Special

he first clear example of time dilation was provided over fifty years ago by an experiment detecting muons.  (David H. Frisch and James A. Smith, Measurement of the Relativistic Time Dilation. These particles are produced at the outer edge of our atmosphere by incoming cosmic rays hitting the first traces of air.  They are unstable particles, with a “half-life” of 1.5 microseconds (1.5 millionths of a second), which means that if at a given time you have 100 of them, 1.5 microseconds later you will have about 50, 1.5 microseconds after that 25, and so on.  Anyway, they are constantly being produced many miles up, and there is a constant rain of them towards the surface of the earth, moving at very close to the speed of light.  In 1941, a detector placed near the top of Mount Washington (at 6000 feet above sea level) measured about 570 muons per hour coming in.  Now these muons are raining down from above, but dying as they fall, so if we move the detector to a lower altitude we expect it to detect fewer muons because a fraction of those that came down past the 6000 foot level will die before they get to a lower altitude detector.  Approximating their speed by that of light, they are raining down at 186,300 miles per second, which turns out to be, conveniently, about 1,000 feet per microsecond.  Thus they should reach the 4500 foot level 1.5 microseconds after passing the 6000 foot level, so, if half of them die off in 1.5 microseconds, as claimed above, we should only expect to register about 570/2 = 285 per hour with the same detector at this level. Dropping another 1500 feet, to the 3000 foot level, we expect about 280/2 = 140 per hour, at 1500 feet about 70 per hour, and at ground level about 35 per hour. (We have rounded off some figures a bit, but this is reasonably close to the expected value.)

To summarize: given the known rate at which these raining-down unstable muons decay, and given that 570 per hour hit a detector near the top of Mount Washington, we only expect about 35 per hour to survive down to sea level.  In fact, when the detector was brought down to sea level, it detected about 400 per hour!  How did they survive?  The reason they didn’t decay is that in their frame of reference, much less time had passed.  Their actual speed is about 0.994c, corresponding to a time dilation factor of about 9, so in the 6 microsecond trip from the top of Mount Washington to sea level, their clocks register only 6/9 = 0.67 microseconds.  In this period of time, only about one-quarter of them decay.

What does this look like from the muon’s point of view?  How do they manage to get so far in so little time?  To them, Mount Washington and the earth’s surface are approaching at 0.994c, or about 1,000 feet per microsecond.  But in the 0.67 microseconds it takes them to get to sea level, it would seem that to them sea level could only get 670 feet closer, so how could they travel the whole 6000 feet from the top of Mount Washington?  The answer is the Fitzgerald contraction.  To them,Mount Washington is squashed in a vertical direction (the direction of motion) by a factor of   the same as the time dilation factor, which for the muons is about 9.  So, to the muons, Mount Washington is only 670 feet high—this is why they can get down it so fast!

But this isn’t the whole story—we must now turn everything around and look at it from Jill’s point of view.  Her inertial frame of reference is just as good as Jack’s.  She sees his light clock to be moving at speed v (backwards) so from her point of view his light blip takes the longer zigzag path, which means his clock runs slower than hers.  That is to say, each of them will see the other to have slower clocks, and be aging more slowly.  This phenomenon is called time dilation. It has been verified in recent years by flying very accurate clocks around the world on jetliners and finding they register less time, by the predicted amount, than identical clocks left on the ground.  Time dilation is also very easy to observe in elementary particle physics, as we shall discuss in the next section.

Suppose now the blip in Jill’s clock on the moving flatbed wagon takes time t to get from the bottom mirror to the top mirror as measured by Jack standing by the track. Then the length of the “zig” from the bottom mirror to the top mirror is necessarily ct, since that is the distance covered by any blip of light in time t. Meanwhile, the wagon has moved down the track a distance vt, where v is the speed of the wagon. This should begin to look familiar—it is precisely the same as the problem of the swimmer who swims at speed c relative to the water crossing a river flowing at v!  We have again a right-angled triangle with hypotenuse ct, and shorter sides vt and w.