Why Does E=mc²? (And Why Should We Care?) - Brian Cox, Jeffrey R. Forshaw (2009)
Chapter 3. Special Relativity
In Chapter 1 we succeeded in establishing that the very intuitive Aristotelian view of space and time was laden with excess baggage. That is to say, we showed that there is simply no need to view space as the fixed, immutable, and absolute structure in which things happen. We also saw how Galileo appreciated the irrelevance of holding on to the notion of absolute space, while firmly maintaining the idea of a universal time. In the last chapter, we took a detour into the nineteenth-century physics of Faraday and Maxwell, where we learned that light is none other than a symbiosis of electric and magnetic fields surging forward in perfect agreement with Maxwell’s beautiful equations. Where does all that leave us? If we are to dismiss the idea of absolute space, with what are we to replace it? And what does it mean when we allude to the breakdown of the notion of absolute time? The aim of this chapter is to provide answers to these questions.
Albert Einstein is undoubtedly the iconic figure of modern science. His white, unkempt hair and sockless demeanor provide the contemporary shorthand for “professor”; ask a child to draw a scientist and she might well produce something that looks like the old Einstein. The ideas in this book are, however, the ideas of a young man. At the turn of the twentieth century, when Einstein was thinking about the nature of space and time, he was in his early twenties, with a young wife and family. He did not have an academic post at a university or research establishment, although he discussed physics regularly with a small group of friends, often late into the night. An unfortunate consequence of Einstein’s apparent isolation from the mainstream is the modern temptation to look upon him as a maverick who took on the scientific establishment and won; unfortunate because it provides inspiration to any number of crackpots who think they have single-handedly discovered a new theory of the universe and cannot understand why nobody will listen to them. In fact, Einstein was reasonably well connected to the scientific establishment, although it is true that he did not have an easy beginning to his academic career.
What is striking is his persistence in continuing to explore the important scientific problems of the day while being overlooked for university-level academic positions. On emerging from the Swiss Federal Institute of Technology (ETH) in Zürich at the age of twenty-one, having qualified as a specialized teacher in science and mathematics, he took a series of temporary teaching positions that allowed him the time to work on his doctoral thesis. During 1901, while teaching at a private school in Schaffhausen in northern Switzerland, he submitted his doctoral thesis to the University of Zürich, which was rejected. Following that setback, Einstein moved to Bern and famously began his career as a technical expert, third class, in the Swiss patent office. The relative financial stability and freedom this afforded resulted in the most productive years of his life, and arguably the most productive years of any single scientist in history.
Most of this book deals with Einstein’s work leading up to and encompassing his golden year of 1905, in which he first wrote down E = mc2, was finally awarded his PhD, and completed a paper on the photoelectric effect, for which he eventually won the Nobel Prize. Remarkably, Einstein was still working at the patent office in 1906, where his reward for changing our view of the universe forever was to be promoted to technical expert, second class. He finally got a “proper” academic position in Bern in 1908. While one might be tempted to wonder what Einstein could have achieved if he had not been forced to relegate physics to a leisure pursuit during these years, he always looked back with immense fondness at his time in Bern. In his book Subtle Is the Lord, Einstein’s biographer and friend, Abraham Pais, described Einstein’s days at the patent office as “the closest he would ever come to paradise on earth,” because he had the time to think about physics.
Einstein’s inspiration on the road to E = mc2 was the mathematical beauty of Maxwell’s equations, which impressed him to such a degree that he decided to take seriously the prediction that the speed of light is a constant. Scientifically this doesn’t sound like too controversial a step: Maxwell’s equations were built on the foundation of Faraday’s experiments, and who are we to argue with the consequences? All that stands in our way is a prejudice against the notion that something can move at the same speed regardless of how fast we chase after it. Imagine driving down a road at 40 miles per hour and suppose a car passes you traveling at a speed of 50 miles per hour. It seems to be pretty obvious that you see the second car pull away at a net speed of 10 miles per hour. Thinking of this as “obvious” is just the kind of prejudice that we have to resist if we are to follow Einstein and accept that light always streams away from you at the same speed regardless of how fast you are moving. Let us for now accept, as Einstein did, that our common sense might be misleading us, and see where a constant speed of light will lead.
At the heart of Einstein’s theory of special relativity lie two proposals, which in the language of physics are termed axioms. An axiom is a proposition that is assumed to be true. Given the axioms, we can then proceed to work out the consequences for the real world, which we can check using experiments. The first part of this method is an old one, dating back to ancient Greece. Euclid most famously deployed it in his Elements, in which he developed the system of geometry still taught in schools to this day. Euclid constructed his geometry based on five axioms, which he took to be self-evident truths. As we shall see later, Euclid’s geometry is in fact only one of many possible geometries: the geometry of a flat space, such as a tabletop. The geometry of the surface of the earth is not Euclidean and is defined by a different set of axioms. Another even more important example for us, as we shall soon learn, is the geometry of space and time. The second part, checking the consequences against nature, was not much used by the ancient Greeks. If it had been, then the world might well be a very different place today. This seemingly simple step was introduced to the world by Muslim scientists as early as the second century and took hold in Europe much later, in the sixteenth and seventeenth centuries. With the anchor of experiment, science was finally able to make rapid progress, and with that came technological advancement and prosperity.
The first of Einstein’s axioms is that Maxwell’s equations hold true in the sense that light always travels through empty space at the same speed regardless of the motion of the source or the observer. The second axiom advocates that we are to follow Galileo in asserting that no experiment can ever be performed that is capable of identifying absolute motion. Armed only with these propositions, we can now proceed as good physicists should and explore the consequences. As ever in science, the ultimate test of Einstein’s theory, derived from his two axioms, is its ability to predict and explain the results of experiments. Quoting Feynman more fully this time: “In general we look for a new law by the following process. First we guess it. Then we compute the consequences of the guess to see what would be implied if this law that we guessed is right. Then we compare the result of the computation to Nature, with experiment or experience, compare it directly with observation, to see if it works. If it disagrees with experiment it is wrong. In that simple statement is the key to science. It does not make any difference how beautiful your guess is. It does not make any difference how smart you are, who made the guess, or what his name is—if it disagrees with experiment it is wrong. That’s all there is to it.” It is a terrific quote from a lecture filmed in 1964, and we recommend looking it up on YouTube.
Therefore, our goal for the next few pages is to work out the consequences of Einstein’s axioms. We will begin by using a technique that Einstein himself often favored: the thought experiment. Specifically, we want to explore the consequences of assuming that the speed of light remains constant for all observers, no matter how they are moving relative to each other. To do this, we are going to imagine a clumsy-looking clock called a light clock. The clock consists of two mirrors, between which a beam of light bounces back and forth. We can use this as a clock by counting each bounce of the light beam as one tick. For example, if the mirrors are 1 meter apart, then it takes light approximately 6.67 nanoseconds to complete one round trip.2 You can check this number for yourself: The light has to travel 2 meters and does so at a speed of 299,792,458 meters every second. This would be a very high-precision clock, with around 150 million ticks corresponding to one heartbeat.
Now, imagine putting the light clock on a train that is whizzing along past someone standing on a station platform. The million-dollar question is: How fast does the clock on the train tick according to the person on the platform? Until Einstein, everybody assumed that it ticks at the same rate—one tick every 6.67 nanoseconds.
Figure 2 shows how one tick of the clock on the train looks according to the person standing on the platform. Because the train is moving, the light must travel farther in one tick, as determined from the platform. Put another way, the starting point of the light beam’s journey is not in the same place as its end point according to the person on the platform, because the clock has moved during the tick. In order for the clock to tick at the same rate as it does when it stands still the light must travel a little bit faster. Otherwise it could not complete its longer journey in 6.67 nanoseconds. This is exactly what happens in Newton’s worldview, because the light is given a helping hand by the motion of the train. But—and this is the crucial step—applying Einstein’s logic means that the light cannot speed up because the speed of light must be the same to everyone. This has the disturbing consequence that the moving clock must genuinely take longer to tick, simply because the light has farther to travel, from the perspective of the person on the platform. This thought experiment teaches us that if we are to assert that the speed of light is a constant of nature, as Maxwell seems to be trying to tell us, then it follows that time ticks at different rates depending on how we are moving relative to someone else. In other words, absolute time is not consistent with the notion of a universal light speed.
It is very important to emphasize that this conclusion is not specific to light clocks. There is no important difference between a light clock and a pendulum clock, which works by “bouncing” the pendulum between two places once every second. Or for that matter an atomic clock, which counts the number of peaks and troughs of a light wave emitted from an atom to generate the ticks. Even the rate of decay of the cells in your body could be used as little clocks, and the conclusions would be the same because all these devices measure the passing of time. The light clock is in fact a bit of an old chestnut in the teaching of Einstein’s theory and provokes no end of confused discussion because it is such an unfamiliar clock. It can be tempting to attribute the weird conclusion we have just reached to this lack of familiarity, rather than to recognize it as an insight into the nature of time itself. To do so would be to make a bad mistake—our sole reason for picking a light clock rather than any other type of clock is that we can exploit Einstein’s bizarre demand that light should travel at the same speed for everyone to draw our conclusions. Any conclusion that we draw from thinking about the light clock must also apply to any other kind of clock, for the following reason. Imagine that we seal ourselves into a box with a light clock and a pendulum clock and set them ticking away in sync. If they are very accurate clocks, they will stay in sync and tell the same time forever. Now, let’s put the box onto the moving train. According to Einstein’s second axiom, we should not be able to tell whether we are moving. But if the light clock behaved differently than the pendulum clock, they would drift out of sync and we could say for certain from inside our sealed box that we were moving.3 So a pendulum clock and a light clock must count time in exactly the same way and that means that if the moving light clock is running slow as determined by the person on the platform, then so too must all other moving clocks run slow. This isn’t some kind of optical illusion: The passage of time is slowed down on the moving train according to someone on the platform.
The upshot is that we must either cling to the comforting notion of absolute time and ditch Maxwell’s equations, or ditch absolute time in favor of Maxwell and Einstein. How should we check which is the correct thing to do? We must find an experiment in which we should, if Einstein is right, observe time actually slowing down for moving objects.
To design such an experiment, first we need to work out how fast something should move in order to reveal the proposed effect. It should be quite clear that moving at 70 mph down the highway in a car does not cause time to slow down very much, because we don’t come home after a trip to the store to find that our children have grown older than us while we were away. Silly as this seems, taking Einstein at face value means that this is exactly what does happen, and we would certainly notice the difference if only we could travel fast enough. So what constitutes fast enough? From the viewpoint of the person on the station platform, the light travels along the two sides of the triangle shown in the diagram. Einstein’s argument is that because this is a greater distance for the light to travel than if the clock were standing still, time will pass more slowly because the tick takes longer. All we have to do now is to calculate how much longer (for a given train speed) and we have the answer. We can do this with a little help from Pythagoras.
If you do not want to follow the maths you can skip over the next paragraph, but then you will have to take our word for it that the numbers all work out. That goes for any other maths we might bump into as the book progresses. It is always an option to skip past it and not worry—the mathematics helps provide a deeper appreciation of the physics but it isn’t absolutely necessary to follow the flow of the book. Our hope is that you will have a go with the maths even if you have no prior experience at all. We have tried to keep things accessible. Perhaps the best way to approach the maths is not to worry about it. The logic puzzles that appear in the daily newspapers are much harder to tackle than anything we will do in this book. That said, here comes one of the trickier bits of maths in the book, but the result is worth the effort.
Take a look at Figure 2 again and suppose that the time taken for half of one tick of the clock on the train as measured by the person standing on the platform is equal to T . It is the time taken for the light to travel from the bottom mirror to the top mirror. Our goal is to figure out what T actually is and double it to get the time for one tick of the clock according to the person on the platform. If we did know T, then we could figure out that the length of the longest side of the triangle (the hypotenuse) is equal to cT, i.e., the speed of light (c) multiplied by the time taken for light to travel from the bottom mirror to the top mirror (T). Remember, the distance something travels is obtained by multiplying its speed by the time of the journey. For example, the distance a car travels in one hour at 60 miles per hour is 60 x 1 = 60 miles. It is not hard to work out the result for a two-hour journey. All we are doing here is invoking the formula “distance = speed x time.” Knowing T, we could also figure out how far the clock moves in half of one tick. If the train is moving at a speed, υ, then the clock moves a distance υT each half-tick. Again we did nothing except use “distance = speed x time.” This distance is the length of the base of a right-angled triangle and because we know the length of the longest side, we can go ahead and figure out the distance between the two mirrors using Pythagoras’ theorem. But we know what that distance actually is already—it is 1 meter. So Pythagoras’ theorem tells us that (cT) 2 = 12 + (υT)2. Note the use of parentheses: In mathematics they are used to indicate which operations to carry out first. In this case (υT) 2 means “multiply υ by T and then square up the answer.” That’s all there is to it.
We are nearly done now. We know c, the speed of light, and let’s presume to know the speed of the train, υ. Then we can use this equation to figure out T. The crudest way to do it would be to guess a value of Tand see if it solves the equation. More often than not the guess will be wrong and we’ll need to try another guess. After a while we might hone in on the right answer. Fortunately, we can avoid that tedious process because the equation can be “solved.” The answer is T2 = 1/ (c2—υ2), which means, “first work out c2—υ2 and then divide 1 by that number.” The forward slash is the symbol we will use to denote “divide by.” So 1/2 = 0.5 and a/b means “a divided by b,” etc. If you know a bit of maths, then you’ll probably feel a little bored by now. If not, then you might wonder how we arrived at T2 = 1/(c2—υ2). Well, this isn’t a book on maths, and you’ll just have to trust that we got it right—you can always convince yourself that we got it right by putting some numbers in. Actually, we have the result for T2, which means “T multiplied by T.” We get T by taking the square root. Mathematically, the square root of a number is such that when multiplied by itself we regain the original number; for example, the square root of 9 is 3 and the square root of 7 is close to 2.646. There is a button on most calculators that computes the square root for you. It is usually denoted by the symbol “√” and one would normally write things like 3 = √9. As you can see, the square root is the opposite of squaring, 42 = 16 and √16 = 4.
Returning to the task at hand, we can now write the time taken for one tick of the clock as determined by someone on the platform: It is the time for light to travel up to the top mirror and back down again—that is 2T. Taking the square root of our equation above for T2, and multiplying by 2, we find that 2T = 2/√c2 υ v2 This equation allows us to work out the time taken for one tick as measured by the person on the platform, knowing the speed of the train, the speed of light, and the distance between the two mirrors (1 meter). But the time for one tick according to someone sitting on the train next to the clock is simply equal to 2/c, because for them the light simply travels 2 meters at a speed c (distance = speed x time, so time = distance / speed). Taking the ratio of these two time intervals tells us by how much the clock on the train is running slow, as measured by someone on the platform; it is running slow by a factor of c/√c2 υ v2 , which can also be written, with a little more mathematical rearranging, as 1/ √1 - v2υc2 . This is a very important quantity in relativity theory, and it is usually represented by the Greek letter γ, pronounced “gamma.” Notice that γ is always larger than 1 as long as the clock is flying along at less than the speed of light, because υ/c will be smaller than 1. When υ is very small compared to the speed of light (i.e., for most ordinary speeds, since in units more familiar to motorists the speed of light is 671 million miles per hour), γ is very close to 1 indeed. Only when υ becomes a significant fraction of the speed of light does γ start to deviate appreciably from 1.
Now we are done with the mathematics—we have succeeded in figuring out by exactly how much time slows down on the train as determined by someone on the platform. Let’s put some numbers in to get a feel for things. If the train is moving at 300 kilometers per hour, then you can check that υ2/c2 is a very tiny number: 0.000000000000077. To get the “time stretching” factor γ we need 1/√0.000000000000077 = 1.000000000000039. As expected, it is a tiny effect: Traveling for 100 years on the train would only extend your lifetime by a matter of 0.0000000000039 years according to your friend on the platform, which is slightly above one-tenth of a millisecond. The effect would not be so tiny if the train could whiz along at 90 percent of the speed of light, however. The time-stretching factor would then be bigger than 2, which means that the moving clock would tick at less than half the rate of the station clock according to someone sitting on the platform. This is Einstein’s prediction and, like all good scientists, we have to test it experimentally if we are to believe it. It certainly seems a little unbelievable at this point.
Before we discuss an experiment that settles the argument, let us pause to reflect upon the result that we just uncovered. Let’s look once again at the thought experiment from the point of view of a passenger on the train sitting beside the clock. For the passenger, the clock is not moving and the light simply bounces up and down, just as it would have for a person sitting with the same clock in a café in the station. The passenger must see the clock tick once every 6.67 nanoseconds and 150 million times for every heartbeat, because she is perfectly correct in deciding that the clock is not moving relative to herself, in the spirit of Galileo. Meanwhile, the person on the platform says that the clock on the train took a little longer than 6.67 nanoseconds to perform one tick. After 150 million ticks of the moving clock, his heart will therefore have made slightly more than one heartbeat. This is astonishing: According to the person on the platform, he is aging faster than the passenger sitting on the train.
As we have just seen, the effect is a tiny one for real trains, which don’t travel anywhere near as fast as the speed of light, but it is real nonetheless. In an imaginary world where the train whizzes along a very long track at close to the speed of light, the effect gets magnified and there would be no doubt about it: The person on the platform would age quicker from his perspective.
In real experiments, if we are to test this breakdown in absolute time, then we need to find a way to investigate objects that can move close to the speed of light, for only then will the time-stretching factor γbecome measurably larger than 1. Ideally we’d also like to study an object that has a lifetime, that is to say that it dies. We could then look to see if we could prolong the lifetime of the object simply by making it move fast.
Fortunately for scientists, such objects do exist; in fact, scientists themselves are built out of them. Elementary particles are tiny subatomic objects that by virtue of their smallness are easy to accelerate to vast speeds. They are referred to as elementary because, as far as we can tell with our current technology, they are the smallest building blocks of everything in the universe. We will have much more to say about elementary particles later in the book. For now, we would like to describe just two: the electron and the muon.
The electron is a particle to which we are all indebted, because we are built out of them. It is also the particle that flows through electric wires to light our bulbs and heat our ovens; the electron is the particle of electricity. The muon is identical to the electron in every way, except it is heavier. Why nature should have chosen to give us a copy of the electron that appears to be redundant if all you want to do is to build planets and people, is not something physicists really understand. Whatever the reason for the existence of the muon, it is very useful indeed to scientists wishing to test Einstein’s theory of relativity because it has a short lifetime and it is very small and easy to accelerate to very high speeds. As far as we can tell, electrons live forever, whereas a muon placed at rest beside you would live for something like 2.2 microseconds (a microsecond is one-millionth of a second). When a muon dies, it nearly always turns into an electron and another pair of subatomic particles called neutrinos, but that is extra information that we don’t need. All we need here is that the muon does die. The Alternating Gradient Synchrotron (AGS) facility at Brookhaven National Laboratory on Long Island, New York, provides a very nice test of Einstein’s theory. In the late 1990s, the scientists at Brookhaven built a machine that produced beams of muons circulating around a 14-meter-diameter ring at a speed of 99.94 percent of the speed of light. If muons live for only 2.2 microseconds when they are speeding around the ring, then they would manage only 15 laps of the ring before they died.4 In reality, they managed more like 400 laps, which means their lifetime is extended by a factor of 29 to just over 60 microseconds. This is an experimental fact. Einstein appears to be on the right track, but just how accurate is he?
Here is where the mathematics we did earlier in this chapter becomes very valuable. We have made a precise prediction for the amount by which a little clock traveling at speed actually slows down relative to a clock standing still. We can therefore use our equation to predict by how much time should slow down when traveling at 99.94 percent of the speed of light, and therefore by how much a muon’s lifetime should be extended. Einstein predicts that the muons in Brookhaven should have their time stretched by a factor of γ = 1/√1—υ2/c2 with υ/c = 0.9994. If you have a calculator handy, then type the numbers in and see what happens. Einstein’s formula gives 29, exactly as the Brookhaven experimenters found.
It’s worth taking a brief pause here to ponder what has happened. Using only Pythagoras’ theorem and Einstein’s assumption about the speed of light being the same for everyone, we derived a mathematical formula that allowed us to predict the lengthening of the lifetime of a subatomic particle called a muon when that muon is accelerated around a particle accelerator in Brookhaven to 99.94 percent of the speed of light. Our prediction was that it should live 29 times longer than a muon standing still, and this prediction agrees exactly with what was seen by the scientists at Brookhaven. The more you think about this, the more wonderful it is. Welcome to the world of physics! Of course, Einstein’s theory was already well established in the late 1990s. The scientists at Brookhaven were interested in studying other properties of their muons—the life-enhancing effects of Einstein’s theory provided a bonus, which meant that they got to observe them for longer.
We must therefore conclude, because experiment tells us so, that time is malleable. Its rate of passage varies from person to person (or muon to muon) depending upon how they move about.
As if this rather unsettling behavior of time isn’t enough, something else is lurking, and the alert reader may have spotted it. Think back to those muons whizzing around the AGS. Let’s put a little finish line in the ring and count how many times the muons cross it as they circulate before they die. For the person watching the muons, they cross 400 times because their lifetimes have been extended. How many times would you cross the finish line if you could speed around the ring with the muons? It has to be 400 as well, of course; otherwise the world would make no sense at all. The problem is that according to your watch, as you fly around the ring with the muons, they live for only 2.2 microseconds, because the muon is standing still relative to you and muons live for 2.2 microseconds when they stand still. Nevertheless, you and the muon must still manage to make 400 or so laps of the ring before the muon finally expires. What has happened? Four hundred laps in 2.2 microseconds doesn’t seem possible. Fortunately, there is a way out of this dilemma. The circumference of the ring could be reduced from the viewpoint of the muon. To be entirely consistent, the length of the ring, as determined by you and the muon, must shrink by exactly the same amount that the muon’s lifetime increases. So space must be malleable too! As with the stretching of time, this is a real effect. Real objects do shrink when they move. As a bizarre example, imagine a 4-meter-long car trying to fit into a 3.9-meter-long garage. Einstein predicts that if the car is traveling faster than 22 percent of the speed of light, then it will just about squeeze into the garage, at least for a split second before it crashes through the back wall. Again, if you have been following the maths, then you can check that 22 percent is the right number. Any faster and the car shrinks to below 2.9 meters; any slower and it doesn’t shrink enough.
The discovery that the passage of time can be slowed down and distances can be shrunk is strange enough when applied to the realm of subatomic particles, but Einstein’s reasoning applies equally well to things the size of humans. One day we may even come to rely on this strange behavior for our survival. Imagine living on the earth in the far future. In a few billion years’ time, the sun will no longer be a stable provider of life-sustaining illumination to our world, but a seething, unstable monster of a star that may well engulf our planet as it swells in its final reddening death throes. If we have not become extinct for some other reason by then, it will be necessary for humans to escape our ancestral home and journey to the stars. The Milky Way, our local spiral island of a hundred billion suns, is 100,000 light-years across. This means that light takes 100,000 years to journey across it, as determined by someone on Earth. Hopefully, the need for the last caveat is clear given all that we have been saying. It might seem that humanity’s possible destinations within the Milky Way will be forever restricted to a tiny portion of the stars very close to our home (on astronomical scales) because we could hardly be expected to undertake a journey to distant corners of the galaxy that would take light itself 100,000 years to reach. But here is where Einstein comes to the rescue. If we could build a spaceship that could whisk us into space at speeds very close to light speed, then the distances to the stars would shrink, and the amount of shrinking would increase the closer to light speed we could travel. If we managed to travel at 99.99999999 percent of light speed, then we could travel out of the Milky Way and all the way to the neighboring Andromeda galaxy, almost 3 million light-years away, in a mere fifty years. Admittedly, that looks like a tall order and indeed it is. The big obstacle is figuring out how to power a spaceship so that it could get up to such high speeds, but the point remains: With the warping of space and time, travel to distant parts of the universe becomes imaginable in a way it never was before. If you were part of humanity’s first Andromeda expedition, arriving in a new galaxy after a fifty-year journey, your children born in space might wish to return to their home world and gaze upon the earth with their own eyes for the first time. For them, the Blue Planet would be nothing more than a bedtime space story. Turning the spaceship around, and traveling back to Earth for fifty years, the entire journey to Andromeda and back would have taken one hundred years. By the time they arrived back in Earth orbit, however, a shocking 6 million years would have passed by for the inhabitants of the earth. Would their progenitor civilization have even survived? Einstein has opened our eyes to a weird and wonderful world.