Time for some Time Dilation? #6

by Whabby

At the end of last week’s installment, you and naked George had agreed to perform the time measurement experiment again, but this time with Shirley moving smartly along at half the speed of light, or 150,000 kilometers per second. After your customary hundred or so repetitions of the experiment, George is starting to climb the walls with boredom, as he keeps measuring the same old two seconds for the light pulse to go up and down Shirley’s shaft. Your watch, though, is measuring something distinctly different: a shade over 2.3 seconds.

Say what? This difference can’t be blamed on naked George. He’s been well trained to switch the searchlight on and off at the exact point he and Shirley are passing over top of you. Thus, you have supreme confidence that you and he are starting your stopwatches at exactly the same time. Where is this extra 0.3 seconds and change coming from?

The figure to the left shows where. In the second it takes the light flash to travel up to the ceiling, Shirley travels exactly 150,000 kilometers along the line to your right, so Line B reaches 150,000 kilometers in length. In George’s frame of reference, the pulse only has to go up and down the equivalent of Line C. For you, it has to climb up and down Line A, the diagonal side of the two right-angle triangles. Looking at the shape of the triangles, it is clear that your “version” of the light pulse has to cover more distance than George’s version.

How much more? Let’s do the standard pythagorian calculation and find out. Remember, to get the length of line A, you multiply the length of Line B by itself (square it), and the length of Line C by itself, add the two together, and get the square root of that summed value. 150,000 (Line B) squared equals 22 billion, five hundred million kilometers (see what happens when you square an already huge number?). We already know that Line C squared equals 90 billion kilometers. Adding these two big values together gives us exactly 922 billion, five hundred million. The square root of that number is just slightly under 336,000 kilometers. Both Line A going up and Line A going down are close to 336,000 kilometers long, for a total distance of almost 672,000 kilometers.

Whoa, owls! That number is roughly 72,000 kilometers longer than the distance the light flash has to travel from George’s perspective! Clearly, your light pulse is going to take more time to cover that extra distance. How much? Easy. Light travels at 300,000 kilometers per second, so covering an additional 72,000 or so kilometers will take your pulse about 0.236 seconds of additional time.

Let’s take stock here. Two different observers, you on the ground and naked George in Shirley, are measuring the identical event and finding that it takes different amounts of time to complete! The conclusion is irrefutable. When you measure the length of an event that occurs in the frame of reference of a spaceship (or anything else) that’s moving very fast with respect to you, the event take more time than it does when the event is measured from within the moving frame of reference itself (in this case, measured from “the moving frame of reference itself” means performing the measurement from onboard Shirley, the way naked George is doing).

Mathematically, it’s the increasing length of side “B”, brought about by Shirley’s motion, that is to blame. The longer Line B is, the longer Line A is, and the more distance the light pulse has to cover from your stationary perspective.

But, there’s actually a slight problem with this analysis. Recall that the average of the values you measured from your stopwatch was actually a little over 2.3 seconds, almost a tenth of a second longer than the 2.236 seconds you should have been measuring if the light pulse was traveling “only” 72,000 extra kilometers. You’re understandably quite eager to sweep this discrepancy under the rug, attributing it to just measurement error perhaps, since it’s so small. But I can tell you now, owls, this difference is not to be trifled with. In fact, as we’ll discover in next Wednesday’s blog, it holds within it the best part of the time dilation phenomenon!

Would you care to speculate in today’s comment section about where the discrepancy is coming from, and why it is so important? Or just wait for next week?

Here’s a hint: During the 0.236 seconds that it takes the light pulse to cover that extra 72,000 kilometers, what is Shirley doing?