In early January 1999, a 57-year-old man driving his BMW in Caputh, Germany, drove into the Havel River. Upon questioning, he revealed that he had been following the driving instructions relayed to him by his car's satellite navigation computer. The computer directed him across the river to his destination, but neither he nor his computer realized in time that the only way across the river was by ferry. Fortunately, the man was not injured; unfortunately, the same couldn't be said of his car.
"Normally, accidents like this shouldn't happen," said a Caputh police spokesman. "This sort of thing can happen only when people rely too much on technology." In other words, one runs into trouble by reasoning that, "If the computer says so, it must be true."
One is reminded of the statement made by the English countess and pioneer of computer theory, Lady Ada Lovelace (1815-1852): "The [computer] has no pretensions to originate anything. It can do whatever we know how to order it to perform." Special conditions often result in strange behavior.
On the other hand, a Swiss-German patent clerk decided to take the strange behavior of light at face value. "If the light says so," he might have said to himself, "it must be true." His name was Albert Einstein (1879-1955), and what the light told him led him to, among other things, the special theory of relativity.
Special relativity looks, at first blush, like nothing so much as a bundle of contradictions. It predicts that objects change shape when they move close to the speed of light, that they change mass as well, that the nature of space and time is intimately related to where we observe them from. Part of the reason that special relativity is so counterintuitive is that light travels so darn fast: very nearly exactly 300,000 km/s, about a million times faster than any man-made object at the time. It was very difficult to listen to what the light said.
So let's not start with light. Let's start with a tennis ball.
Suppose you're sitting in a boxcar in a train. Say that the boxcar is 9 feet wide. You sit on one side of the boxcar, and idly throw a tennis ball off the other side, 9 feet away. It bounces back to you. If you throw the ball at 45 feet per second (about 30 mph), the ball takes 0.2 seconds to reach the other side, which we're able to calculate very easily, based on the following simple formula:
(1) t = d / v
which simply states that the time t that it takes an object to travel a distance d is simply d divided by the speed or velocity, v. We'll use this formula a lot. In this case, the time taken is 9 feet, divided by 45 feet per second, or 0.2 seconds. Then it takes 0.2 seconds to bounce back to you -- the whole thing takes 0.4 seconds.
Now let's start the train in motion. Suppose it gets up to a rate of 60 feet per second (40 mph). Anyone who has ever played dodgeball with a sibling in the back seat of the family station wagon knows that the speed of the train won't affect the behavior of the ball. It continues to travel 18 feet round trip, taking 0.4 seconds to do so.
Consider, however, a stationary observer sitting by the side of the track. He agrees that it takes 0.4 seconds from the time you throw the ball to the time you catch it again. On the other hand, he doesn't agree that the ball only travels 18 feet round trip, because to him, the ball doesn't travel straight forward and back. Instead, he sees it take a zig-zag path, as shown in Figure 1.
Figure 1. Path of ball as seen by outside observer.
In the 0.2 seconds that it takes the ball to travel from one side of the boxcar to the other, the train travels 12 feet. The distance that the ball travels, from the point of view of the stationary observer, is the length of the diagonal of the right triangle, which is sqrt(92 + 122), or 15 feet. The ball then travels another 15 feet on the way back, again from the point of view of the stationary observer. To this observer, therefore, the ball travels 30 feet in the same 0.4 seconds, and 30 feet divided by 0.4 seconds equals 75 feet per second (50 mph). From the point of view of someone on the ground, that's how fast the tennis ball is moving.
There's nothing terribly peculiar about this. The thing it depends on is the Newtonian principle that time is absolute. If you clock the round trip time of the tennis ball at 0.4 seconds, then so does everyone else, no matter how fast they're moving. In the everyday experience of Newton and his contemporaries, there was nothing to contradict that common sense rule.
In 1873, the Scottish physicist James Clerk Maxwell (1832-1879) formally set down the four equations that govern the transmission of electromagnetic waves through a vacuum. He noticed that if you combined the four equations, you could derive the speed of those waves -- it was the square root of the product of two constants, both of which could be measured in the laboratory. Given the then best-known values for those constants, he came up with a speed of just about 300,000 km/s (about 186,000 miles per second).
That is very close to the speed of light, and Maxwell decided that was too much of a coincidence. He concluded that light itself was an electromagnetic wave. But what was waving? Ocean waves are waves in water, sound waves are waves in air or some other sound-transmitting medium, but a light beam can go through a vacuum just as well as it can through anything else -- better, in fact. Maxwell couldn't bring himself to conceive of light waves just "waving themselves," so he proposed what came to be known as the luminiferous aether. The aether was a mysterious medium, which had no mass, no energy, nothing -- except that it was necessary in order for light to move anywhere at all.
For light to get to us from all across the universe, this aether had to be everywhere, in our houses, out in the fields, in buildings, in the solar system, throughout the galaxy -- everywhere. Since the aether was the medium for all electromagnetic waves, it made sense to say that the speed of light was 300,000 km/s relative to the aether. In those days, the only massive objects that were known to move an appreciable fraction of that speed were astronomical: stars, planets, galaxies, and so forth. That raised the interesting question: what was our own motion -- that is, the motion of the earth -- relative to the aether?
In 1887, the American physicist Albert Michelson (1852-1931) and the American chemist Edward Morley (1838-1923) conducted an experiment to detect the earth's motion through the aether. They set up an intriguing apparatus designed to measure small variations in the speed of light. As the apparatus was rotated, they expected that the instrument would show small changes reflecting the alignment of the apparatus with the earth's motion through the aether. Light moving with the aether would be faster than light moving against the aether, and light moving across the aether would be somewhere in between.
What they found startled them: no matter how the apparatus was rotated, no change at all was detected. This seemed to imply that the earth was stationary with respect to the aether, or, in other words, that the aether moved with the earth! That seemed completely unreasonable, but it occurred to them that perhaps, just by coincidence, the earth happened to be moving with the aether. Six months later and half a revolution around the sun later, it ought to be moving the opposite direction, and then the results ought to show motion relative to the aether.
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.
Using Equation 8, we can also see why it takes an enormously fast train, or whatever object, to yield a detectable time dilation. Our first train, travelling at an excruciatingly slow 60 feet per second, moved 16 million times slower than the speed of light. If we plug v / c = 1/16 million into Equation 8, we get a time dilation of only 1 part in 500 trillion. No wonder no one ever noticed this effect before the 20th century!
One more thing: there is a fundamental difference between light and ordinary objects such as tennis balls. The ball in Figure 1 and the light burst in Figure 2 take exactly the same path, but you don't get time dilation on the slower train in Figure 1, because it's not a basic law of nature that tennis balls always travel at 45 feet per second. Once the train gets moving, you accept that the tennis ball travels at 75 feet per second (from the point of view of the observer on the ground). It has to, because it travels a greater distance in the same amount of time.
In Figure 2, the light also travels a greater distance, but in this case, it is a basic law of nature that light always travels at the same speed. Something has to give, and Einstein -- equipped with the results of experiment and theory -- decided it was time that had to blink first.
Now, let's once again return to our train, moving at 0.8c. This time, let's put the mirror at the front of the boxcar, and the laser and the detector at the back, so that from your perspective, riding the train, the burst of light now travels the length of the car twice. If the car is 45 feet long, it travels twice 45 feet, or 90 feet. That takes 90 nanoseconds -- this should be getting easy by now!
So -- how long does it take from the outside observer's perspective?
From the moment the burst of light leaves the source at the back of the boxcar, it travels, of course, at the speed of light, c. But since the boxcar itself is travelling at 0.8c, the light is only "gaining" on the front of the boxcar at c - 0.8c = 0.2c, one-fifth the speed of light. Ordinarily, at the speed of light, it would take 45 nanoseconds to catch up to the front of the boxcar, but at only one-fifth that speed, it takes 5 times longer, or 225 nanoseconds.
After bouncing off the mirror, the light now heads toward the back of the boxcar. But instead of the light having to chase the boxcar, this time the boxcar is rushing headlong to intercept the light, and the outside observer sees detector and light meet at c + 0.8c = 1.8c, or nine-fifths the speed of light. Again, ordinarily, it would take 45 nanoseconds, but now, at nine-fifths that speed, it takes five-ninths as long, or 25 nanoseconds.
The round trip time is therefore 225 nanoseconds plus 25 nanoseconds, or 250 nanoseconds. Now, we know that 250 nanoseconds as measured by the outside observer doesn't take 250 nanoseconds on board the train. No, on the train, clocks are slowed down according to Equation 8; inside the boxcar, you should measure the interval as
(250 nanoseconds) sqrt(1 - 0.82) = 150 nanoseconds
But wait -- that's not the time that you actually measured. As we said, you measure it as 90 nanoseconds. So, despite taking into account the time dilation effect, we still have a discrepancy. Where did we go wrong?
Einstein again decided that the analysis was just fine, it was another Newtonian assumption at fault. Which one was it this time? Again, put yourself in Einstein's shoes and see if you can figure out which assumption to abolish.
He decided that it was the notion of absolute length that was the problem. Aboard the train, you measure the length of the boxcar as the old 45 feet. But in order for the times to match, it must somehow be the case that from the outside observer's perspective, the boxcar is compressed in the direction of motion! Compressed by how much? In this case, to correct the 150 nanoseconds down to 90 nanoseconds, the length must also be compressed to 90/150 times its former value. Since the original length was 45 feet, the compressed length must be (90/150)(45 feet) = 27 feet. That must be the length of the boxcar from the outside observer's perspective.
How long would it be in general? In our example, 90 nanoseconds was the time it took for light to travel twice the length of the boxcar as measured on the train itself. Call the length of the boxcar at rest, Lo. Then, the time it takes light to travel from the back to the front, and then back to the back is
(9) to = 2Lo / c
If the train is in motion, however, then we've shown that it must be shortened to some length L, and we have to figure out what L is in terms of Lo. If it's moving at velocity v, then the light catches up with the front of the boxcar at speed c - v. The time it takes to do that is just L / (c - v). (This is just Equation 1 again!) Then, after bouncing off the mirror, the outside observer sees the light meet the back of the boxcar at speed c + v, and the time it takes to do that is L / (c + v). The total round trip time is therefore
(10) t = L / (c - v) + L / (c + v) = 2L / c (1 - (v / c)2)
From Equation 8, we know that this time is reduced on board the train to
(11) to = t / (t / to) = t sqrt(1 - (v / c)2)
Combining Equations 10 and 11, we get
(12) to = 2L / c sqrt(1 - (v / c)2)
Finally, since the to in Equation 9 has to be equal to the to in Equation 12, we can write
(13) 2Lo / c = 2L / c sqrt(1 - (v / c)2)
or, by simplifying and rearranging terms,
(14) L / Lo = sqrt(1 - (v / c)2) = 1 / y
And this is another part of the Lorentz transform equations, so Einstein had another hint here for length compression. As a matter of fact, Einstein wasn't the first to suggest length compression. Lorentz and the Irish physicist George Fitzgerald (1851-1901) tried to sustain the aether theory by supposing that as objects such as the earth plowed through the aether, they were dragged and thereby compressed. Everything was compressed, including the Michelson-Morley apparatus, so -- as Lorentz and Fitzgerald claimed -- the speed of light really did slow down in the face of the "aether headwind," but since the apparatus was shortened by aether drag by exactly the same amount, no change in the speed of light was recorded!
Aether drag was therefore very similar to what Einstein proposed, but it differed in one significant respect. Aether drag still required that the compression take place relative to an all-pervasive aether, whereas in Einstein's formulation, compression took place between any two different frames of reference. Experiment eventually proved Einstein right, and aether drag went the way of the dodo.
Again, this isn't an optical illusion caused by the train moving so fast past the observer on the ground that he underestimates the length. For the laws of nature to make sense, and for light to always have the same speed everywhere, the train must actually shrink in the direction of motion! Does it make any sense? No, but logically that's what has to happen, and experiment has demonstrated this effect time and again.
Something that we've left unstated here, but which you may have guessed at, might be making you a bit uncomfortable. We've been assuming all along that it's the observer on the ground who is "at rest," and you on the train moving, relative to the ground. But isn't the whole deal with special relativity that, well, it's all relative?
Consider this old conundrum. Suppose you're running at a high speed -- again, say, four-fifths the speed of light -- carrying a 10-foot pole parallel to the ground. You're running toward a 10-foot shack with front and back doors, both of which are controlled by a remote actuator, which I control. At a push of a button, I can make both doors close at precisely the same time.
My goal is to trap you and your pole in the shack. Since your pole is 10 feet long, this seems like a tricky task, requiring absolutely precise timing. But I take advantage of the fact that you're running so fast. At a rate of four-fifths the speed of light, Equation 14 tells me that your 10-foot pole will be compressed to a length of 6 feet, so I have plenty of time to capture you and your pole in the shack. (I have quick reflexes.) No problem!
But you see things differently. From your point of view, it's the shack that's moving relative to you, and is therefore compressed to a depth of 6 feet. Your pole is safely longer than that, so there's no way I can possibly succeed. Again, no problem! Of course there's no problem. But who's the one with no problem?
Amazing as it may sound, neither of us has a problem -- we're both right. What Einstein discovered is that the weird effects that he had already deduced from the constancy of the speed of light forced him, as it will force us, to reject yet another cherished Newtonian notion -- that of absolute simultaneity. The principle of absolute simultaneity says that if you see two things happen at the same time, I'll see them happen at the same time, also. But Einstein discovered that wasn't the case most of the time; he discovered that we only agree if either (a) we're at rest with respect to one another, or (b) the two events also occur at the same place, with respect to our relative motion.
In our little chestnut of a problem, neither (a) nor (b) is true. We're not at rest with one another, and the two events -- the front door closing and the back door closing -- don't occur at the same place, relative to your direction of motion (from one door to the other). So what "really" happens? As I see it, you and your 6-foot pole go into the shack, I close both doors at the same time, and then you and your pole continue on, busting through the back door.
What you see, instead, is the following. You and your 10-foot pole go partway into the 6-foot shack. Then the doors close, but from your point of view, they don't close at the same time. Instead, the back door closes first, then your pole busts through it, and then the front door closes behind you. By rejecting the notion of absolute simultaneity, Einstein explained how either observer could see relativistic effects experienced by the other, relatively moving observer. In general, events ahead in the direction of travel are advanced in time, events behind us in the direction of travel are retarded in time.
This also explains how you could see my clock slowed down, and I could see your clock slowed down. Isn't that some kind of impossible cycle? But it turns out that in order to synchronize clocks, you have to agree on simultaneous events. So long as you and I remain moving at a constant relative rate of speed, we can't do that, and we can't decide which one of us is "really" right -- because we both are! It's only when we come to a rest with respect to one another that we can again compare clocks. In a certain sense, one clock only "really" slows down with respect to another because it accelerates with respect to the first -- inertial clocks don't slow down. But that's covered in Einstein's general theory of relativity --and therefore a matter for another essay.
Is that all there is to special relativity? No! There is yet another transform that involves the same gamma factor, and that is an even more mysterious effect, that of mass dilation. Suppose that while you're in the boxcar, moving along at 0.8c and tossing the tennis ball at 45 feet per second, it heads straight out of the boxcar, perpendicular to the train tracks. Meanwhile, the stationary observer on the ground throws a tennis ball back at you, also at 45 feet per second. Just by chance, the two balls happen to collide in mid-air. If the tennis balls are identical, what should happen? Does the observer's ball knock your ball back toward the tracks, or do you knock his ball away from the tracks?
I'll explain it one way to make you think it goes back toward the tracks, and then I'll explain it another way to make you think it goes away from the tracks. The first way is, your ball, which travels 45 feet in each second from your point of view, takes longer to travel those same 45 feet from the observer's point of view. In fact, Equation 8 says it should take 5/3 of a second to travel 45 feet, so from the observer's point of view, it's only 45 feet, divided by 5/3 second, or 27 feet per second.
Actually, that's only the motion of the ball in the direction of the observer. It has a high sideways velocity -- 0.8c, and this speed is imparted by the train -- but that speed is precisely irrelevant to the ability of your ball to knock his ball. Since your ball is going slower in the direction of the observer, his ball should knock yours back toward the tracks.
The problem is, you can reason the same exact way, but in reverse. From your point of view, it's his ball that's slowed down in time, and going only 27 feet per second. Therefore, your ball should knock his ball away from the train tracks. So what really happens?
Symmetry demands that neither ball knocks the other one back -- both rebound equally from the mid-air collision. The ability of a ball to knock around other things depends on its momentum p, which is defined as
(15) p = mv
If its measured velocity is only 3/5 of what it "should" be, then in order to compensate, in order to maintain the same momentum p, its mass must be greater -- 5/3 of what it "should" be. In other words, the observer sees your ball as both slower (at least with respect to its motion toward him) and more massive than his ball, and the two factors exactly compensate for one another. You see the same effects with respect to his ball. To put it more generally, mass is increased in the same proportion that time is dilated:
(16) m / mo = 1 / sqrt(1 - (v / c)2) = y
And that is the third and last of the Lorentz transform equations.
So now you know a little about how special relativity is derived. But do we understand why special relativity works? That is, why does time dilate? Why do objects compress in the direction of motion? And how on earth do objects somehow gain mass just by virtue of moving rapidly?
The answer is, nobody knows why! We know that it must all be true, because of the constancy of the speed of light and because the predictions of special relativity have been proven time and time again in experiments. But no one really knows why space and time work that way. It turns out that physics, and science in general, is not very good at answering those kinds of "Why?" questions. The best we can do is take nature's clues and figure out as much as we can, and leave it at that.
Adapted from Astronomical Games: April 2001.
Brian Tung is a computer scientist by day and avid amateur astronomer by night. He is an active member of the Los Angeles Astronomical Society and runs his own astronomy Web site. His previous publications in Strange Horizons can be found in our archives.
You must log in to post a comment.