Shirley's Bottom #4

by Whabby

Last week, owls, I left you with a burning question. You’re trying to measure the duration of an event, the journey of a pulse of light up and down the shaft of your spacecraft. You yourself are stationary, looking up from the ground, while your spacecraft, which I’ve decided to call “Shirley”, is traveling at a speed of one kilometer per second along the line formed by your outstretched right arm. You and George (who, by popular demand, is naked) start your stopwatches at the same time. George starts his when he toggles the searchlight on and off, and you start yours right then and there because you immediately see the flash of light coming from the searchlight through the floor. But, do you and George still measure the same amount of time for the light flash to travel up to the ceiling and back?

To answer that question, we have to deal with an obvious problem that arises now that Shirley is moving and you are stationary. Since it takes the flash two seconds to travel up and down Shirley’s shaft, when the flash gets back to the floor, Shirley is no longer directly above you; she’s two kilometers down the line to your right. That’s no problem for George; being inside Shirley, he still sees the flash just as he always has. But you, being far away, can no longer see the flash when it passes through the floor, so you don’t have any signal on which to stop your stopwatch.

To deal with that little complication, I’d like to take some poetic license with Shirley’s glass bottom. Let’s assume that her bottom is programmed to automatically rotate (angle itself) just precisely enough that the light flash, when it hits the floor (and is seen by naked George), is reflected at just the right angle to travel directly back to your eyes. That way, both you and George see the return flash of light, and you both have an event to base the stopping of your stopwatches on.

Physically, the situation is illustrated in the figure above. From George’s perspective, the light goes straight up to the ceiling and straight back down, just like it would if he was shining a flashlight on the ceiling from the floor of a moving train. Meanwhile, from your perspective, Shirley is moving at one kilometer a second. In the second it takes the light pulse to reach the ceiling, Shirley has moved one kilometer away; while the light’s reflection comes back down, she has an opportunity to move another kilometer away. When the flash finally reaches the floor, it is reflected immediately along the straight dotted line right back to you. In this way, you continue to see the flash, too.

OK, now, what are the results? Still pretty disappointing, I’m afraid. Light takes only a tiny amount of extra time, 1/150,000 of a second to be exact, to cross the two kilometers that separate Shirley’s bottom from you. Although the flash is reaching you just a teensy, tiny bit later than it reaches George, the difference is far too tiny to show up with a measuring device as crude as you hitting a stopwatch.

Still, you’re not satisfied. You know that, compared to George, the light had to travel two additional kilometers. Since you know this distance precisely, you decide to simply program your stopwatch to, not only automatically subtract your brain processing time, but also automatically subtract the additional time required for the light flash to travel back from Shirley to you.

As I said, with Shirley just two kilometers away, that extra time is so negligible that you won’t notice it. You and George continue to measure the same two-second duration for the event. It would appear that Einstein has still not shown up to the party!

Still, something about this entire setup is beginning to bother you. The figure above provides a strong hint that there’s a geometric complication to this situation. Now that the spacecraft is moving, there’s something about the path that the light has to take from your perspective that’s different from the path it takes from naked George’s perspective (which is just straight up and down).

What is that something, owls? Just Lin zeroed in on it in last Wednesday’s comment section, and even figured out what it means as far as the duration of the event is concerned. Feel free to speculate about it yourself in today’s comments. Why? Well, the more you “stay ahead” of our little thought experiment, the less disruptive it’s going to be later on when Einstein finally makes his grand entrance!