So six months passed, and Michelson and Morley duly ran their experiment again. And once again, no variation in the speed of light was found. This was a simply astonishing result—not only had the aether previously moved in the same direction as the earth, but it had then followed it in its circular orbit around the sun! That was too much to take, and although physicists would try to resuscitate the aether through a number of gambits, by the turn of the century they reluctantly concluded that the experiment proved the nonexistence of that which it had set out to measure—the luminiferous aether. Light was not the undulating motion of any aether; light waves could just wave themselves. Maxwell, it turned out, was wrong in this regard.
But if there is no aether, then there is no preferred frame of reference for measuring the speed of light, either—the Michelson-Morley experiment, as it came to be known, proved that as surely as it disproved the existence of the aether. The speed of light must be the same in any inertial frame of reference. (An inertial frame of reference is simply one in which a stationary object remains stationary so long as nothing pushes or pulls on it.) For some years, this was regarded as a fascinating principle of nature, but no one could have guessed the way in which it would revolutionize the future of physics.
In 1905, Einstein considered this principle—that light has the same speed no matter what the motion of the observer or the source—and took it further than anyone else previously had. Let’s go back to our boxcar in the train. Suppose that instead of bouncing a tennis ball, we bounce a burst of light. We’ll put a laser, capable of emitting very short bursts of light, on one side of the boxcar, and a mirror on the other side. Our purpose is to measure the speed of light by timing the delay between the time the light is emitted to the time its return bounce is picked up by a detector.
As before, let’s start our experiment on a stationary train. The light burst travels 9 feet across and 9 feet back, a total of 18 feet. This round trip takes about 18 nanoseconds, and measuring this enables you to correctly measure the speed of light—18 feet, divided by 18 nanoseconds, equals 1 foot per nanosecond. (This isn’t the exact value, but it makes computations convenient, so let’s pretend for the time being. It doesn’t change the train of thought, if you’ll pardon the expression.)
The observer on the ground sees things no differently, since the train isn’t moving. He also sees the light burst travel 18 feet, also clocks it as taking 18 nanoseconds, and therefore derives an identical value for the speed of light.
Now let’s put the train in motion again. To reveal the effects of Einstein’s special theory of relativity, it isn’t enough to travel at everyday speeds—light travels far too fast for the effects to be easily detectable. No, let’s move the train a large fraction of the speed of light: let’s say, four-fifths the speed of light—that is, 0.8c, where c is the speed of light.
Back inside the train, you see nothing different. The boxcar is still 9 feet from side to side, so the round trip distance is still 18 feet. Since our immutable principle is that the speed of light is the same, no matter what, it must take 18 nanoseconds, even when the train is moving at 0.8c.
Now, let’s look at things back from the point of view of the stationary observer on the ground. Just as with the tennis ball on the slower train, the light burst no longer travels straight forward and back, but instead takes a zig-zag path, as shown in Figure 2.
Figure 2. Path of light on the fast train.
In fact, it takes exactly the same path as the tennis ball did, but only because the train is moving so fast. In the span of time that it takes for light to get from one side to the other, the train has moved forward 12 feet, and the light has moved 15 feet. That’s absolutely right, since the train is moving 0.8c, and 12 feet is 0.8 of 15 feet—the train has moved four-fifths as far as the burst of light. The same thing happens on the return bounce: the train moves forward another 12 feet, and the light travels another 15 feet. In total, from emission to detection, the train moves 24 feet and the burst of light moves 30 feet.
But how is that possible? It sure looks as though, from the point of view of the stationary observer, the burst of light has travelled 30 feet in 18 nanoseconds, meaning that the speed of light, as measured by that observer is 30 feet divided by 18 nanoseconds, or 5/3 feet per nanosecond. The burst of light has exceeded the speed of light!
Einstein decided that was an untenable state of affairs. Maxwell’s four equations convinced him that the speed of light was a fundamental constant of nature, and the Michelson-Morley experiment convinced him that it must not vary no matter what frame of reference you measure it in. Therefore, one of the other assumptions must be wrong. But which one?
Put yourself in Einstein’s position for a moment, and ask yourself if you can figure out what was wrong with the old way of looking at things. In hindsight, it doesn’t seem so difficult to imagine, but it was a large leap of faith for Einstein in particular, and physics in general.
Einstein decided, for aesthetic reasons, as well as another reason we’ll discuss later, that it was the Newtonian principle that time is absolute that was at fault. He decided that it must not be the case that everyone everywhere sees the whole sequence taking 18 nanoseconds. In particular, the observer on the ground must see it as taking longer. In order for the speed of light to remain constant, it must take exactly as long as it should for the speed of light to remain 1 foot per nanosecond. Since the distance travelled is 30 feet, the stationary observer must clock the sequence at 30 nanoseconds. To put it another way, 30 nanoseconds have passed on the stationary earth, while only 18 nanoseconds have passed on the train.
It’s important to emphasize that this is not some sort of psychological effect that requires a person on board. It isn’t the case that you on board the boxcar get “speed sickness” near the speed of light and only subjectively experience 0.6 seconds per “real” second. Time actually moves slower on the moving train—that is a necessary conclusion, once you admit that light travels at the same speed no matter how fast you’re moving.
What’s more, this effect happens no matter what the speed of the train—it’s only the magnitude of the effect that changes. Here, time on the train is slowed down to 0.6 seconds per second, but that’s only because the train is travelling fast enough that the light that spans 18 feet on the train travels 30 feet from the point of view of someone on the earth. Simple algebra can predict the time dilation effect for any train speed. Those of you who aren’t interested in looking at how we derive the equation for this can skip over Equations 2 through 8.
If we look only at the right triangle ABC in Figure 2, we see that the speed of the train, expressed as a fraction of the speed of light, is the distance the train moves (AC) divided by the distance the light moves (AB). That is,
(2) v / c = AC / AB
where v is the speed of the train and c is the speed of light. On the other hand, the time dilation to / t, expressed in seconds per seconds, is the distance the light travels as measured by you on the train (BC) divided by the distance as measured by an observer on the ground (AB), on account of the constancy of the speed of light. That is,
(3) to / t = BC / AB
Since ABC is a right triangle, we have, from the Pythagorean theorem,
(4) AC2 + BC2 = AB2
Dividing both sides by AB2, we get
(5) (AC / AB)2 + (BC / AB)2 = 1
Substituting Equations 2 and 3 into Equation 5, we get
(6) (v / c)2 + (to / t)2 = 1
(7) (to / t)2 = 1—(v / c)2
or at last,
(8) t / to = 1 / sqrt(1—(v / c)2)
which we can rewrite more simply as
(8a) t / to = y
if we define y (actually, the Greek letter gamma) to be the factor
(8b) y = 1 / sqrt(1—(v / c)2)
Equation 8 is one of the famous Lorentz transform equations, which the Dutch physicist Hendrik Lorentz (1853-1928) devised to express the time dilation in certain reactions experienced by charged particles. This was the other reason that Einstein violated the absoluteness of time the way he did; he knew of Lorentz’s equations, and this line of reasoning led him to the same answer. The difference is, Lorentz thought his equations only worked for charged particles—Einstein showed that all objects, charged or uncharged, experience the same time dilation effect. Various particles decay slower when they’re moving fast, and at a rate precisely predicted by Equation 8.