Why Does E=mc²? (And Why Should We Care?) - Brian Cox, Jeffrey R. Forshaw (2009)
Chapter 1. Space and Time
What do the words “space” and “time” mean to you? Perhaps you picture space as the blackness between the stars as you turn your gaze toward the sky on a cold winter’s night. Or maybe you see the void between earth and moon sailed by spacecraft clad in golden foil, bedecked with the stars and stripes, piloted into magnificent desolation by shaven-headed explorers with names like Buzz. Time may be the tick of your watch or the reddening of the leaves as the earth’s yearly circuit of the sun tilts northern latitudes toward shade for the 5 billionth time. We all have an intuitive feel for space and time; they are part of the fabric of our existence. We move through space on the surface of our blue world as time ticks by.
During the late years of the nineteenth century, a series of scientific breakthroughs in apparently unrelated fields began to force physicists to reexamine these simple and intuitive pictures of space and time. By the early years of the twentieth century, Albert Einstein’s colleague and tutor Hermann Minkowski was moved to write his now-famous obituary for the ancient arena within which planets orbit and great journeys are made: “From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right.”
What could Minkowski have meant by a blend of space and time? To understand this almost mystical-sounding statement is to understand Einstein’s special theory of relativity—the theory that introduced the world to that most famous of all equations, E = mc2, and placed forever center-stage in our understanding of the fabric of the universe the quantity with the symbol c, the speed of light.
Einstein’s special theory of relativity is at its heart a description of space and time. Central to the theory is the notion of a special speed, a speed beyond which nothing in the universe, no matter how powerful, can accelerate. This speed is the speed of light; 299,792,458 meters per second in the vacuum of empty space. Traveling at this speed, a flash of light beamed out from Earth takes eight minutes to pass by the sun, 100,000 years to cross our own Milky Way galaxy, and over 2 million years to reach our nearest galactic neighbor, Andromeda. Tonight, the largest telescopes on Earth will gaze outward into the blackness of space and capture ancient light from distant, long-dead suns at the edge of the observable universe. This light began its journey over 10 billion years ago, several billion years before the earth was formed from a collapsing cloud of interstellar dust. The speed of light is fast, but nowhere near infinitely so. When faced with the great distances between the stars and galaxies, light speed can be frustratingly slow; slow enough that we can accelerate very small objects to within a fraction of a percent of the speed of light with machines like the 27-kilometer Large Hadron Collider at the European Center for Particle Physics (CERN) in Geneva, Switzerland.
The existence of such a special speed, a cosmic speed limit, is a strange concept. As we will discover later in this book, linking this special speed with the speed of light turns out to be something of a red herring. It has a much deeper role to play in Einstein’s universe, and there is a good reason why light travels at the speed it does. We will get to that later on. For now, suffice to say that when objects approach the special speed, strange things happen. How else could an object be prevented from accelerating beyond that speed? It’s as though there were a universal law of physics that prevented your car going faster than seventy miles per hour, no matter how large the engine. Unlike a speed restriction, however, this law is not something that needs to be enforced by some kind of ethereal police force. The very fabric of space and time is constructed in such a way that it is absolutely impossible to break the law, and this turns out to be extremely fortunate, for otherwise there would be unpleasant consequences. Later, we shall see that if it were possible to exceed the speed of light, we could construct time machines capable of transporting us backward through history to any point in the past. We could imagine journeying back to a time before we were born and, by accident or design, preventing our parents from ever meeting. This makes for excellent science fiction, but it is no way to build a universe, and indeed Einstein found that the universe is not built like this. Space and time are delicately interwoven in a way that prevents such paradoxes from occurring. However, there is a price to pay: We must jettison our deeply held notions of space and time. Einstein’s universe is one in which moving clocks tick slowly, moving objects shrink, and we can journey billions of years into the future. It is a universe in which a human lifetime can be stretched almost indefinitely. We could watch the sun die, the earth’s oceans boil away, and our solar system be plunged into perpetual night. We could watch the birth of stars from swirling dust clouds, the formation of planets and maybe the origins of life on new, as yet unformed worlds. Einstein’s universe allows us to journey into the far future, while keeping the doors to the past firmly locked behind us.
By the end of this book, we will see how Einstein was forced to such a fantastical picture of our universe, and how this picture has been shown to be correct in many scientific experiments and technological applications. The satellite navigation system in your car, for example, is designed to account for the fact that time ticks at a different rate on the orbiting satellites than it does on the ground. Einstein’s picture is radical: Space and time are not what they seem.
But we are getting ahead of ourselves. To understand and appreciate Einstein’s radical discovery, we must first think very carefully about the two concepts at the heart of relativity theory, space and time.
Imagine you are reading a book while riding on an aircraft. At 12:00 you glance at your watch, decide to put your book down, leave your seat, and walk down the aisle to chat with your friend ten rows in front of you. At 12:15 you return to your seat, sit down, and pick up your book. Common sense tells you that you have returned to the same place. You had to walk the same ten rows to get back to your seat, and when you returned your book was where you left it. Now think a little more deeply about the concept of “the same place.” This might seem a little pedantic, because it’s intuitively obvious what we mean when we describe a place. We can call a friend and arrange to meet up for a drink in a bar, and the bar won’t have moved by the time we both arrive. It will be in the same place that we left it, quite possibly the night before. Many things in this opening chapter will appear at first sight to be pedantic, but stick with it. Thinking carefully about these apparently obvious concepts will lead us in the footsteps of Aristotle, Galileo Galilei, Isaac Newton, and Einstein. How, then, could we go about defining precisely what we mean by “the same place”? We already know how to do this on the surface of the earth. A globe has a set of grid lines, lines of latitude and longitude, drawn onto its surface. Any place on the earth’s surface can be described by two numbers, representing the position on this grid. For example, the city of Manchester in the UK is located at 53 degrees 30 minutes north, and 2 degrees 15 minutes west. These two numbers tell us exactly where to find Manchester, given that we all agree on the locations of the equator and the Greenwich Meridian. Therefore, by simple analogy, one way to pin down the location of any point, whether on the earth’s surface or not, would be to picture an imaginary three-dimensional grid, extending upward from the earth’s surface and into the air. Indeed, the grid could also carry on downward through the center of the earth and out the other side. We could then describe where everything in the world sits relative to the grid, whether in the air, on the surface, or below ground. In fact, we needn’t stop with just the world. The grid could extend outward beyond the moon, past Jupiter, Neptune, and Pluto, beyond even the edge of the Milky Way galaxy to the farthest reaches of the universe. Given our giant, possibly infinitely large, grid we can work out where everything is, which to paraphrase Woody Allen, is very useful if you’re the kind of person who can never remember where you put things. Our grid therefore defines an arena within which everything exists, a kind of giant box containing all objects in the universe. We may even be tempted to call this giant arena “space.”
Let’s get back to the question of what is meant by “the same place” and return to the aircraft example. You might suppose that at 12:00 and 12:15 you were at the same point in space. Now imagine what the sequence of events looks like to a person sitting on the ground watching the plane. If she sees the plane fly overhead at 600 miles per hour, she would say that between 12:00 and 12:15 you had moved 150 miles. In other words, you were at different points in space at 12:00 and 12:15. Who is correct? Who has moved, and who has stood still?
If you can’t see the answer to this apparently simple question, then you are in good company. Aristotle, one of the greatest minds of ancient Greece, got it dead wrong. Aristotle would have answered unequivocally that it is you, the passenger on the aircraft, who is moving. Aristotle believed that the earth stands still at the center of the universe. The sun, moon, planets, and stars rotate around the earth attached to fifty-five concentric crystalline spheres, stacked inside each other like Russian dolls. He therefore shared our intuitively satisfying concept of space: the box or arena in which the earth and the spheres are placed. To modern ears, this picture of the universe consisting only of the earth and a set of spinning spheres sounds rather quaint. But think for a moment about what conclusion you would draw if nobody had told you that the earth rotates around the sun and that the stars are distant suns, some many thousands of times brighter than our nearby star but billions and billions of miles away. It certainly doesn’t feel like the earth is adrift in an unimaginably large universe. Our modern worldview was hard-won and is often counterintuitive. If the picture of the universe we have developed through thousands of years of experiment and thought was obvious, then the greats of the past, such as Aristotle, would have worked it out for themselves. This is worth remembering if you find any of the concepts in this book difficult; the greatest minds of antiquity may well have agreed with you.
To find the flaw in Aristotle’s answer, let us accept his picture for a moment and see where it leads. According to Aristotle, we should fill space with imaginary grid lines centered on the earth and work out where everything is, and who is doing the moving. If we accept this picture of space as a box filled with objects, with the earth fixed at its center, then it is obvious that you, the passenger on the plane, have changed your position in the box, while the person watching you fly by is standing still on the surface of the earth, hanging motionless in space. In other words, there is such a thing as absolute motion and therefore absolute space. An object is in absolute motion if it changes its position in space, as measured against the imaginary grid fixed to the center of the earth, as time ticks by.
A problem with this picture, of course, is that the earth is not standing motionless at the center of the universe; it is a spinning ball in orbit around the sun. In fact, the earth is moving at about 67,000 miles per hour relative to the sun. If you go to bed at night and sleep for eight hours, you’ll have traveled over half a million miles by the time you wake up. You could even claim that, in about 365 days, your bedroom would have returned to exactly the same point in space since the earth would have completed one full orbit around the sun. You might therefore decide to change your picture a little, while keeping the spirit of Aristotle’s view intact. Why not center the grid on the sun? It’s a simple enough thought, but it’s wrong too because the sun itself is in orbit around the center of the Milky Way galaxy. The Milky Way is our local island of over 200,000 million suns, and as you can probably imagine it’s very large and takes quite a while to get around. The sun, with the earth in tow, is traveling around the Milky Way at 486,000 miles per hour, at a distance of 156,000 trillion miles from the center. At this speed, it takes 226 million years to complete one orbit. And so, perhaps one more step might be sufficient to save Aristotle. Center the grid at the center of our Milky Way galaxy and you could be led to another evocative thought: As you lie in your bed, imagine what the world looked like the last time the earth was “here” at this very point in space. A dinosaur was grazing in the early morning shadows, eating prehistoric leaves at the place where your bedroom now stands. Wrong again. In fact, the galaxies themselves are racing away from each other, and the more distant the galaxy, the faster it recedes from us. Our motion among the myriad galaxies that make up the universe appears to be extremely difficult indeed to pin down.
So Aristotle has a problem, because it seems to be impossible to define exactly what is meant by the words “standing still.” In other words, it seems impossible to work out where to center the imaginary grid against which we can work out where things are, and thereby decide what is standing still and what is moving. Aristotle himself never had to face this problem because his picture of a stationary Earth surrounded by rotating spheres was not seriously challenged for almost 2,000 years. Perhaps it should have been, but as we have already said, these things are far from obvious even to the greatest of minds. Claudius Ptolemaeus, known today as Ptolemy, worked in the great Library of Alexandria in Egypt in the second century. He was a careful observer of the night sky, and he worried about the apparently strange motion through the heavens of the five then-known planets, or “wandering stars,” from which the word “planet” is derived. When viewed from Earth over many months, the planets do not follow a smooth path across the starry background, but appear to perform loop-the-loops in the sky. This strange motion is known as retrograde motion and had in fact been known for many thousands of years before Ptolemy. The ancient Egyptians described Mars as the planet “who travels backward.” Ptolemy agreed with Aristotle that the planets were rotating around a stationary Earth, but to explain the retrograde motion he was forced to attach them to smaller off-center rotating wheels, which in turn were attached to the spinning spheres. This rather complicated model was able to explain the motion of the planets across the night sky, although it is far from elegant. The true explanation of the retrograde motion of the planets had to wait for the mid-sixteenth century and Nicholas Copernicus, who proposed the more elegant (and more correct) explanation that the earth is not stationary at the center of the universe, but in fact orbits around the sun along with the rest of the planets. Copernicus’s work was not without its detractors and was removed from the Catholic Church’s banned list only in 1835. Precision measurements by Tycho Brahe, and the work of Johannes Kepler, Galileo, and Newton, finally established not only that Copernicus was correct, but led to a theory of planetary motion in the form of Newton’s laws of motion and gravitation. Those laws stood unchallenged as our best picture of the motion of the wandering stars and indeed the motion of all objects under gravity, from spinning galaxies to artillery shells, until Einstein’s general theory of relativity came along in 1915.
This constantly shifting view of the position of the earth, the planets, and their motion through the heavens should serve as a lesson to anyone who is absolutely convinced that they know something. There are many things about the world that appear at first sight to be self-evidently true, and one of them is that we are standing still. Future observations can always surprise us, and they often do. Perhaps we should not be too surprised that nature sometimes appears counterintuitive to a tribe of observant, carbon-based ape descendants roaming around on the surface of a rocky world orbiting an unremarkable middle-aged star at the outer edge of the Milky Way galaxy. The theories of space and time we discuss in this book may well—in fact, probably will—turn out to be approximations to an as yet undiscovered deeper theory. Science is a discipline that celebrates uncertainty, and recognizing this is the key to its success.
Galileo Galilei was born twenty years after Copernicus proposed his sun-centered model of the universe, and he thought very deeply about the meaning of motion. His intuition would probably have been the same as ours: The earth feels to us as though it is standing still, although the evidence from the motion of the planets across the sky points very strongly to the fact that it is not. Galileo’s great insight was to draw a profound conclusion from this seeming paradox. It feels like we are standing still, even though we know we are moving in orbit around the sun, because there is no way, not even in principle, of deciding what is standing still and what is moving. In other words, it only ever makes sense to speak of motion relative to something else. This is an incredibly important idea. It might seem obvious in some sense, but to fully appreciate its content requires some thought. It might seem obvious because, clearly, when you sit on the plane with your book, the book is not in motion relative to you. If you put it down on the table in front of you, it stays a fixed distance from you. And of course, from the point of view of someone on the ground, the book moves through the air along with the aircraft. The real content of Galileo’s insight is that these statements are the only ones that can be made. And if all you can do is speak of how the book moves relative to you as you sit in your aircraft seat, or relative to the ground, or relative to the sun, or relative to the Milky Way, but always relative to something, then absolute motion is a redundant concept.
This rather provocative statement sounds superficially profound in the way that Zenlike utterances from fortune-tellers often do. In this case, however, it does turn out to be a great insight; Galileo deserves his reputation. To see why, let’s say that we want to establish whether Aristotle’s grid, which would allow us to judge whether something is in absolute motion, is useful from a scientific perspective. Useful in a scientific sense means that the idea has observable consequences. That means it has some kind of effect that can be detected by carrying out an experiment. By “experiment,” we mean any measurement of anything at all; the swing of a pendulum, the color of light emitted by a burning candle flame, or the collisions of subatomic particles in the Large Hadron Collider at CERN (we’ll come back to this experiment later on). If there are no observable consequences of an idea, then the idea is not necessary to understand the workings of the universe, although it might have some sort of chimerical value in making us feel better.
This is a very powerful way of sorting out the wheat from the chaff in a world full of diverse ideas and opinions. In his china teapot analogy, the philosopher Bertrand Russell illustrates the futility of holding on to concepts that have no observable consequences. Russell asserts that he believes there is a small china teapot orbiting between Earth and Mars, which is too small to be discovered by the most powerful telescopes in existence. If a larger telescope is constructed and, after an exhaustive and time-consuming survey of the entire sky, finds no evidence of the teapot, Russell will claim that the teapot is slightly smaller than expected but still there. This is commonly known as “moving the goalposts.” Although the teapot may never be observed, it is an “intolerable presumption,” says Russell, on the part of the human race to doubt its existence. Indeed, the rest of the human race should respect his position, no matter how preposterous it appears. Russell’s point is not to assert his right to be left alone to his personal delusions, but that devising a theory that cannot be proved or disproved by observation is pointless in the sense that it teaches you nothing, irrespective of how passionately you may believe in it. You can invent any object or idea you like, but if there is no way of observing it or its consequences, you haven’t made a contribution to the scientific understanding of the universe. Likewise, the idea of absolute motion will mean something in a scientific context only if we can devise an experiment to detect it. For example, we could set up a physics laboratory in an aircraft and carry out high-precision measurements on every conceivable physical phenomenon, in a last valiant attempt to detect our movement. We could swing a pendulum and measure the time it takes to tick, we could conduct electrical experiments with batteries, electric generators, and motors, or we could watch nuclear reactions take place and make measurements on the emitted radiation. In principle, with a big enough aircraft, we could carry out pretty much any and every experiment that has ever been conducted in human history. The key point that underpins this entire book and forms one of the very cornerstones of modern physics is that, provided the aircraft is not accelerating or decelerating, none of these experiments will reveal that we are in motion. Even looking out the window doesn’t tell us this, because it is equally correct to say that the ground is flying past beneath us at six hundred miles per hour and that we are standing still. The best we can do is to say, “we are stationary relative to the aircraft,” or “we are moving relative to the ground.” This is Galileo’s principle of relativity; there is no such thing as absolute motion, because it cannot be detected experimentally. This probably won’t come as much of a shock, because we really do know it already at an intuitive level. A good example is the experience of sitting on a stationary train as the train on the next platform slowly pulls out of the station; for a split second it feels like we are the ones doing the moving. We find it difficult to detect absolute motion because there is no such thing.
This may all seem rather philosophical, but in fact such musings lead to a profound conclusion about the nature of space itself, and they allow us to take the first step along the path to Einstein’s theories of relativity. So what conclusion about space can be drawn from Galileo’s reasoning? The conclusion is this: If it is in principle impossible to detect absolute motion, it follows that there is no value in the concept of a special grid that defines “at rest,” and therefore no value in the concept of absolute space.
This is important, so let us investigate it in more detail. We have already established that if it were possible to define a special Aristotelian grid covering the whole universe, then motion relative to that grid could be defined as absolute. We have also argued that since it is not possible to design an experiment that can tell us whether we are in motion, we should jettison the idea of that grid, simply because we can never work out to what it should be fixed. But how then should we define the absolute position of an object? In other words, where are we in the universe? Without the notion of Aristotle’s special grid, these questions have no scientific meaning. All we can speak of are the relative positions of objects. There is therefore no way of specifying absolute positions in space, and that is what we mean when we assert that the notion of absolute space itself has no meaning. Thinking of the universe as a giant box, within which things move around, is a concept that is not required by experiment. We can’t overemphasize how important this piece of reasoning is. The great physicist Richard Feynman once said that no matter how beautiful your theory, no matter how clever you are or what your name is, if it disagrees with experiment, it’s wrong. In this statement is the key to science. Turning this statement around, if a concept is not testable by experiment, then we have no way of telling whether it’s right or wrong, and it simply doesn’t matter either way. Of course, that means we could still assume that an idea holds true, even if it isn’t testable, but the danger is that in so doing we run the risk of hindering future progress because we are holding on to an unnecessary prejudice. So, without any possible means to identify a special grid, we are freed from the notion of absolute space, just as we have been freed from the concept of absolute motion. So what?! Well, being freed from the millstone of absolute space played a crucial role in allowing Einstein to develop his theory of space and time, but this will have to wait until the next chapter. For now, we have established our freedom, but we haven’t acted as liberated scientists just yet. To whet the appetite, let us merely state that if there is no absolute space, then there is no reason why two observers should necessarily agree on the size of an object. That really should strike you as bizarre—surely if a ball has a diameter of 4 centimeters that is the end of the matter, but without absolute space it need not be.
So far we have discussed in some detail the connection between motion and space. What, then, of time? Motion is expressed as speed, and speed can be measured in miles per hour—that is, the distance traveled through space in a particular interval of time. In this way, the notion of time has in fact already entered into our thinking. Is there anything to be said of time? Is there some experiment we can do to prove that time is absolute, or should we also jettison this even more deeply held concept? Although Galileo dispensed with the notion of absolute space, his reasoning has nothing at all to teach us about absolute time. Time is immutable, according to Galileo. Immutable time means that it is possible to imagine perfect little clocks, all synchronized to show the same time, ticking away at every point in the universe. One clock could be on a plane, one on the ground, one (a tough one) at the surface of the sun, and one in orbit around a distant galaxy, and providing they are perfect timekeepers, they will read the same time as each other now and forever. Astonishingly, this seemingly obvious assumption turns out to be in direct contradiction with Galileo’s statement that no experiment can tell us whether we are in absolute motion. Unbelievable as it may seem, the experimental evidence that finally destroyed the notion of absolute time emerged from the type of experiments many of us remember from school physics classes: batteries, wires, motors, and generators. To address the notion of absolute time, we must first take a detour into the nineteenth century, the golden age of discovery for electricity and magnetism.