At Last! Some Time Dilation!

In our last blog, we discovered that when Shirley is moving and you are stationary, in your frame of reference the light pulse has to trace out a path up and down the hypotenuse of two right-angle triangles, and so the total distance traveled is more than in George’s frame of reference, where the pulse simply goes up and down in a straight line.  As Shirley was traveling only 1 kilometer per second, the additional distance was too short for you to reliably measure the increase in time it took for the pulse to cover it.  However, as we left our story, you and George were about to perform the time measurement experiment again, this time with Shirley moving smartly along at fully half the speed of light (150,000 km per second).  After your customary hundred or so repetitions of the experiment, George is climbing the walls with boredom, as he keeps measuring the same two seconds for the light pulse to go up and down Shirley’s shaft.  Your watch, though, is measuring a shade over 2.3 seconds.

Say what? How can the same event be taking different amounts of time?  The difference can’t be blamed on George.  He’s been well trained to toggle the searchlight on and off at the exact point he and Shirley pass over top of you.  Thus, you have supreme confidence that you and he are starting your stopwatches at the same time.  So where is the extra 0.3 seconds and change coming from?

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

How much more?  Again, let’s work through the standard Pythagorean equation to find out.  Remember, to get the length of line A, you multiply the length of Line B by itself (square it), multiply the length of Line C by itself, add the resultant values together, and take the square root of that summed value. I’ll spare you having to go to your calculator:  150,000 (Line B) squared equals exactly 22 billion, five hundred million kilometers (again, see what happens when you square an already huge number?). We know already that squaring Line C gives 90 billion kilometers.  Adding these two big values together yields 112 billion, five hundred million kilometers.   The square root of that number is just slightly under 336,000 kilometers.  Therefore, 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!  That is roughly 72,000 kilometers longer than the 600,000 kilometers the light flash travels from George’s perspective!  Clearly, your version is going to have to take more time to cover that extra distance.  How much more?  Easy.  Light travels at 300,000 kilometers per second, so covering an additional 72,000 or so kilometers takes your pulse about 0.236 seconds of additional time.

The figure illustrates another important way to imagine this situation.  For both you and George, the pulse travels at exactly the same rate: 300,000 kilometers per second.  For George, that’s enough to get all the way to the ceiling in just one second.  For your version of the pulse, traveling up Line A, one second of travel time puts the pulse 300,000 kilometers up the line.  However, as you can see from the position of the pulse along the line, that’s only 88% of the way to the top; the pulse is still 36,000 kilometers short of the ceiling.  Covering that extra distance is simply going to take more time.  There’s no getting around it.

Let’s pause for a minute to take stock.  Two different observers, you on the ground and George onboard Shirley, are measuring the duration of an identical event and finding that it takes different amounts of time to complete!   The general conclusion from your measurements is irrefutable.  When you determine 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 covers a longer duration 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).  Mathematically, it’s the increasing length of side “B”, brought about by Shirley’s rapid motion, that is “causing” the whole thing.  The longer Line B becomes, the longer Line A is, and the more distance the light pulse has to cover.

But guess what?  There’s actually a slight problem with this analysis.  Recall that the average of the values you measured from your stopwatch was a little over 2.3 seconds, almost a tenth of a second longer than the 2.236 seconds you should have measured if the light pulse was traveling “only” 72,000 extra kilometers (36,000 km on the way up and 36,000 km on the way down).   You’re understandably quite eager to sweep this discrepancy under the rug, attributing it to just measurement error perhaps, since it’s pretty small.  But I can tell you now, blog readers, this difference is not to be trifled with.  As we’ll discover in the next blog, it holds within it the key to the most mind-blowing aspect of the whole time dilation phenomenon!

Would you care to speculate in today’s comment section about where this discrepancy is coming from, and why it is so important? Or just wait?  If you want to take a shot, here’s a hint in the form of a question: During the 0.236 seconds that it takes the light pulse to cover that extra 72,000 kilometers, what is Shirley doing?