take 13 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 a line stretching off to your right at 299,999 kilometers per second. Geometrically, after one second Shirley’s movement has created a Line B that’s 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. I’ll spare you the numerical details: Line A turns out to be 424,263 kilometers long (a full 124,263 kilometers longer than Line C, which is the path the light pulse takes for George). The resulting triangle is shown below.

After one second of George’s time, his light pulse has reached the ceiling, and is about to start the journey back. 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 to go before it reaches the ceiling.

That’s a lot of extra distance, owls. It’s about 10 times the diameter of the Earth. Even traveling at 300,000 kilometers per second, 124,263 kilometers is going to take the pulse a little less than half a second - 0 .414 seconds to be exact - to traverse. In 0.414 seconds, though, Shirley, zipping along at a constant 299,999 kilometers per second, extends Line B by an additional 124,262 kilometers (0.414 times 299,999). By the Pythagorean theorem, that, in turn, lengthens Line A by a further 95,000 kilometers or so.

Just like in last week’s example, successive cycles of space/time creation have commenced. The extra time needed for your light pulse to cover that additional 95,000 kilometers is about 0.317 seconds. But in 0.317 seconds, Shirley moves 80,000 kilometers further away. And, of course, the same movement is happening while the light pulse is traveling back down Shirley’s shaft, stretching Line B for the mirror-image triangle by the same amount. These numbers are big, so it’s already clear that the time you measure for the light flash to reach Shirley’s floor will be larger - quite a bit larger – than last week, when Shirley was traveling at half the speed of light. Still, just like last week, these numbers are shrinking rapidly. You would be forgiven for thinking that the cycles would quickly shrink to insignificance.

But you’d be wrong! As the cycles start to pile on, an important aspect of the geometry of the situation rears its ugly head and starts to exert an increasing influence. To understand this, we need to revisit Pythagoras’ theorem (sorry, owls):

According to the theorem, to get the length of Line A, you first square the lengths of Lines B and C. Then, you add the squared values together. Finally, you take the square root of that sum. Remember the blog where Shirley was moving only one kilometer per second? And we talked about the fact that squaring the lengths of Lines B and C and then adding them together inflates any initial difference in their size? When the lengths of Lines B and C are very discrepant, with one relatively long and the other relatively short, squaring the numbers ensures that the length of Line A is almost completely controlled by the length of the longer side.



A useful way of stating this relation in the context of our current triangle is: The greater the discrepancy between the lengths of Lines B and C, the greater the proportion of the length of the longer side gets “donated “ to line A.

We encountered this situation when Shirley was moving only one kilometer per second (slow, relative to the speed of light, that is) in Blog Five, making Line C vastly longer than Line B. You can see this in the figure above: with Line C so much more in control of the length of Line A than Line B, Line A was virtually the same length as Line C. This time around, with Shirley moving at 299,999 kilometers per second, Line B is almost equal in length to Line C after just one second. Therefore, the amount that Line B and Line C contribute to the length of A is now virtually equal (for those who care, Line C supplies just a tiny trifle more than 71% of its length, Line B just a tiny trifle less).

In just the first cycle of time/space creation, though, we’ve seen that Line B increased by 124,262 kilometers, making Line B quite a bit 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, Line B begins to donate a larger and larger proportion of its overall length to Line A, and that means it donates a larger and larger proportion of the portion of its length that was just added in the latest cycle. The top triangle of the three triangles in the figure below shows what’s happened by the 20^th cycle of space/time creation, close to the number of cycles that completely “closed things out” when Shirley traveled at 150,000 kilometers per second. This time around, Shirley has moved almost 1.5 million kilometers away, 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, most of that additional length is donated to Line A. In fact, Line A lengthens by about 30,303 kilometers, only a fraction less than the latest increase in Line B.

It takes just slightly more than a tenth of a second for the light flash to cover the extra 30,000 or so kilometers along A. In that amount of time, though, Shirley moves another 29,999 or so kilometers away, and donates almost all of that extra length to Line A. As these iterations of space/time creation continue to mount, and Line B grows ever longer compared to Line C, the proportion of the additional length on Line B that’s donated to A grows every larger, until they’re almost identical. This situation is illustrated in the bottom triangle of the figure with the large number of evenly spaced light pulses through the midsection. Eventually, things almost reach 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, the light pulse makes very little headway on its journey toward the top of the triangle (Shirley’s mirrored ceiling), or (after one second of George’s time) back down to the floor.

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

Eventually, the cycle IS choked off, and the light pulse DOES reach naked (and, don’t worry, still fully erect) George. As shown by the pile-up of light pulses to the right side of the triangle, though, it takes many, many, many extra cycles for that to happen. The distances along the sides say it all: by the time the pulse reaches the ceiling, Line B has stretched to over 116 million kilometers. By the time the pulse reaches the floor, Shirley has covered a total of about 232 million kilometers, putting her well on her way to Jupiter! Compared to that colossal distance, the length of line C, which remains fixed at 300,000 kilometers, has become a trivial pittance. This is why the shape of the triangle is so squashed, and 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 and take stock. We have exactly the opposite of the situation in Blog Five, when the spacecraft was traveling at only one kilometer per second, and it was Line B that was insignificant compared to Line C. See 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 cover. And, of course, light always takes time to cover distance.

With this description, we’ve virtually finished the quest to understand the time in time dilation! The key to the entire story 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 don’t get too 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 creaction cycles acquire a life of their own.

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 when Shirley travels at very close to light speed, she covers over 230 million kilometers before the light pulse returns to her floor. Of course, that distance puts her way out there in outer space: in the context of our solar system, 230 million kilometers away is 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. That is only about one and a half times the distance to the Moon; it’s 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 in George’s world, and where she is in yours’.

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 erect 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 George and his equipment (I was talking about his searchlight and stop watch). But from George’s perspective, Shirley doesn’t travel nearly far enough to enter the asteroid belt. The hide-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 (and his marvelous erection) both live and die? How can Shirley get pulverized and not get pulverized? Speculate on how to resolve this paradox in today’s comment section! Or, if you like, just sit tight and wait until next week for the (in my humble opinion) mind-boggling answer!