Why Does E=mc²? (And Why Should We Care?) - Brian Cox, Jeffrey R. Forshaw (2009)

Chapter 8. Warping Spacetime

Thus far we have thought of spacetime as fixed and unchanging—something akin to a four-dimensional stage or the arena within which “things happen.” We have also come to appreciate that spacetime has a geometry and that the geometry is most certainly not that of Euclid. We have seen how the idea of spacetime leads naturally to E = mc2 and how this simple equation and the physics it represents has become a foundation stone of both our modern theories of nature and the industrial world. Let us move toward the final twist in our story by asking one last curiosity-driven question: Is it possible that spacetime could be warped and curved differently from place to place in the universe?

The idea of curved space should not be new to us, of course. Euclidean space is flat and Minkowski space is curved. By which we mean that Pythagoras’ theorem doesn’t apply in Minkowski spacetime. Instead, the minus-sign version of the distance equation applies. We also know that the distance between two points in spacetime is analogous to the distance between different places on a map of the earth, in that the shortest distance between two points is not a straight line in the usual sense of the word. So Minkowski spacetime and the surface of the earth are examples of curved spaces. Having said that, the distance between two points in Minkowski spacetime does always satisfy s2 = (ct)2x2, and this means that it curves in the same way everywhere. The same can be said for the surface of the earth. Might it, however, make sense to speak of a surface that curves differently from place to place? What would spacetime look like if this were allowed, and what would the implications be for clocks, rulers, and the laws of physics? To explore this admittedly rather arcane-sounding possibility, we shall once again take a step down from the mind-bending four dimensions to the commonplace two dimensions and focus our attention on the surface of a sphere.

A smooth ball is curved the same way everywhere—that much is obvious. But a golf ball, with dimples in it, is not. Likewise, the earth’s surface is not a perfect sphere. As we zoom in, we see valleys and hills, mountains and oceans. The law for the distance between two points on the earth’s surface is only approximately the same everywhere. For a more precise answer we need to know how the earth’s undulating surface changes as we journey over the mountains and through the valleys between the start and finish of any journey. Could spacetime have dimples in it like a golf ball or mountains and valleys like the earth? Might it “warp” from place to place?

When we first derived the distance equation in spacetime, it seemed that we had no flexibility to change it from place to place. Indeed we argued that the precise form of the distance equation was forced upon us by the constraints of causality. But we did make a very big assumption. We assumed that spacetime is the same everywhere. It is true enough to say that this turns out to be an assumption that works remarkably well and the experimental evidence is largely in its favor, for this assumption was a crucial one on the road to E = mc2. But maybe we have not looked carefully enough. Might spacetime not be the same everywhere, and might this lead to consequences that we can observe? The answer is emphatically yes. To arrive at this conclusion, let us follow Einstein on one last journey. It was a journey that caused him ten years of hard struggle before he finally arrived at yet another majestic destination: the theory of general relativity.

Einstein’s journey to special relativity was triggered by a simple question—what would it mean if the speed of light were the same for all observers? His rather more tortuous journey to general relativity began with an equally simple observation that impressed him so much that he could not rest until he had recognized its true significance. The fact is this: All things fall to the ground with the same acceleration. That’s it . . . that is what excited Einstein so much! It takes a mind like Einstein’s to recognize that such an apparently benign fact could be of very deep significance.

Actually, this is a famous result in physics, known long before Einstein came along. Galileo is credited with being the first to recognize it. Legend has it that he climbed up the Leaning Tower of Pisa, dropped two balls of different masses off the top, and observed that they hit the ground at the same time. Whether he actually carried out the experiment does not really matter; what is important is that he correctly recognized what the outcome would be. We do know for sure that the experiment was eventually performed, not in Pisa but on the moon in 1971 by Apollo 15 commander David Scott. He dropped a feather and a hammer and both hit the ground at the same time. We can’t do that experiment on earth because a feather gets caught by the wind and slows down, but it is quite spectacular when performed in the high vacuum of the lunar surface. There isn’t much need to go all the way to the moon to check that Galileo was right, of course, but that doesn’t detract from the drama of the Apollo 15 demonstration, and the video is well worth watching. The important fact is that everything falls at the same rate, if complicating factors such as air resistance can be removed. The obvious question is why? Why do they fall at the same rate, and why are we making it out to be such a big deal?

Imagine you are standing in a stationary elevator. Your feet press firmly on the ground and your head pushes down on your shoulders. Your stomach rests in place inside your body. Now imagine you have the misfortune to be inside an elevator that is plummeting toward the ground because the cables have been cut. Since everything falls at the same rate, your feet no longer push onto the floor of the lift, your head no longer pushes onto your shoulders, and your stomach floats freely inside your body. In short, you are weightless. This is a big deal because it is exactly as if someone had turned off gravity. An astronaut floating freely in outer space would feel just the same. To be a little more precise, as the lift falls there are no experiments that you can do inside the lift that are able to distinguish between the possibilities that you are plummeting toward earth or floating in outer space. Of course you know the answer because you walked into the elevator, and perhaps the floor counter is whizzing toward “ground” at an alarming rate, but that is not the point. The point is that the laws of physics are identical in the two cases. That is what affected Einstein so deeply. The universality of free fall has a name. It is called the principle of equivalence.

Generally speaking, gravity changes from place to place. Its pull is stronger the closer to the center of the earth you are, although there isn’t that much difference between sea level and the top of Mount Everest. It is much weaker on the moon, because the moon is less massive than the earth. Likewise, the gravitational pull of the sun is much stronger than that of the earth. But wherever you happen to be in our solar system, the force of gravity will not vary too much within your immediate locality. Imagine standing on the ground. The gravity at your feet will be slightly stronger than the gravity at your head but it will be a very small difference. It will be smaller for a short person and bigger for a tall person. You might imagine a tiny ant. The difference in the gravitational pull on its feet compared to its head will be smaller still. Let’s travel the well-worn pathways of the thought experiment one more time and imagine smaller and smaller things, all the way down to a tiny “elevator.” So small is our elevator that the gravity can be assumed to be the same everywhere inside it. The tiny elevator is populated by even tinier physicists whose job it is to carry out scientific experiments within their elevator. Now we can imagine that the little elevator is in free fall. In this case, none of the tiny physicists would ever utter the word “gravity.” A description of the world in terms of observations made by this group of tiny falling physicists has the astonishing virtue that gravity simply does not exist. Nobody would utter the word “gravity” in their tiny squeaky voices because there is no observation that could be made within the elevators that would indicate that there was such a thing. But hang on a second! Clearly something makes the earth orbit the sun. Is this just some clever sleight of hand or are we onto something important?

Let’s leave gravity and spacetime for a moment and return to the analogy of the curved surface of the earth. A pilot planning a trip from Manchester to New York clearly needs to recognize that the earth’s surface is curved. In contrast, when moving between your dining room and your kitchen you can safely ignore the curvature of the earth and assume that the surface is flat. In other words, the geometry is (very nearly) Euclidean. This is ultimately why it took awhile for humans to discover that the earth is not flat but spherical; the radius of curvature is very much bigger than the day-to-day distances that we are used to dealing with. Let’s imagine chopping up the earth’s surface into lots of little square patches, as illustrated in Figure 25. Each patch is pretty near flat, and the smaller we make the patches, the nearer to flat each one is. On each patch, Euclid’s geometry holds sway: Parallel lines don’t cross and Pythagoras’ theorem works. The curvature of the surface becomes evident only when we try to cover large areas of the earth’s surface with our Euclidean patches. We need lots of little patches sewn together to faithfully construct the curved surface of the sphere.

Now let’s return to our little elevator in free fall and imagine it is accompanied by many other little elevators, one at each point in spacetime, in fact. The spacetime inside each is approximately the same everywhere, and the approximation gets better as the elevators get smaller. Now, recall that in Chapter 4 we were very careful to point out our assumption that spacetime should be “unchanging and the same everywhere,” and this was critical in allowing us to construct Minkowski’s spacetime distance formula. Since the spacetime within each tiny elevator is also “unchanging and the same everywhere,” it therefore follows that we can use Minkowski’s distance formula inside each individual little elevator.



Hopefully, the analogy with the sphere is beginning to emerge. For “flat patch on the earth’s surface,” read “falling elevator in spacetime,” and for “curved surface of the earth,” read “curved spacetime.” In fact, physicists often refer to Minkowski spacetime as “flat spacetime” for this very reason. Minkowski spacetime plays the role of flat Euclidean space in the analogy. In this book, we’ve reserved the use of the word “flat” for Euclidean geometry, and the minus sign in the Minkowskian version of Pythagoras’ theorem motivated us to use the term “curved.” Sometimes the use of language is not as straightforward as we might like it to be! So the assembly of little elevators is to spacetime as the assembly of little patches is to the sphere. In each little elevator, gravity has been banished, but we could imagine sewing all the little Minkowski patches together to form a curved spacetime in exactly the same way that we constructed the curved surface of the earth from flat Euclidean patches. If there were no gravity, then we could get by with one big elevator within which the geometry is that of Minkowski. So what we have just learned is that if there is gravity around, we can make it go away but only at the expense of making spacetime curved. What a remarkable conclusion.

Turn this around, and it looks like we have discovered that the force of gravity is actually nothing more than a signal to us that spacetime itself is curved. Is this really true, and what causes the curving? Since gravity is found in the vicinity of matter, we might conclude that spacetime is warped in the vicinity of matter and, since E = mc2, energy. The amount of warping is something we have so far said nothing at all about. And we don’t intend to say very much because it is, to use a well-worn physics phrase, nontrivial. In 1915, Einstein wrote down an equation that was able to quantify exactly how much warping there should be in the presence of matter and energy. His equation improves upon Newton’s age-old law of gravity in that it is automatically in accord with the special theory of relativity (Newton’s law is not). Of course, it gives very similar results to Newton’s theory for most cases we encounter in everyday life, but it does expose Newton’s theory as an approximation. To illustrate the different ways of thinking about gravity, let’s see how Newton and Einstein would describe the way in which the earth orbits the sun. Newton would say something like this: “The earth is pulled toward the sun by the force of gravity, and that pull prevents it from flying off into space, constraining it instead to move in a big circle.”13 It is similar to whirling a ball on a string around your head. The ball will follow a circular path because the tension in the string prevents it from doing otherwise. If you cut the string, the ball would head off in a straight line. Likewise, if you suddenly turned off the sun’s gravity, Newton would say that the earth would then head off into outer space in a straight line. Einstein’s description is quite different and goes like this: “The sun is a massive object and as such it distorts spacetime in its vicinity. The earth is moving freely through spacetime but the warping of spacetime makes the earth go in circles.”

To see how an apparent force might be nothing more than a consequence of geometry, we can consider two friends walking on the earth’s surface. They are told to begin at the equator and to walk due north parallel to each other in perfect straight lines, which they dutifully do. After a while, they will notice that they are coming closer together and, if they carry on walking for long enough, they will bump into each other at the North Pole. Having established that neither of them cheated and wandered off course, they may well conclude that a force acted between them that pulled them together as they walked northward. This is one way to think about things, but there is of course another explanation: The surface of the earth is curved. The earth is doing much the same thing as it moves around the sun.

To get a better feel for what we are talking about, let’s return to one of our intrepid walkers on the surface of the earth. As before, he is told always to walk in a straight line. Locally, that is an instruction he can follow without any confusion because at any point on the earth he can assume Euclidean geometry works just fine and, as a result, the idea of a straight line is clear to him. Even so, he ends up walking in a circular path, although we can think of the circle as being build up of lots of little straight lines. Now let’s return to the case of gravity and spacetime. The notion of straight lines through curved spacetime is entirely analogous to the notion of straight lines on the earth’s surface. The complication arises because spacetime is a four-dimensional “surface,” while the earth’s surface is only two-dimensional. But once again the complication is more to do with our limited imagination rather than any increase in mathematical complexity. In fact, the mathematics of geometry on the surface of a sphere is no harder than the mathematics of geometry in spacetime. Armed with the idea of straight lines (they are also known as geodesics) in spacetime we might be so bold as to suggest how gravity works. We have seen that gravity can be banished in exchange for curved spacetime and that locally the spacetime is the “flat” spacetime of Minkowski. We know very well by this point in the book how things move in such an environment. For example, if a particle is at rest it will remain so (unless something comes along and gives it a push or pull). That means it follows a spacetime trajectory that moves only along the time axis. Likewise, objects that are moving with a constant speed will carry on moving in the same direction and at the same speed (again, unless something comes and knocks them off course). In this case they will follow straight lines on the spacetime diagram that are tilted away from the time axis. So, on each tiny patch of spacetime everything should follow a straight line unless acted upon by some external influence. The whole appearance of gravity emerges when we sew all of the little patches together; for only then do the individual straight lines join together into something more interesting, like the orbit of a planet around the sun. We have not said how to join up the patches in order to build the warping of spacetime, and it is Einstein’s equation of 1915 that determines exactly how we are to do that. But the bottom line could not be much simpler—gravity has been banished in exchange for pure geometry.

So gravity is geometry and all things move along straight lines in spacetime unless they are knocked off course. But at any given point in spacetime there is an infinite number of geodesics, just as there is an infinite number of straight lines passing through any point on the earth’s surface (or any other surface, for that matter). So how are we to figure out which spacetime trajectory an object will move along? The answer is simple enough: Circumstances dictate it. For example, the person on the trek around the earth could start out in any number of directions. He chooses which route to take. Likewise, an object dropped from rest near to the earth will start out on one spacetime geodesic while one that is thrown will start out on a different geodesic. By specifying the direction an object moves through spacetime at any particular point, we therefore know its complete trajectory. Moreover, all objects heading off in that particular direction necessarily follow the same trajectory, irrespective of their internal properties (like mass or electric charge). They just follow a straight line, and that’s all there is to it. In this way the curved spacetime view of gravity beautifully expresses the principle of equivalence that so captivated Einstein.

Our musings on the nature of space and time have led us to understand that the earth is doing nothing more than falling in a straight line around the sun. It is just that the straight line is in a curved spacetime, which manifests itself as a (nearly) circular orbit in space. We have not gone ahead and proved that the sun warps spacetime such that the earth falls along a geodesic whose shadow in three-dimensional space is (nearly) a circle. We haven’t done it simply because it involves too much mathematics. It also involves us making some statement as to how objects actually warp spacetime, and we have been ducking that issue. The mathematical complexity is the main reason why it took Einstein ten years to develop the theory. General relativity is conceptually rather simple but mathematically difficult, although the difficulty most definitely does not obscure its beauty. Indeed many physicists consider Einstein’s theory of general relativity to be the most beautiful of all our theories of nature.

You may well have noticed that nothing we have said has singled out one type of object over another. In particular, light itself should also move through spacetime along a geodesic. In each spacetime patch that it passes over, the light travels along one of the 45-degree lines we introduced in Chapter 4 but, upon sewing all the patches together, we will find a trajectory that bends through space. The bending simply reflects the way in which the spacetime is warped by the presence of mass and energy. Just as for the case of the earth in orbit around the sun, its path through space is a shadow of its four-dimensional geodesic. The power of the equivalence principle and the implied bending of light can be illustrated nicely by another thought experiment.

Imagine that you are standing on the earth and you fire a laser beam horizontally. What happens to it? The principle of equivalence tells us what happens. The light falls toward the ground at exactly the same rate as would an object that is released from rest at the precise moment that the laser is fired. If Galileo had access to a laser and he fired it horizontally off the Leaning Tower of Pisa at the same time as dropping a cannonball, then Einstein predicts that the laser beam would hit the ground at the same time as the cannonball. The problem with this experiment in reality is that the earth’s surface curves away very quickly and the laser would never actually hit the ground because it would run out of earth. If we imagine instead that we are standing on a flat earth, then that problem goes away and we would expect the laser beam to hit the ground at exactly the same time as the cannonball, only a very great deal farther away. In fact, if the cannonball took a second to hit the ground, then the laser would hit the ground one light-second from the tower, which is just over 186,000 miles away.

The description of gravity as geometry is certainly immensely satisfying and it leads to quite startling conclusions but, as we have emphasized throughout this book, it is ultimately useless unless it leads to predictions that can be tested against experiment. Fortunately for Einstein, he had to wait only four years for his exotic predictions to be confirmed.

The first great test of Einstein’s new theory came in 1919 when Arthur Eddington, Frank Dyson, and Charles Davidson wrote a paper titled “A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919.” The paper was published in the Philosophical Transactions of the Royal Society of London and contains the immortal words “both of these point to the full deflection of 1.”75 of Einstein’s generalized relativity theory.” Overnight, Einstein became a global superstar. His esoteric theory of curved spacetime had been vindicated following the not inconsiderable efforts of Eddington, Dyson, and Davidson: To see the eclipse, they had to make expeditions to Sobral in Brazil and Principe, off the western coast of Africa. The eclipse allowed them to look at stars lurking very close to the sun that would otherwise be obscured by its light. This is the starlight best suited to testing Einstein’s theory, because it should be deflected the most since the spacetime curvature is greater the closer you get to the sun. In essence, Eddington, Dyson, and Davidson were looking to see whether the stars shifted their position in the sky as the sun passed by. Quite literally, the sun bends spacetime and acts like a lens, distorting the pattern of stars on the sky.

Today Einstein’s theory has been tested to a high accuracy using some of the most remarkable objects in the universe: spinning neutron stars called pulsars. We met neutron stars and pulsars at the end of Chapter 6, and they are abundant in the universe. Of all the objects we can study accurately from the earth using telescopes, spinning neutron stars are special in that they provide us with large distortions of spacetime and a precise time stamp that rivals the stability of the world’s best atomic clocks. If you wanted to dream up an object that would provide the perfect environment in which to test general relativity, you might well come up with a pulsar. Pulsars deliver their time stamp by beaming out radio waves as they spin. You might like to imagine a lighthouse, shining out a narrow beam that scans around once every second or so. These wonderfully useful objects were discovered quite by accident in 1967 by Jocelyn Bell Burnell and Tony Hewish. If you’re wondering how it is possible to stumble across a spinning neutron star by accident, Bell Burnell was looking for fluctuations in the intensity of radio waves emitted by distant objects known as quasars. The fluctuations were known to be caused by the solar winds in interstellar space. Being a good scientist, however, she was always on the lookout for interesting things in her data and, one November night, she detected a regular signal that she and her supervisor, Hewish, naturally thought was of man-made origin. Subsequent observations convinced them that this could not be the case and that the signal must come from a source beyond our planet. “I went home that evening very cross,” Bell Burnell later said of her observations. “Here was I trying to get a PhD out of a new technique, and some silly lot of little green men had to choose my aerial and my frequency to communicate with us.”

Although pulsars are fairly commonplace in the universe, there is only one known instance where two pulsars are circling each other. The existence of this double pulsar was established by radio astronomers in 2004, and subsequent observations have led to the most precise test to date of Einstein’s general theory.

The double pulsar is a remarkable thing. We now know that it consists of two neutron stars separated by a distance of around 1 million kilometers. Imagine the violence of this system. Two stars, each with the mass of the sun compressed into the size of a city, spinning hundreds of times a second and careering around each other at a distance only three times greater than that from the earth to the moon. The advantage of having two pulsars for Einstein-testers is that the radio waves from one of them sometimes pass very close to the other pulsars. This means that the ultraregular radio beam passes through a region of heavily curved spacetime, which delays its transit. Careful observations can measure the delay and in that way confirm the correctness of Einstein’s theory.

Another virtue of the double pulsar system is that as the stars orbit around each other, they induce ripples in spacetime that propagate outward. The ripples take energy away from the rotational motion of the pair and cause them to slowly spiral inward. The ripples have a name. They are called gravitational waves and their existence is also a prediction of Einstein’s theory (they do not exist in Newtonian gravity). In one of the greatest achievements in experimental science, astronomers using the 64-meter Parkes telescope in Australia, the 76-meter Lovell telescope at Jodrell Bank in the UK, and the 100-meter Green Bank telescope in West Virginia have measured the rate at which the pulsars spiral inward to be just 7 millimeters each day, which is in accord with the prediction of general relativity. The achievement is breathtaking. These are spinning neutron stars orbiting around each other at a distance of a million kilometers and located 2,000 light-years from earth. Their behavior was predicted to millimeter precision using a theory developed in 1915 by a man who wanted to understand why two lumps of stuff dropped off a leaning tower in Pisa three centuries previously hit the ground at the same time.

Ingenious and arcane as the double pulsar measurements are, general relativity makes its presence felt here on Earth too in a much more commonplace phenomenon. The GPS satellite system is ubiquitous throughout the world, and its successful functioning depends upon the accuracy of Einstein’s theories. A twenty-four-strong network of satellites circle the earth at an altitude of 20,000 kilometers, each performing two complete circuits every day. The satellites are used to “triangulate” locations on Earth using precise onboard clocks. In their high-altitude orbits the clocks experience a weaker gravitational field, which means that spacetime is warped differently for them compared to similar clocks on Earth. The effect is that the clocks speed up at a rate of 45 microseconds each day. Apart from the gravitational effect, the satellites are also whizzing around at pretty high speeds (around 14,000 kilometers per hour) and the time dilation predicted by Einstein’s special theory amounts to a slowing down of the clocks by 7 microseconds each day. Taken together, the two effects amount to a net speeding up of 38 microseconds per day. That doesn’t sound like much but ignoring it would lead to a complete failure of the GPS system within a few hours. Light travels around 30 centimeters in 1 nanosecond, which is 1,000-millionth of a second. Thirty-eight microseconds is therefore equivalent to over 10 kilometers in position per day, which wouldn’t make for accurate navigation. The solution is simple enough: The satellite clocks are made to run slow by 38 microseconds per day, which allows the system to work to accuracies of meters rather than kilometers.

The faster running of the GPS satellite clocks relative to the clocks on the ground can be quite easily understood using what we’ve learned in this chapter. In fact, the speeding up of clocks is really a direct consequence of the principle of equivalence. To understand how it comes about, let us travel back in time to 1959 to a laboratory at Harvard University. Robert Pound and Glen Rebka have set about designing an experiment to “drop” light from the top of their laboratory to the basement, 22.5 meters below. If the light falls in strict accord with the principle of equivalence, then, as it falls, its energy should increase by exactly the same fraction that it increases for any other thing we could imagine dropping.14 We need to know what happens to the light as it gains energy. In other words, what can Pound and Rebka expect to see at the bottom of their laboratory when the dropped light arrives? There is only one way for the light to increase its energy. We know that it cannot speed up, because it is already traveling at the universal speed limit, but it can increase its frequency. Remember, light can be thought of as a wave motion; a series of peaks and troughs rather like the water waves emanating outward when a stone is thrown into a still pond. The frequency of the waves is simply the number of peaks (or troughs) that pass a particular point every second, and these peaks and troughs can be used as the ticks of a clock. In particular, in the Pound-Rebka experiment you might imagine that Pound is sitting beside the light source at the top of the tower. He can count how many peaks of light are emitted for every beat of his heart. Now suppose that down in the basement Rebka is sitting beside an identical light source. He too can count how many peaks correspond to each beat of his heart and he should get the same answer as his colleague because they are identical light-source clocks and identical hearts. Okay, they will get exactly the same number only if they really have identical hearts, and that isn’t going to be the case, but we can imagine for the sake of this argument that their hearts do beat as one. Now, let’s think about how Rebka, sitting in the basement, sees the light that is arriving from Pound’s light source at the top. Because the light has gained energy and thereby increased its frequency, it follows that Rebka finds that the peaks are arriving more frequently than they would if the light source were beside him. But the peaks are synchronized to his colleague’s heartbeat. That means that according to Rebka down in the basement, Pound’s heart would be beating faster and so he would age more quickly. The effect is a tiny one, corresponding to a speeding up of one second every 13 million years. It is testament to the skill and ingenuity of Pound and Rebka that they managed to devise an experiment capable of detecting the effect. This speeding up of time is precisely what is happening with the GPS satellite clocks. They are at a much higher altitude than the 22.5 meters of the Harvard laboratory but the basic idea is just the same: Clocks run faster in weaker gravitational fields.

Einstein’s general theory of relativity, confirmed beautifully by experiment, has led us to view spacetime not as a forever-fixed blend of space and time but instead as a more dynamical entity—one that can be manipulated by the presence of matter and, since through E = mc2 we know that mass and energy are interchangeable, energy too. In turn, the dynamical structure of spacetime controls the way objects move through it. No longer are we to think of space as an inert arena within which things happen and of time as the immutable and absolute ticking of a giant clock in the sky. Perhaps the most important lesson to learn in the face of this radical revision is that it is not wise to extrapolate experience beyond its realm. Why should fast-moving things behave according to the same laws as the slow-moving things we encounter in everyday life? Likewise, why should we have a right to infer the behavior of very massive objects by studying only the lighter ones?

Certainly our everyday experiences prove to be a pretty poor guide and, as Einstein has shown us, the deeper level of understanding is so much more elegant. Bringing together as it does such disparate concepts as mass and energy, space and time, and ultimately gravity, Einstein’s special and general theories will stand forever as two of the greatest achievements of the human mind. In the years to come, new understanding built upon new observations and experiments may well lead to a revision in the ideas we have presented here. Indeed many physicists are already anticipating a new order in their quest for more accurate and more widely applicable theories. This humbling lesson not to extrapolate beyond the evidence is not confined to relativity— the other great leap forward in twentieth-century physics was the discovery of the quantum theory, which underpins the behavior of all things at atomic scales and smaller. Nobody ever would have figured out how nature works at small distances based purely on everyday experience. To human beings, whose direct observations are confined to the “big things,” the quantum theory is ridiculously counterintuitive, but in the twenty-first century it underpins so much of our modern lives, from medical imaging to the latest computing technologies, that we must accept it whether we feel comfortable about it or not.

Today physicists are faced with a dilemma. Einstein’s general relativity, our best theory of gravity, cannot be meshed with quantum theory. Either one or both must be revised. Does spacetime “break up” at tiny distance scales? Maybe it does not really exist at all but is instead only an illusion formed by the ever-increasing set of “things that happen.” Are the fundamental objects in nature tiny vibrations of energy known as strings? Or does the solution lie in some other theory yet to be uncovered? This is the frontier of fundamental physics, and those standing on the edge are both thrilled and inspired to be looking out into the unknown.

At the end of a book on Einstein’s theories of relativity, it is all too easy to contribute to an unfortunate cult of personality surrounding the great man, and this is not our intention. Indeed, such a cult probably inhibits future progress because it gives the impression that science is the preserve of supermen in possession of a unique insight inaccessible to the rest of us. Nothing could be farther from the truth. Relativity was not the work of one man, although in a book about relativity this can sometimes appear to be the case. Einstein was undoubtedly one of the great practitioners of the art of science, but as we have emphasized throughout this book, he was led to his radical revision of space and time by the curiosity and skill of many. He was not a freak of nature and his intellect was not supernatural. He was simply a great scientist who did what scientists do: He took simple things seriously and followed through the consequences logically. His genius lay in taking seriously the constancy of the speed of light, as implied by Maxwell’s equations, and the equivalence principle, first appreciated by Galileo.

Our hope is to have written a book that allows nonscientists to understand Einstein’s beautiful theories. This understanding is within reach for nonexperts because science is really not that difficult. Given the right starting point, the road to a deeper understanding of nature is traveled in small steps, carefully taken. Science is at its heart a modest pursuit, and this modesty is the key to its success. Einstein’s theories are respected because they are correct as far as we can tell, but they are no sacred tomes. They will stand, to put it bluntly, until something better comes along. Likewise the great scientific minds are not revered as prophets but as diligent contributors to our understanding of nature. There are certainly those whose names are familiar to millions, but there are none whose reputations can protect their theories from the harsh critique of experiment. Nature is no respecter of reputations. Galileo, Newton, Faraday, Maxwell, Einstein, Dirac, Feynman, Glashow, Salam, Weinberg . . . all are great, the first four were only approximately correct, and the rest may well meet the same fate during the twenty-first century.

Having said all that, we have absolutely no doubt that Einstein’s special and general theories of relativity will forever be remembered as two of the greatest achievements of the human intellect, not least in the way they show how powerful imagination can be. From an inspired mix of pure thought and a little experimental data, a man was able to change our understanding of the very fabric of the universe. That Einstein’s physics is both aesthetically and philosophically pleasing while also being extremely useful delivers an important lesson, the true significance of which is all too rarely appreciated. Science at its best is driven by inquiring minds afforded the freedom to dream, coupled with the technical ability and discipline to think. If the society in which Einstein flourished had decided that it needed a new power source to provide for the needs of its citizens, it is impossible to imagine that some enlightened politician would have channeled public funding into an exploration of the nature of space and time. But as we have seen, it was precisely this road that led to E = mc2 and delivered the keys to unlock the power of the atomic nucleus. From the simplest of ideas—that the speed of a beam of light is one thing upon which everyone in the universe should agree—a box of riches was discovered. “From the simplest of ideas” . . . if there were ever to be an epitaph written for humanity’s greatest scientific achievements, it might begin with these five words. Taking delight in observing and considering the smallest and seemingly most insignificant details of nature has led time and again to the most majestic of conclusions.

We walk in the midst of wonders, and if we open our eyes and minds to them, the possibilities are boundless. Albert Einstein will be remembered for as long as there are humans in the universe both as an inspiration and an example to all those who are captivated by a natural curiosity to understand the world around them.


There have been many attempts, since Michelson and Morley, to detect the ether and all have yielded null results.


A nanosecond is one thousandth of a microsecond, or 0.000000001 seconds.


The sealed box is just to stop us from being distracted by the idea that we could look out of the window of the train to determine whether we are moving. Of course that is irrelevant; looking out of the window only ascertains that we are moving relative to the ground outside.


You can check this for yourself once you know that the circumference of a circle is equal to pi multiplied by the diameter, where pi equals approximately 3.142.


Supposed points on Earth that resonate “psychic energy.”


There is nothing special about it being a ball; it could be any object.


Strictly speaking, this is not true. There is another possible mass lying just 0.000006 eV/c2 above the smallest mass. That tiny difference is very important to radio astronomers, but we will assume it is so close to the smallest mass that it makes no difference.


The energy taken away by the photon is equal to 13.6 eV minus 10.2 eV, which is 3.4 eV.


101 = 10, 102 = 100, etc. So 1026 is equal to 100000000000000000000000000 and you can see why the more compact notation was invented.


10-1 = 0.1, 10-2 = 0.01, etc. So 10-27 has twenty-six zeros after the decimal point.


The largest mass of a neutron star can be estimated in a manner similar to Chandrasekhar’s limit for the largest possible mass of a white dwarf—i.e., by assuming that the neutrons do not travel close to the speed of light if they are to form a neutron star.


Strictly speaking, it is an electron neutrino, because it is produced in conjunction with an antielectron.


Actually, it moves in an ellipse, a slightly squashed circle, but it is pretty close to a circle.


If you know that the potential energy is equal to “mgh,” then you can easily see that this fractional increase is equal to gh/c2 where g is the acceleration due to gravity and h is the height of the drop.