Shirley's Magic Bottom

Let’s recap. You are trying to measure the duration of an event, the journey of a pulse of light up and down the shaft of your spacecraft, Shirley.  In the current version of the experiment, you’re stationary, looking up from the ground, while Shirley is moving at one kilometer per second along the line formed by your outstretched right arm.  You and George start your stopwatches at the same time, when George toggles the searchlight on and off while Shirley is directly above you.  The question is, do you and George measure the same amount of time for the light flash to travel up to the ceiling and back?

Before tackling that question, we first have to deal with an obvious problem that rears its ugly head now that Shirley is moving.  We know that from George’s perspective it takes the flash two seconds to travel up and down Shirley’s shaft.  By the time the flash gets back to the floor, therefore, 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 as he always has.  But obviously, you can no longer see it because, when it passes through the glass floor, you’re not there!  As things, stand, you don’t have any event on which to stop your stopwatch. 

To deal with that little complication, I’d like to take some poetic license and add a rather advanced capability to Shirley’s bottom.  Specifically, the molecules of the glass that make up her ass automatically realign such that, when the light pulse hits the floor, it is reflected at just the right angle to travel directly back into your eyes.  That way, both you and George see the return flash of light, and you both have an event to stop your stopwatches with.

 

Physically, this situation is illustrated in the figure above.  Since George is inside the spacecraft, he’s moving right along with the pulse.  From his perspective, the pulse goes straight up to the ceiling and straight back down, just like it did when Shirley (and George) weren’t moving. The white arrows going straight up and down inside Shirley illustrate the path that the pulse takes for George.  Meanwhile, when the flash reaches the floor it is steered immediately along the dotted line right back to you.  In this way, you continue to see the flash, too.

OK, how does the value on your stopwatch compare to the value on George’s?  Light takes only a tiny amount of extra time, 1/150,000 of a second to be exact, to cross the two kilometers separating you from Shirley.  Although the flash is reaching you just a teensy, tiny bit later than it reaches George, the difference is far too small to show up with a measuring device as crude as a human operating a stopwatch. Still, you’re a stickler for accuracy.  You know that, compared to George, the light has had to travel a longer distance, which means that you are not… quite… measuring the duration of the same physical event that George is.  But your smart stopwatch comes to the rescue! You simply program the watch to automatically subtract the additional time required for the flash to travel from Shirley to you (recall that your watch is already automatically subtracting your brain processing time).

Does the reconfigurable glass floor solve all the measurement problems that accompany the fact that Shirley is now moving?  Possibly not, because something else about this whole situation is starting to gnaw at you.  The figure provides a strong hint to the problem.  Now that Shirley is moving, what is the exact path through space that the light has to take from your perspective in order to get to the ceiling and back? Is it possible that this path is different from the path that it takes from George’s perspective (which is just straight up and down)?

If so, how is it different?  And why? The more you take a crack at answering that question, and “stay ahead” of the blogs, the less pain you’re going to have later on when Einstein finally crashes our little party!  Anybody care to tackle this issue in the comments section?