Does Motion Affect The Speed At Which Time Passes? (Part 2)

In Part 1, we uncovered the time dilation phenomenon using an imaginary, or “thought” experiment. Einstein himself utilized this technique to deduce many of the basic principles of special relativity. But actual proof of the validity of SR had to wait some years until the technology to enable ‘real world’ experiments to be carried out came into existence. Indeed, the fact that the special theory was laid out based only, in essence, on Einstein’s two fundamental postulates, is a tribute to his intellectual prowess.

The phenomenon of time dilation introduced by special relativity may seem all too unbelievable at this point, but nature has provided us with dramatic and measurable evidence showing that “time indeed slows down for objects that are in relative motion”, in the form of subatomic particles called muons (pronounced “mew-ons”). Muons are similar to electrons but are incredibly unstable, meaning that they are very short lived particles, from conception to self-destruction. They are created well above the surface of the earth, when cosmic rays coming from outer space collide with the atoms that make up the earth’s atmosphere. From there, the newly formed muons traverse the 22,000 feet down to the earth’s surface at the hair-scorching speed, relative to the earth, of 0.998c, that is, 99.8% of the speed of light! As a consequence, these minuscule particles are constantly detected bombarding the surface of the earth. But muons can also be created in the laboratory, where they remain more or less at rest, unlike their ‘high on speed’ brothers. Physicists have measured the average lifetime of these stationary ‘lab’ muons, from creation to natural disintegration or decay, to be a mere 2.2 microseconds long (microseconds is short for “millionths of a second”!). Hence, the moving muons that travel from the top of the earth’s atmosphere to the earth’s surface must also live an average of only 2.2 microseconds. But in this very short time span, they should disintegrate well before they hit the earth – and here’s the math to prove it: Moving at a fantastic relative speed of 0.998c, but for only 2.2 microseconds, means that the distance traveled by these muons before going kaboom equals (9.823 x 10⁸ feet/second) x (2.2 x 10^-6 seconds), or only 2,161 feet! So the obvious question is, how in the name of science do these muons manage to travel the extra 19,839 or so feet that is needed to reach the surface of the earth, if they disintegrate after traveling only 2,161 feet!?

Enter relativistic time dilation. Because the muons are moving, relative to us here on earth, at a very large fraction of light speed, then time will run more slowly for them, and in fact very noticeably, just as time slowed down for Scotty by a factor of 2 (which, if we recall, is called the time dilation factor), thus making him age 2 times slower than normal. But since the muons are moving much closer to the speed of light than Scotty was, the rate at which their ‘internal’ clocks tick, or equivalently, the rate at which they age, will be even slower, as observed and measured by us from our vantage point on earth. Specifically, at a relative speed of 0.998c, the time dilation factor for the muons has a value of approximately 15.82! This means that for every 2.2 microseconds that we ‘see’ elapse on the muons’ internal clocks, a much greater corresponding value of (2.2 microseconds) x (15.82) or 34.8 microseconds of normal time will elapse on our clocks. Hence, just as we observed Scotty aging in ‘slow motion’ at a rate 2 times slower than normal, so will we also observe the moving muons aging more slowly, but at a rate that is 15.82 times slower than normal!

And so, from our point of view, time dilation is the mechanism by which the muons are able to traverse the 22,000 feet of atmosphere and strike the surface of the earth before they decay, because, as we have found out, the 2.2 microseconds of ‘muon’ time that is their lifespan, is stretched out, or dilated, over the course of 34.8 microseconds of our normal time here on earth. From our perspective, then, we observe that the muons live a much longer life of 34.8 microseconds, as measured by our clocks, before decaying. Knowing this, let’s do the revised, or correct math for the distance, as calculated from our vantage point here on earth: Moving at a relative speed of 0.998c for 34.8 microseconds means that the distance traveled by the muons before disintegrating equals (9.823 x 10⁸ feet/second) x (34.8 x 10^-6 seconds), or 34,184 feet, which is more than enough for the muons to reach the surface of the earth! “The defense rests its case, your Buffoonship…”

Let’s take a quick detour, and figure out what this journey looks like from the point of view of the muons. By the principle of relativity, the situation can very justifiably be reversed, and each individual muon can claim that it is at rest and the earth is the one moving toward it at the relative speed of 0.998c. Furthermore, from the muon’s perspective, time is passing by at the normal rate, and so its internal clock is not running slowly at all. Thus, after a normal 2.2 microseconds have elapsed on its internal clock, the muon will say, “Adios, amigos”, and disintegrate. But in this extremely short time span (okay, we’ve done this calculation before, just not from the muon’s point of view – until now), the earth will move a total distance, towards the muon, equal to (9.823 x 10⁸ feet/second) x (2.2 x 10^-6 seconds), or only 2,161 feet! So now, the even more perplexing question is, how does the earth manage to travel the extra 19,839 or so feet that is needed to reach the muon, if the muon disintegrates after the earth has traveled only 2,161 feet towards it!? Holy contradictus maximus, dude wearing the bat costume! We have just demonstrated that from our point of view, the muon hits the earth. But from the muon’s point of view, it seems like the earth doesn’t even come close to hitting it! What in the hell !?

But the muons must hit the earth from both points of view, because our detectors are constantly bombarded by them! Now we have already correctly calculated, using the muon’s relative speed (which is 0.998c) and our clocks, that the muon travels 34,184 feet before disintegrating (though it only needs 22,000 feet to reach the earth). But the muon, on the other hand, has also correctly calculated, using the earth’s relative speed (which must also be 0.998c) and its clocks, that the earth travels only 2,161 feet towards it before disintegration occurs. Therefore, the only way that the muon can strike the earth from both points of view is if, from the muon’s perspective, the entire 34,184 feet of distance that we have measured is compressed, or contracted into the 2,161 feet of distance that the muon has measured! And that’s exactly what happens. Thus, the muon will ‘see’ the full 34,184 foot distance that we see contracted to a distance of only 2,161 feet! Consequently, the muon will see a much shorter travel distance than we do, specifically, a travel distance that is (34,184 ft) / (2,161 ft), or 15.82 times shorter than what we see (at this point, we carefully note that 15.82 is also the value of the time dilation factor! This is not a mere coincidence whatsoever, and if we pay close attention to our previous calculations, we will see why this is so). And it is this amount of distance contraction that will allow the earth, which the muon sees traveling towards it at a relative speed of 0.998c for only 2.2 microseconds of normal time, to reach the muon well before disintegration occurs!

This is yet another mind-boggling phenomenon that results from the special theory of relativity, known as the “Lorentz-Einstein contraction”. And, as the muons have just revealed, it is a direct consequence of the time dilation phenomenon. Stating it more formally in the lingo of special relativity means that: Distance is contracted ‘in the moving frame’ along the direction of relative motion, and the greater the relative speed of the moving frame, the greater the amount of contraction. And so, from the muon’s point of view, the moving frame is the earth and its atmosphere, which, to the muon, will therefore appear to be “squished” or compressed along the direction of relative motion. And just as with the phenomenon of time dilation, distance contraction becomes perceivable only at relative speeds that are significant fractions of the speed of light.

Hence, from our point of view, relativistic time dilation is what enables the muons to strike the earth, while from the muon’s point of view, relativistic distance contraction is what enables the earth to strike the muons. Therefore, the apparent paradox which we faced earlier has been resolved, since the muons now strike the earth, or equivalently, since the earth now strikes the muons, from both perspectives. The following diagram will help to visualize the situation more easily (and we also note that it is definitely not drawn to the proper scale!).

We now also take note of the fact that, from our point of view, we must also observe, due to the symmetry of relativity, that “distance is contracted in the moving frame”. But to us, the moving frame is the muon itself, hurtling towards earth at 0.998c, and the distance that we will observe to be compressed or contracted is therefore the diameter of the muon. Hence the muon, to us, will appear to be a “squashed” sphere, or more formally, an ellipsoid, whose diameter will be 15.82 times shorter than normal along the direction of relative motion, as shown in the diagram above.

Let’s check in with Scotty once again. Recall that his train was moving at a relative speed of 0.866c, and that as a direct consequence we observed that for him, time was passing by at a rate that was twice as slow as normal. But we now know that we will also see the length of his train to be contracted along the direction of relative motion, so that it is twice as short as its normal length. Scotty, however, has every right to claim that he, instead, is the one at rest while we and the train crossing are whizzing by at the relative speed of 0.866c. Therefore, because we are the ones who are moving, according to Scotty, he will see the vicinity of the train crossing, which includes us, to be contracted by a factor of 2 along the direction of relative motion. But furthermore, Scotty will claim that our clocks are the ones that are ticking more slowly, and that for us, time is passing by at a rate that is twice as slow as normal! How can we and Scotty each declare that time is running slower for the other!? The following diagrams summarize the current situation. Whose, then, is the correct point of view?