Why Does E=mc²? (And Why Should We Care?) - Brian Cox, Jeffrey R. Forshaw (2009)
Chapter 4. Spacetime
In the previous chapters we followed the historical road to relativity, and in fact our reasoning was not too far from what Einstein originally presented. We have been forced to accept that space is not the great stage upon which the events of our lives are played out. Likewise, time is not something universal and absolute. Instead we moved toward a picture of space and time that is much more malleable and subjective. The great clock in the sky, and in some sense the sky itself, has been banished. It might feel to us like the world is a box within which we go about our business, because that picture allows us to make sense of it quickly and efficiently. The ability to map the movement of things against an imaginary grid is what we might call spatial awareness, and it is clearly important if you are to avoid predators, catch food, and survive in a dangerous and challenging environment. But there is no reason why this model, buried deep within our brains and reinforced over millions of years by natural selection, should be anything other than a model. If a way of thinking about the world confers a survival advantage, then that way of thinking will become ubiquitous. The scientific correctness of it is irrelevant. The important thing is that, because we chose to accept the results of experiments conducted on Faraday’s mottled benchtop and the explanations codified by Maxwell, we have acted like scientists and rejected the comfortable model of space and time that allowed our ancestors to survive and prosper on the ancient plains of Africa. This model has been embedded and reinforced deep within our psyche by our experiences over many millions of years, and discarding it may well be disorientating. That dizzying feeling of confusion, if (hopefully) followed by an epiphany of clarity, is the joy of science. If the reader is feeling the former, we hope to deliver the latter by the end of the book.
This is not a history book. Our aim is to describe space and time in the most enlightening way we can, and it is our view that the historical road to relativity does not necessarily provide the best path to enlightenment. From a modern perspective, over a century after Einstein’s revolution, we have learned that there is a deeper and more satisfying way to think about space and time. Rather than dig any deeper into the old-fashioned textbook view, we are going to start again from a blank canvas. In so doing we will come to understand what Minkowski meant when he said that space and time must be merged together into a single entity. Once we have developed a more elegant picture, we will be well placed to achieve our principal goal—we shall be able to derive E = mc2.
Here is the starting point. Einstein’s theories can be constructed almost entirely using the language of geometry. That is, you don’t need much algebra, just pictures and concepts. At the heart of the matter, there lie only three concepts: invariance, causality, and distance. Unless you are a physicist, two of these will probably be unfamiliar words, and the third familiar but, as we shall see, subtle.
Invariance is a concept that lies at the core of modern physics. Glance up from this book now and look out at the world. Now turn around and look in the opposite direction. Your room will look different from different vantage points, of course, but the laws of nature are the same. It doesn’t matter whether you are pointing north, south, east, or west, gravity still has the same strength and still keeps your feet on the ground. Your TV still works when you spin it around, and your car still starts whether you’ve left it in London, Los Angeles, or Tokyo. These are all examples of invariance in nature. When put like this, invariance seems like little more than a statement of the obvious. But imposing the requirement of invariance on our scientific theories proves to be an astonishingly fruitful thing to do. We have just described two different forms of invariance. The requirement that the laws of nature will not change if we spin around and determine them while facing different directions is called rotational invariance. The requirement that the laws will not change if we move from place to place is called translational invariance. These seemingly trivial requirements turned out to be astonishingly powerful in the hands of Emmy Noether, whom Albert Einstein described as the most important woman in the history of mathematics. In 1918 Noether published a theorem that revealed a deep connection between invariance and the conservation of particular physical quantities. We will have more to say about conservation laws in physics later on, but for now let us just state the deep result Noether discovered. For the specific example of looking at the world in different directions, if the laws of nature remain unchanged irrespective of the direction in which we are facing, then there exists a quantity that is conserved. In this case, the conserved quantity is called angular momentum. For the case of translational invariance, the quantity is called momentum. Why should this be important? Let’s pull an interesting physics fact out of the metaphorical hat and explain it.
The moon moves 4 centimeters farther away from the earth every year. Why? Picture the moon in your mind’s eye as being stationary above the surface of the spinning earth. The water in the oceans directly beneath the moon will bulge out a tiny bit toward the moon because the moon’s gravity is pulling it, and the earth will rotate once a day beneath this bulge. This is the cause of the ocean tides. There is friction between the water and the surface of the earth, and this friction causes the earth’s rate of spin to slow down. The effect is tiny but measurable; the earth’s day is gradually lengthening by approximately two-thousandths of a second per century. Physicists measure the rate of spin using angular momentum, so we can say that the angular momentum of the earth is reducing over time. Noether tells us that because the world looks the same in every direction (to be more precise, the laws of nature are invariant under rotations), then angular momentum is conserved, which means that the total amount of spin must not change. So what happens to the angular momentum the earth loses by tidal friction? The answer is that it is transferred to the moon, which speeds up in its orbit around the earth to compensate for the slowing down of the earth’s rotation. This causes it to drift slightly farther away from the earth. In other words, to ensure that the total angular momentum of the earth and moon system is conserved, the moon must drift into a wider orbit around the earth to compensate for the fact that the earth’s rate of spin is slowing down. This is a very real and quite fantastic effect. The moon is big, and it is drifting farther away from the earth as every year goes by to conserve angular momentum. Italian novelist Italo Calvino found it so wonderful that he wrote a short story called “The Distance of the Moon,” in which he imagined a time in the distant past when our ancestors could sail each night across the ocean in boats to meet the setting moon and clamber onto its surface using ladders. As the moon drifted farther away over the years, there came a night when the moon lovers had to make a choice between becoming trapped on the moon forever or returning to Earth. This surprising and, in the hands of Calvino, strangely romantic phenomenon has its explanation in the abstract concept of invariance and the deep connection between invariance and the conservation of physical quantities.
It is difficult to overstate the importance of the idea of invariance in modern science. At the heart of physics is the desire to produce an intellectual framework that is universal and in which the laws are never a matter of opinion. As physicists, we aim to uncover the invariant properties of the universe because, as Noether well knew, these lead us to real and tangible insights. Identifying the invariant properties is far from easy, however, because nature’s underlying simplicity and beauty are often hidden.
Nowhere in science is this truer than in modern particle physics. Particle physics is the study of the subatomic world; the quest for the fundamental building blocks of the universe and the forces of nature that stick them together. We have already met one of the fundamental forces, electromagnetism. Understanding it led us to an explanation for the nature of light that has launched us on the road to relativity. In the subatomic world there are two other forces of nature that hold sway. The strong nuclear force sticks the atomic nucleus together at the heart of the atom, and the weak nuclear force allows stars to shine and is responsible for certain types of radioactive decay; the use of radiocarbon dating to measure the age of things, for example, relies on the weak nuclear force. The fourth force is gravity, the most familiar perhaps, but by far the weakest. Our best theory of gravity today is still Einstein’s general theory of relativity and, as we shall see in the final chapter, it is a theory of space and time. These four forces act between just twelve fundamental particles to build everything in the world we can see, including the sun, moon, and stars, all the planets in our solar system, and indeed our own bodies. This all constitutes an astonishing simplification of what at first glance appears to be an almost infinitely complicated universe.
Glance out your window. You may be faced with the distorted reflections of a city, as the afternoon light scatters off sheets of steel and glass, or black and white cattle grazing in neatly fenced green fields. But whether cityscape or farmland, the most astonishing thing about practically every window view in the world is the evidence of human intervention. Our civilization is all-pervasive, and yet twenty-first-century physics tells us that, at its heart, it is all a mathematical dance involving a handful of subatomic particles, organized by only four forces of nature over 13.7 billion years. The complexity of human brains and the products of the powerful synthesis between consciousness and dexterous skill that we glimpse outside our windows mask the underlying simplicity and elegance of nature. The scientist’s task is to hunt for those properties that act as a Rosetta stone, to allow us to decipher the language of nature and reveal its beauty.
The tool that allows us to search for and exploit these properties of nature is mathematics. In itself, this is a sentence that throws up deep questions, and entire books have been written attempting to advance plausible reasons as to why it may be so. Quoting Eugene Wigner again: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” Perhaps we will never understand the true nature of the relationship between mathematics and nature, but history has shown that mathematics allows us to organize our thinking in a way that proves to be a reliable guide to a deeper understanding.
As we have been at pains to emphasize, to proceed in this spirit, physicists write down equations, and equations do nothing more than express relationships between different real-world “things.” An example of an equation is speed = distance/time, which we met in the last chapter when we were discussing light clocks: in symbols υ = x/t, where υ is the speed, x is the distance traveled, and t is the time taken to travel the distance x. Very simply, recall that if you travel 60 miles in 1 hour, then you have traveled at a speed of 60 miles per hour. Now, the most interesting equations will be those that are capable of furnishing a description of nature that is agreed upon by everyone. That is, they should deal only in invariant quantities. We could all then agree on what we are measuring, irrespective of our perspective in the universe. According to common sense, the distance between any two points in space is such an invariant quantity, and pre-Einstein it was. But we saw in the previous chapter that it is no such thing. Remember: Common sense is not always reliable. Similarly, the passage of time has become a subjective thing and it varies depending on how fast clocks are moving relative to each other. Einstein has upset the order of things, and we cannot even rely on distance and time to build a reliable picture of the universe. From the point of view of a physicist looking for the deep laws of nature, the equation υ = x/t is therefore of no fundamental use, because it does not express a relationship between invariant quantities. By undermining space and time, we have shaken the very foundations of physics. What, then, are we to do?
One option is to try and reestablish order by making a conjecture. Conjecture is a fancy word for “guess,” and scientists do it all the time—there are no prizes for how smart we are in figuring out the underlying theory; a successful educated guess will do just fine so long as it agrees with experiment. The conjecture is radical: Space and time can be merged into a single entity that we call “spacetime,” and distances in spacetime are invariant. This is a bold assertion and its content will become clearer as we go. When you think about it for a moment, it is perhaps less bold than it seems at first sight. If we are to lose the age-old certainties of absolute, unvarying distances in space and the unchanging tick-tock of time as the great clock in the sky marks the passing of the years, then maybe the only thing to do is to search for some kind of unification of the two seemingly separate concepts. Therefore, our immediate challenge is to search for a new measure of distance in spacetime that does not change depending on how we move around relative to each other. We will need to tread carefully to understand how the spacetime synthesis works. But what exactly does it mean to search for a distance in spacetime?
Suppose I get out of bed at 7 a.m. and finish my breakfast at 8 a.m. The following statements are true given what we know from experiment: (1) I may measure the distance in space from my bed to my kitchen to be 10 meters, but someone whizzing by at high speed will measure a different distance; (2) My watch indicates that I took 1 hour to eat breakfast, but the high-speed observer will record a different time. Our conjecture is that the distance in spacetime between my getting out of bed and my finishing breakfast is something we can all agree upon—i.e., it is invariant. The existence of this consensus is crucial because we want to build up a set of natural laws using only this type of object. Of course, we just guessed that this might be how things are and we certainly haven’t proven anything yet. We haven’t even decided how to calculate distances in spacetime. But to proceed further, we must first explain what is meant by the second of our three key words, causality.
Causality is another seemingly obvious concept whose application will have profound consequences. It is simply the requirement that cause and effect are so important that their order cannot be reversed. Your mother caused your birth, and no self-consistent picture of space and time should allow you to be born before your mother. To construct a theory of the universe in which you could be born first would be nonsense and lead to contradictions. When put in these terms, nobody could argue with the requirement of causality.
It is worth reflecting, however, that humans seem capable of ignoring it on a daily basis. Take prophesy, for example. Figures like Nostradamus are revered to this day for allegedly being able to see events that happen in the future, either in dreams or some other mystical trancelike state. In other words, events that happened centuries after Nostradamus’ death were visible in his lifetime, at least to him. Nostradamus died in 1566, but he is credited with observing the Great Fire of London in 1666, the rise of Napoleon and Hitler, the September 11, 2001, attacks on the United States, and, our own personal favorite, the rise of the Antichrist in Russia in 1999. The Antichrist hasn’t appeared yet but perhaps he/she is still rising and if he/she does appear before this book goes to print, then we stand corrected.
Putting amusing drivel aside, we need to introduce some important terminology. Nostradamus’s death was an “event,” as were the birth of Adolf Hitler and the Great Fire of London. For Nostradamus to observe an event such as the Great Fire that happened after his death would require the ordering of the two events to be reversed. To say this explicitly is almost a tautology; Nostradamus died before the Great Fire, and therefore he could not have observed it. To observe it, the event that is the Great Fire must have been available for viewing before the event that is Nostradamus’s death, and therefore the order of the events must have been reversed. There is an important subtlety: Nostradamus could have caused the Great Fire. We could imagine that he left a sum of money in a bank account that encouraged someone to light a fire in Pudding Lane shortly after midnight on September 2, 1666. This would establish a causal link between the events associated with the life and death of Nostradamus and the events associated with the Great Fire of London. As we shall see later, it is in fact only the ordering of such connected events (called causally connected events) that cannot be reversed—cause and effect are sacred in Einstein’s universe.
Other events occur far enough away from each other in space and time that they could not have any possible influence on each other. Remarkably, the ordering of these can be reversed. Einstein’s theory exploits a loophole that allows the order of events to be switched provided that doing so makes absolutely no difference to the workings of the universe. We shall explain what we mean by “far enough away” later on. For now, we have introduced the concept of causality as an axiom that we shall use to build our theory of spacetime. The success of the theory in predicting the outcome of experiments will of course be the ultimate arbiter. As an aside, Nostradamus did get one prediction right. While suffering from a particularly acute bout of gout, he apparently told his secretary, “You will not find me alive at sunrise.” The next morning he was found dead on the floor.
What has causality got to do with spacetime and, in particular, distances in spacetime? Well, we will soon discover that insisting on a causal universe constrains the structure of spacetime to such an extent that we are left with no choice in the matter. There will be only one way in which we can merge space and time together to manufacture spacetime while simultaneously preserving the causal order of things. Any other way would violate causality and allow us to do fantastical things like going back in time to prevent our own birth or, in Nostradamus’s case, perhaps avoiding a lifestyle that made him susceptible to gout.
Time now to return to the challenge of developing the concept of distance in spacetime. To get warmed up we will set time to one side for the moment and think about the idea of distance in ordinary three-dimensional space, a concept with which we are all familiar. Suppose we try to measure the shortest distance between two cities on a flat map of the earth. As will be very familiar to anyone who has flown on a long-haul flight and watched her progress on the map on the aircraft entertainment system, the shortest distance between any two points on the earth’s surface appears as a curve. This line is known as a great circle. Figure 3 shows a map of the earth, and drawn on it is a line that corresponds to the shortest distance between Manchester and New York. On a globe, this line can be understood but at first glance it is a surprise to see a curved line representing the shortest distance between two points. This occurs because the earth’s surface is not flat, but curved. To be specific, the earth is a sphere. The curved nature of the earth’s surface is also the reason why, on some flat maps, Greenland looks much bigger than Australia, when in reality it is much smaller. The message is clear: Straight lines represent the shortest distance between two points only in flat space. The geometry of flat space is often called Euclidean geometry. What Euclid didn’t know at the time, however, and in fact it did not become clear until the nineteenth century, was that his geometry of flat space is only a specific example of a whole family of different possible geometries, each of which are mathematically consistent and some of which can be used to describe nature. A very good example is the surface of the earth, which is curved and therefore described using a geometry that is non-Euclidean. Specifically, the shortest distance between two points is not a Euclidean straight line.
There are other familiar Euclidean properties that are not obeyed on the surface of the earth. For example, the interior angles of a triangle no longer add up to 180 degrees, and lines that are parallel and point north-south at the equator cross at the poles. If Euclid is no use anymore, we need to figure out how to calculate distances in a curved space, such as on the earth’s surface. One way would be to work directly with a globe and measure out the distances using a piece of string. Now we would be correctly accounting for the curvature of the earth. An airline pilot could stretch a piece of string between two cities on the globe, measure its length with a ruler, and then simply multiply the answer by the ratio in size of the globe and the earth. But maybe we don’t have a globe on hand, or maybe we need to write the computer software that helps airplanes navigate. In either case, we need to do better than a piece of string and figure out an equation that tells us the distance between any two points on the earth’s surface given only their latitude and longitude, and the shape and size of the earth. Such an equation is not too hard to find and if you know a little mathematics you might even try to find it. We don’t need to write it down here, but the point is that an equation exists and it hasn’t got much to do with the Euclidean geometry of a flat tabletop. It does, however, allow one to calculate the shortest distance between two points on a sphere, in much the same way that Pythagoras’ theorem is a recipe for calculating the shortest distance between two points (the hypotenuse) on a tabletop if we know the distances from one corner as measured along the edges of the table. Since straight lines belong in the domain of Euclid, we shall introduce a new term for the shortest distance between two points that applies whether the space is curved or flat. This line is called a geodesic: A great circle is a geodesic on the surface of the earth and a straight line is a geodesic in flat space. So much for distances in three-dimensional space. Now we must decide how to measure distances in spacetime, so let’s go ahead and complicate matters by adding time into the mix.
We already introduced the concepts we will need when we thought about getting out of bed and finishing breakfast in the kitchen. There is no problem in saying that the distance in space between the bed and the kitchen is 10 meters. We could also say, although it sounds rather strange, that the distance in time between getting out of bed and finishing breakfast is 1 hour. This is not how we naturally think about time, because we are not used to describing it in the language of geometry. We would rather say “one hour passed between my getting out of bed and finishing my breakfast.” In the same way, we would not normally say “10 meters have passed since I got out of bed and sat down in the kitchen.” Space is space, and time is time, and never the twain shall be intermingled. But we have set ourselves the task of trying to merge space and time together, because we suspect that this is the only way to rebuild things in a way that fits with Maxwell and Einstein. So let us proceed and see where it leads us. If you are not a scientist, then this may be the most difficult part of the book so far because we are operating in a purely abstract fashion. The capacity for abstract thought is what gives science its power, but also perhaps gives it a reputation as being difficult because it is not a faculty we generally need too much in everyday life. We have already encountered a difficult abstract concept in the form of the electric and magnetic fields, and in fact the abstraction needed to merge space and time together is probably less challenging than that.
What we are doing implicitly in speaking of “the distance in time” is treating time as an additional dimension. We are used to the phrase “3-D,” as in three-dimensional, referring to the fact that space has three dimensions: up and down; left and right; forward and backward. When we try to add time into the framework, so that we can define distances in spacetime, we are in effect creating a four-dimensional space. To be sure, the time dimension behaves differently than the space dimensions. We have complete freedom of movement in space, whereas we go only one way in time, and time doesn’t feel anything like space. But that need not be an insurmountable hurdle. Thinking of time as “just another dimension” is the abstract leap we have to take. The trick, if it sounds too confusing, is to imagine how you might feel if you were a creature that could only ever move forward, backward, left and right. You have never experienced up and down—you live in a flat world. If someone asked you to imagine a third dimension, your flat mind would not be able to grasp it. But if you had a mathematical bent, you might be happy to accept the possibility and, in any case, you could still do the maths even if you couldn’t picture the mysterious extra dimension in your mind’s eye. Likewise for human beings and four-dimensional space. It should become more natural to think of time as “just another dimension” as our story unfolds. If there is one thing we try to teach our students when they first arrive at the University of Manchester, ready to learn to be physicists, it is that everyone gets confused and stuck. Very few people understand difficult concepts the first time they encounter them, and the way to a deeper understanding is to move forward with small steps. In the words of Douglas Adams: “Don’t panic!”
Let us continue in a gentler vain for a moment by noticing something very simple: Things happen. We wake up, we make breakfast, we eat breakfast, and so on. We’ll call the occurrence of a thing “an event in spacetime.” We can uniquely describe an event in spacetime by four numbers: three spatial coordinates describing where it happened and a time coordinate describing when it happened. Spatial coordinates can be specified using any old measuring system. For example, longitude, latitude, and altitude will do if the event is occurring in the vicinity of the earth. So your coordinates in bed might be N 53° 28’ 2.28”, W 2° 13’ 50.52”, and 38 meters above sea level. Your time coordinates are specified using a clock (because time is not universal, we’ll have to say whose clock in order to be unambiguous) and might be 7 a.m. GMT when your alarm goes off and you wake up. So we have four numbers that uniquely locate any event in spacetime. Notice that there is nothing special about the particular choice of coordinates. In fact, these particular coordinates are measured relative to a line passing through Greenwich in London, England. This convention was agreed upon in October 1884 by twenty-five nations, with the only dissenting voice being San Domingo (France abstained). It is a very important concept that the choice of coordinates should make absolutely no difference.
Let’s take the moment when I wake up in bed as our first event in spacetime. The second event could be the event that marks the end of breakfast. We have said that the spatial distance between the two events is 10 meters and the distance in time is 1 hour. To be unambiguous we’d need to say something like “I measured the distance between my bed and my breakfast table using a tape measure whose ends were stretched directly from bed to table” and “I measured the time interval using my bedside clock and the clock sitting in my kitchen.” Don’t forget that we already know that these two distances, in space and in time, are not universally agreed upon. Someone flying past your house in an aircraft would say that your clock runs slow and the distance between your bed and your breakfast table shrinks. Our aim is to find a distance in spacetime upon which everyone agrees. The million-dollar question is then “how do we take the 10 meters and the 1 hour to construct an invariant distance in spacetime?” We need to tread carefully and, just like in the case of distances on the earth’s surface, we shall not assume Euclidean geometry.
If we are to compute distances in spacetime, then we have an immediate problem to resolve. If distance in space is measured in meters and distance in time in seconds, how can we even begin to contemplate combining the two? It is like adding apples and oranges, because they are not the same type of quantity. We can, however, convert distances into times and vice versa if we use the equation we met earlier, υ = x/t. With a miniscule bit of algebra we can write time t = x/υ, or distance x = υt. In other words, distance and time can be interchanged using something that has the currency of a speed. Let us therefore introduce a calibrating speed; call it c. We can then measure time in meters provided we take any time interval and multiply it by our calibrating speed. At this point in our reasoning c really can be any old speed and we have not committed ourselves at all as to its actual value. Actually, this trick of interchanging time and distance is very common in astronomy, where the distance to stars and galaxies is often measured in light-years, which is the distance light travels in one year. This doesn’t seem so strange because we are used to it, but it really is a distance measured in years, which is a unit of time. In the astronomy case, the calibrating speed is the speed of light.
This is progress; we now have time and distance intervals in the same currency. For example, they could both be given in meters, or miles or light-years or whatever. Figure 4 illustrates two events in spacetime, denoted by little crosses. The bottom line is that we want a rule for figuring out how far apart the two events are in spacetime. Looking at the figure, we want to know the length of the hypotenuse given the lengths of the other two sides. To be a little more precise, we shall label the length of the base of the triangle as x while the height is ct. It means that the two events are a distance x apart in space and a distance ct apart in time. Our goal, then, is to answer the question “what is the hypotenuse, s, in terms of x and ct?” Making contact with our earlier example x = 10 meters is the distance in space from bed to kitchen table, and t = 1 hour is the distance in time. So far, since c was arbitrary, ct can be anything and we appear to be treading water. We shall press onward nonetheless.
We have to decide on a means of measuring the length of the hypotenuse, the distance between two events in spacetime. Should we choose Euclidean space, in which case we can use Pythagoras’ theorem, or something more complicated? Perhaps our space should be curved like the surface of the earth, or maybe some other more complicated shape. There are in fact an infinite number of ways that we might imagine calculating distances. We’ll proceed in the way that physicists often do and we will make a guess. Our guess will be guided by a very important and useful principle called Occam’s razor, named after the English thinker William of Occam, who lived at the turn of the fourteenth century. The idea is simple to state but surprisingly difficult to implement in everyday life. It might be summarized as “don’t overcomplicate things.” Occam stated it as “plurality must never be posited without necessity,” which does beg the question: Why didn’t he pay more attention to his own rule when constructing sentences? However it is stated, Occam’s razor is very powerful, even brutal, when applied to reasoning about the natural world. It really says that the simplest hypothesis should be tried first, and only if this fails should we add complication bit by bit until the hypothesis fits the experimental evidence. In our case, the simplest way to construct a distance is to assume that at least the space part of our spacetime should be Euclidean; in other words, space is flat. This means that the familiar way of working out the distance in space between objects in the room in which we are seated reading this book is carried over into our new framework intact. What could be simpler? The question, then, is how we should add time. Another simplifying assumption is that our spacetime is unchanging and the same everywhere. These are important assumptions. In fact, Einstein did eventually relax them and doing so allowed him to contemplate the mind- (and space-) bending possibility that spacetime could be constantly changed by the presence of matter and energy. It led to his general theory of relativity, which is to this day our best theory of gravity. We will meet general relativity in the final chapter, but for the moment we can ignore all these twists and turns. Once we follow Occam and make these two simplifying assumptions, we are left with only two possible choices as to how to calculate distances in spacetime. The length of the hypotenuse must be either s2 = (ct)2 + x2 or s2 = (ct)2—x2. There is no other option. Although we did not prove it, our assumption that spacetime should be unchanging and the same everywhere leads to only these two possibilities and we must pick either the plus sign or the minus sign. Of course, proof or no proof, we can be pragmatic and see what happens when we try each one on for size.
Flipping the sign means that the mathematics is not much of an extension over the by now familiar equation of Pythagoras. Our task is to figure out whether we should stick with the plus-sign version of Pythagoras, or shift to the minus-sign version of the distance equation. This may look at first sight to be a rather odd thing to investigate. What possible reason could there be for even considering Pythagoras with a minus sign? But that is not the right way to think about things. The formula for distances on a sphere looks nothing like Pythagoras either, so all we are doing is entertaining the idea that spacetime might not be flat in the sense of Euclid. Indeed, since the minus-sign version is the only option other than the plus-sign version (given our assumptions), we have no logical reason to throw it out at this stage. We should therefore keep it and investigate the consequences. If neither the plus- nor the minus-sign versions do the job, and we fail in constructing a workable distance measure in spacetime, then we must go back to the drawing board.
We are now about to plunge into a very elegant but perhaps tricky piece of reasoning. We will stick to our promise of using nothing more complicated than Pythagoras, but you might find that you have to read it twice. It should be worth it, because if you follow closely you might experience a feeling described by biologist Edward O. Wilson as the Ionian Enchantment. It derives from the work of Thales of Miletus, who is credited by Aristotle, two centuries later, as laying the foundations of the physical sciences in Ionia in the sixth century BCE. This poetic term describes the belief that the complexity of the world can be explained by a small number of simple natural laws because at its heart it is orderly and simple (we are reminded of Wigner’s essay). The scientist’s job is to strip away the complexity we see around us and to uncover this underlying simplicity. When the process works out, and the simplicity and unity of the world are revealed, we experience the Ionian Enchantment. Imagine for a moment cradling a snowflake in the palm of your hand. It is an elegant and beautiful structure, possessed of a jagged crystalline symmetry. No two snowflakes are alike, and at first sight this chaotic state of affairs seems to defy a simple explanation. Science has taught us that the apparent complexity of snowflakes hides an exquisite underlying simplicity; each is a configuration of billions of molecules of water, H2O. There is nothing more to a snowflake than that, and yet an overwhelming complex of structure and form emerges when those H2O molecules get together in the atmosphere of our planet on a cold winter’s night.
To settle the question of the plus or minus sign, we need to turn our attention to causality. Let us first suppose that Pythagoras’ is the right equation for distances in spacetime—i.e., s2 = (ct) 2 + x2. Yet again we return to our two events: waking up in bed at 7 a.m. and finishing breakfast in the kitchen at 8 a.m. We’ll do something that may send shivers up your spine as you remember sitting in mathematics classes at school and gazing out the window across the football fields, pristine and inviting in the spring afternoon sunlight—let the waking-up event be called O and the finishing-breakfast event be called A. We do this purely for reasons of brevity, without wishing to don tweed and cover ourselves in chalk dust.
We know that the spatial distance between O and A is x = 10 meters and the distance in time between the two events is t = 1 hour, where x and t are measured by me. We haven’t decided what c is yet, but when we do we will know ct and we can then go ahead and use the distance equation to calculate s, the distance in spacetime between events O and A. Our hypothesis is that while x and t can and will be different if they are measured by someone flying past at close to the speed of light, the distance s will stay the same. In other words, x and t can and will change but they must change in such a way that s never changes. To risk overemphasizing the point, we want to remind you that our goal is always to build the laws of physics using invariant objects in spacetime and the distance s is just such an object. If that sounds too abstract, then we can say it again but this time using less mathematically fancy language: Nature’s rules must express relationships between real things, and those things live in spacetime. A thing living in spacetime is akin to an object sitting in a room. Spacetime (or the room) is the arena in which the thing lives. The nature of real things is not a matter of opinion and in that sense we say they are invariant. A three-dimensional example of something that is not an invariant might be the flickering shadow of an object sitting in a room illuminated by a warming fire. Clearly the shadow varies depending on how the fire is burning and where the fire is but we are never in any doubt that a real, unvarying object is responsible for it. Using spacetime, our plan is to lift physics out of the shadows and hunt down relationships between real objects.
The fact that two different observers can disagree on the values of x and t, provided s is the same, has a very important consequence, which can be visualized quite simply. Figure 5 shows a circle centered on O, the waking-up event, with a radius s. Because we are, for the moment, using the Pythagorean form of the distance equation, every point on the circumference of the circle is the same distance s away from O. This is a pretty obvious statement: The distance s is the radius of the circle. Points outside the circle are farther away from O while points inside are closer to O. But our hypothesis is that s is the distance in spacetime between events O and A. In other words, the event A could lie anywhere on the circumference of the circle and still be a distance s in spacetime from O. At what point on the circle should event A lie? That depends on who is measuring x and t. For me in the house, we know exactly where it should be since x = 10 meters and t = 1 hour. This is what we have drawn on the diagram and labeled A. For a person flying past in a high-speed rocket, the distance x in space and the distance t in time will change, but if s is to remain the same, then the event must still lie somewhere on the circle. So different observers record different positions in space and time separately for the same event, but subject to the constraint that we only slide the point around on the circle. We’ve labeled two possible positions A’ and A”. For position A’, nothing particularly interesting has happened, but look carefully at position A”. Something very dramatic indeed has happened. A” has a negative distance in time from O. In other words, A” happened before O. It is now in the O’s past. This is a world where you finish your breakfast before you wake up! Such a circumstance is a clear violation of our cherished axiom of causality.
As an aside, pictures like the ones shown in Figures 4 and 5 are called “spacetime diagrams” and they often help us work out what is going on. They really are simple things. Crosses on a spacetime diagram denote events and we can drop a line down onto the line marked “space” (the space axis) from the event to work out how far apart in space the event lies from the event O. Likewise, a horizontal line drawn to the line marked “time” (the time axis) tells us the time difference between the event and the event O. We can interpret the area above the space axis as the future of O (because t is positive for any event in this region) and the area below as the past (because t is then negative). The problem we have encountered is that we have constructed a definition of the distance in spacetime s between the events O and A that allows for A to be in either the future or the past of O, depending on how the person who observes the events is moving. In other words, we have discovered that the requirement of causality is intimately related to the way that we define the distance in spacetime, and the simple Pythagorean definition with the plus sign is no good.
We are faced with what the English biologist Thomas Henry Huxley famously described as “the great tragedy of science—the slaying of a beautiful hypothesis by an ugly fact.” Huxley, known as Darwin’s bulldog for his sterling defense of evolution, was once asked by William Wilberforce whether it was from his grandfather or grandmother that he claimed his descent from a monkey. Huxley is said to have replied that he would not be ashamed to have a monkey for his ancestor, but he would be ashamed to be connected with a man who used his great gifts to obscure the truth. The tragic truth in our case is that we must reject the simplest hypothesis if we are to preserve causality, and move on to something a little more complicated.
Our next and in fact only remaining hypothesis is that the distance between points in spacetime is to be calculated using s2 = (ct)2—x2. In contrast to the plus-sign version, this is a world where Euclidean geometry does not apply, as in the case of geometry on the surface of the earth. Mathematicians have a name for a space in which the distance between two points is governed by this equation: It is called hyperbolic space. Physicists have a different name for it. They call it Minkowski spacetime. The reader might take this to be a clue that we are on the right track! Our top priority must be to establish whether Minkowski spacetime violates the demands of causality.
To answer this question we need once again to take a look at the lines in spacetime that lie a constant distance s from O. That is, we want to consider the analogue of the circles in Euclidean spacetime. The minus sign makes all the difference. Shown in Figure 6 are the same old events, O and A, along with the line of points that lie the same spacetime distance s from O. Crucially, these points no longer lie on a circle. Instead they lie on a curve known to mathematicians as a hyperbola. Mathematically speaking, all the points on the curve satisfy our distance equation—i. e., s2 = (ct)2—x2. Notice that the curve tends toward the dotted straight lines that lie at 45 degrees to the axes. Now the situation as viewed by observers in rocket ships is completely different from the plus-sign version because event A always stays in the future of event O. We can slide A around but never into O’s past. In other words, everyone agrees that we wake up before we finish our breakfast. We can breathe a sigh of relief: Causality is not violated in Minkowski spacetime.
It’s worth repeating this because it is one of the most important points in the book. If we decide to define the distance in spacetime between the two events O and A using Pythagoras’ equation but with a minus sign, then no matter how anyone views the two events, A never crosses into O’s past; it just moves around on the hyperbola. This means that if event A is in O’s future according to one observer, then every other observer will also agree that A is in O’s future too. Because the hyperbola never ever crosses into O’s past, everyone agrees that eating breakfast comes after waking up.
We’ve just completed a subtle piece of reasoning. It certainly does not mean that we are correct in our original hypothesis that there should be an “invariant” distance in spacetime that is agreed upon by all observers. What it does mean, though, is that our hypothesis has survived an important test—it has survived the demands of the requirement of causality. We are not finished, however, because we are not just playing around with mathematics. We are physicists, and we are trying to construct a theory that describes how the world works. The ultimate and decisive test of our theory will be whether it can produce predictions that agree with experiment, and we are not yet ready to make a prediction, because we don’t know what the calibrating speed c is. Without a number, we simply can’t do the sums.
Remember, we needed c in order to have any chance of defining the notion of distance in spacetime, because we had to measure space and time in the same currency, but so far we have no idea what it actually represents. Is it the speed of anything interesting? The key to the answer lies in an intriguing property of the Minkowski spacetime we have just constructed. Those lines at 45 degrees are important. In Figure 7we’ve drawn several other curves, each of constant spacetime distance from O. The important point is that there are in fact four types of curve that we can draw. One lies wholly in the future of event O, one lies always in the past, and two others lie to the left and right. They look a little bit worrying because they cross the horizontal line in just the same way that our circles crossed it in the case of the plus-sign version of Pythagoras. In the plus-sign case, this led us to reject the hypothesis because it meant that causality was violated. Are we in the same boat with the minus-sign version? Are we sunk? Well, no, there is a way out. Figure 7 shows an event B sitting in the troubling region. It lies in O’s past according to the figure. But the hyperbola of constant distance from O for this event crosses the space axis, with the implication that it is possible for some observers to consider event B as occurring in O’s future, while for others it is in O’s past. Don’t forget: Every observer must agree on the spacetime distance between events even if they do not agree on the distances in space and time separately. It looks like a breakdown of causality, but fortunately that is very definitely not the case.
How are we to restore causality to our theory of spacetime? To answer this question, we need to think a little more carefully about what we mean by causality. This next piece will involve rocket ships and lasers, so if the abstract reasoning of the previous sections has left you drained, then you can relax for a while. Let’s think about event O again: waking up in bed in the morning. To be a little more precise, the event could correspond to my alarm clock going off. Shortly beforehand, on a planet in the Alpha Centauri system, the nearest star system to Earth at a distance of just over 4 light-years, a spaceship lifts off and heads toward Earth. Must everyone agree that the spaceship started its journey before I woke up? From the point of view of causality the issue depends critically upon whether information can travel infinitely fast or not. If information can travel infinitely fast, then the alien spaceship might conceivably be able to fire a laser beam that travels in an instant to the earth and destroys my alarm clock. The result is that I oversleep and miss breakfast. Missing breakfast might be the least worrying issue given this particular scenario, but we are doing a thought experiment, so let us ignore the emotional consequences of having our alarm clock vaporized by an alien laser and continue. The firing of the spaceship’s laser caused me to miss breakfast, and therefore the ordering cannot be swapped without violating our doctrine of the protection of causality. This is easy to see because if some observer were able to conclude that the spaceship took off after I woke up, then we would have a contradiction because I cannot oversleep if I have already woken up. We are forced to conclude that if information can travel at arbitrarily high speeds, then it can never be permissible to switch the time ordering of any two events without violating the law of cause and effect. But there is a loophole in our reasoning that permits the time ordering of certain pairs of events to be flipped, but only if they lie outside the 45-degree lines. These lines are beginning to look very important indeed.
Let us imagine the alien-laser-exploding-alarm-clock incident again, but now subject it to a cosmic speed limit. That is to say, we will not allow the laser beam to travel infinitely fast from the spaceship to our alarm clock. Covering ourselves in a thin mist of chalk dust for the last time, we call the laser-firing event B, as illustrated in Figure 7. If the spaceship fired the laser (event B) very shortly before the alarm clock-ringing event O, from a very great distance away, then there is no way the spaceship could possibly prevent me from waking up because the laser beam simply hasn’t got enough time to travel from the ship to my clock. This must be the case if the laser beam is constrained to travel at or below some kind of cosmic speed limit. If this is the situation, the events O and B are said to be causally disconnected.
As illustrated in the figure, we are supposing that B happens just before O such that it lies in the right-hand wedge region, which is the “dangerous” region for causality. Different observers will generally disagree on whether B happens before or after O, because their different points of view correspond to moving B around on the hyperbola, which crosses the space axis from the future to the past. This is unavoidable, but cause and effect can still be protected if there is absolutely no way that event B can influence event O. In other words, who cares whether B happened in O’s past or future, if it makes no difference to anything because B and O cannot influence each other? There are four distinct regions in Minkowski spacetime, separated from each other by the 45-degree lines. If we are to protect causality, then any event that occurs in either of the left-hand or right-hand wedges must never be able to send a signal that can possibly reach O.
To interpret the delineating lines, look again at our spacetime diagrams. The horizontal axis represents distance in space, and the vertical axis represents distance in time. The 45-degree lines therefore correspond to events that have a distance in space from O that is equal to the distance in time (ct). How fast must a signal travel from O if it is to influence an event lying exactly on the 45-degree line? Well, if the event is 1 second in O’s future, then the signal must travel a distance c x 1 second. If it’s 2 seconds in the future, then it must travel a distance c x 2 seconds. In other words, it must travel at the speed c. For a signal to travel between B and O, therefore, it must travel faster than the speed c. Conversely, for any events that lie between the 45-degree lines but in the upper and lower wedges, it is possible to communicate between them and the event at O using signals that travel at speeds slower than c.
We have finally managed to interpret the speed c: It is the cosmic speed limit. Nothing can travel faster than c because if it did it could be used to transmit information that could violate the principle of cause and effect. Notice also that if everyone is to agree on the distance in spacetime between any two events, then they must also agree that the cosmic speed limit is c, regardless of how they are moving around in spacetime. The speed c therefore has an additional interesting property: No matter how two different observers are moving, they must always measure c to be the same. The speed c is beginning to look a lot like another special speed we have encountered in this book: the speed of light, but we haven’t proved the connection yet.
Our original conjecture is still very much alive. We have managed to build a theory of space and time that looks capable of reproducing the physics we met in the last chapter. Certainly, the existence of a universal speed limit offers promise, especially if we can interpret it as the speed of light. We also have a spacetime in which space and time are no longer absolutes. They have been sacrificed in favor of absolute spacetime. To convince ourselves that we have constructed a possible description of the world, let’s see if we can obtain the slowing down of moving clocks that we met in Chapter 3.
Imagine that you are back on the proverbial train, sitting down in a carriage wearing a wristwatch. For you, it is convenient to measure distances relative to your own position and times using your wristwatch. Your train journey takes two hours from station to station. Since you never leave your seat throughout the journey, you have traveled a distance x = 0. This is the principle we established right at the start of the book. It is not possible to define who is moving and who is standing still, and therefore it is perfectly acceptable for you, seated on a train, to decide that you are not moving. In this case, only time passes. Since your journey takes two hours, then, from your perspective, you have traveled only in time. In spacetime, therefore, you have traveled distance s given by s = ct where t = 2 hours (because the distance in space as measured by you is x = 0). That is all straightforward. Now consider your journey from the standpoint of your friend, who is not on the train but who instead is sitting on the ground somewhere (it does not matter where he actually is, just that he is at rest relative to the earth while you are whizzing by on the train). Your friend would prefer to measure times using his own wristwatch and distances relative to himself. To simplify things a little bit, let us suppose your train journey is on a perfectly straight track. If you travel for 2 hours at a speed of υ = 100 miles per hour, then your friend notes that, at the end of the journey, you have traveled a distance X = υT. We are using capital letters when we talk about distances or times measured by your friend in order to distinguish them from the corresponding quantities measured by you (i.e., x= 0 and t = 2 hours). So, according to your friend, you have traveled a spacetime distance s given by s2 = (cT)2 - (υT) 2 .
Here is the crucial part of the whole argument: You must both agree on the spacetime distance of your journey. According to your measurements, you did not move (x = 0) and your journey took 2 hours (t = 2 hours), while your friend says that you have traveled a distance of υT (where υ = 100 miles per hour) and your journey takes a time T. Well, we are obliged to equate the corresponding distances in spacetime and so (ct) 2 = (cT)2—(υT)2. This formula can be jiggled around to give us T = ct/ √c2—υ2. So, although your wristwatch registers that your journey lasted for 2 hours, according to your friend your journey lasted a little longer. The enhancement factor is equal to c/√c2—υ2 = 1/√1—υ2/c2, which is exactly what we got in the last chapter but only if we interpret c as the speed of light.
Are you beginning to feel the Ionian Enchantment? We have deduced the same formula that emerged from thinking about light clocks and triangles in the previous chapter. Then, we were motivated to think about light clocks because Maxwell’s brilliant synthesis of the experimental results of Faraday and others strongly suggested that the speed of light should be the same for all observers. This conclusion was supported by the experimental work of Michelson and Morley, and taken at face value by Einstein. In this chapter we arrived at exactly the same conclusion but with no reference to history or experiment. We didn’t even need to give light a special role. Instead, we introduced spacetime and, as a result, insisted that there should exist the notion of an invariant distance between events. On top of that we demanded that cause and effect be respected. We then constructed the simplest possible distance measure and remarkably arrived at the same answer as Einstein. This reasoning is perhaps one of the most beautiful examples of the unreasonable effectiveness of mathematics in the physical sciences. Thales would be so enchanted that he would already be reclining in a bath of asses’ milk having been scrubbed by eunuchs. For his concubines to enter his bathroom carrying wine and figs, all we have to do is establish that c must be the speed of light using an argument that is entirely independent of the historical reasoning we encountered in the last chapter. That climax will arrive in the next chapter, for now we can take a rest from the maths, leave Thales poised in anticipation, and revel in the fact that we have succeeded in uncovering a whole new way of thinking about Einstein’s theory. Spacetime really does seem to work—the notion of a unified space and time makes sense, just as Minkowski said.
How are we to picture spacetime? Real spacetime is four-dimensional but the four-dimensional nature poses a stumbling block to our imagination, because human brains cannot directly picture objects in higher than three dimensions. In addition, the fact that time makes up one of the dimensions just sounds plain weird. A picture that might help make it all a little less mystical is to imagine a motorcycle roaming over an undulating countryside. Roads criss-cross the landscape, allowing our motorcyclist to wander this way and that. Spacetime is rather like the rolling countryside. The analogue of our motorcyclist traveling due north might be an object moving only in the time direction through spacetime. In other words, the object would be stationary in space. Of course, statements like “stationary in space” are subjective and so it is to be understood that the identification of “due north” with “the time direction” implies a particular point of view, but that is okay; we just need to bear it in mind. Now, the roads criss-crossing the spacetime landscape are all restricted to lie within a bearing of 45 degrees of north; roads due east and west are disallowed because to travel along them our spacetime “motorcyclist” would have to exceed the cosmic speed limit through space. Think of it this way: If the motorcyclist could travel due east, then he could go as far as he wanted in the easterly direction without any time passing at all, because he would not travel any distance up the northerly time direction. This would correspond to an infinite speed through space; he would get from a to b instantaneously. The roads have therefore been built so that the motorcyclist cannot travel too fast in an easterly or westerly direction.
The analogy can be pushed even further. We will very soon show that everything moves over spacetime at the same speed. It is just as if our motorcyclist has a device that fixes the throttle on his bike so that he always travels at the same speed over the spacetime landscape. We do need to be a little bit careful here, for when we talk about a speed in spacetime, it is not the same as a speed through space. A speed through space can be anything provided it does not exceed the cosmic speed limit—e.g., our motorcyclist might take a road close to a bearing of northeast, and in doing so he would be pushing as close to the cosmic speed limit as he could. In contrast, a road bearing close to due north would not lead to much movement east or west and consequently a journey that is well within the speed limit. The statement that everything moves at the same speed through spacetime sounds rather profound and perhaps a little baffling. It means that as you sit reading this book you are whizzing over the spacetime landscape at exactly the same speed as everything else in the universe. Viewed like that, motion through space is a shadow of a more universal motion through spacetime. In a very real sense, as we will now show, you are exactly like the motorcyclist with the fixed throttle. You are moving over the spacetime landscape with your throttle fixed open as you read this book. Because you are sitting still, your journey is entirely up the northerly time road. If you glance at your watch, you’ll see the distance in time ticking by. This is a very strange-sounding claim, so let’s go through it carefully.
Why does everything move at the same speed through spacetime? Consider our motorcyclist again and imagine 1 second passes according to the watch on his wrist. In that time, he will have traveled through spacetime by a certain distance. But everyone must agree on how far that distance is, because distances in spacetime are universal and not a matter for debate. That means we can ask the motorcyclist how far he thinks he has traveled over the spacetime landscape and the answer he gives will be the right answer. Now, the motorcyclist can choose to calculate distances in spacetime relative to himself, and from this point of view he has not moved in space. It is just like the person sitting on the airplane in Chapter 1 who doesn’t stray from her airplane seat and who therefore states that she has not moved. She may have moved relative to someone else—for example, someone standing on the ground watching the plane fly by—but that is not the point. So from our motorcyclist’s point of view, he has not moved in space and yet 1 second in time has passed. He can therefore use the spacetime distance equation s2 = (ct)2—x2 with x = 0 (because he hasn’t moved in space) and t = 1 second to figure out how far in spacetime he has actually traveled: The answer is a distance equal to c multiplied by 1 second. So the motorcyclist tells us that he is traveling a distance of c (multiplied by 1 second) for every second that passes on his watch, and that is just another way of saying that his speed through spacetime is equal to c. If you have been following closely, then you might object that the passage of 1 second was measured on the motorcyclist’s wristwatch and that a different amount of time will pass according to someone else who is moving relative to the motorcyclist. That is true enough, but there is something special about the motorcyclist’s watch, because the motorcyclist does not move relative to himself (a trivial statement).We are therefore free to put x = 0 in the distance equation and so the time that passes on his wristwatch is a direct way to measure the spacetime distance s. This is a nice result: The time that passes on the motorcyclist’s watch is equal to the spacetime distance traveled divided by c. In a sense, his watch is a device for measuring distances in spacetime. Since both the spacetime distance and c are agreed upon by everyone, it follows that the motorcyclist has unwittingly used his watch to measure something that everyone can agree upon. The spacetime speed c that he deduces is therefore also a quantity that everyone can agree upon.
So the speed through spacetime is a universal upon which everyone agrees. This newfound way of thinking about how things move through spacetime can help us get a different handle on why moving clocks run slow. In this spacetime way of thinking, a moving clock uses up some of its fixed quota of spacetime speed because of its motion through space and that leaves less for its motion through time. In other words, a moving clock doesn’t move so fast through time as a stationary one, which is just another way of saying that it ticks more slowly. In contrast, a clock sitting at rest whizzes along in the time direction at the speed c with no motion through space. It therefore ticks along as fast as is possible.
Armed with spacetime, we are ready to contemplate one of the wonderful puzzles of Special Relativity: the Twins Paradox. Earlier in the book we showed that Einstein’s theory allows us to contemplate the possibility of traveling to distant places in the universe. Speeding within a whisker of the speed of light, we imagined journeying off to the Andromeda galaxy within a human lifetime regardless of the fact that it takes light nearly 3 million years to make the journey. There is a paradox lurking here that we previously glossed over. Imagine twins, one of whom trains to be an astronaut and heads off on humanity’s first mission to Andromeda, leaving her twin back home on Earth. The astronaut twin is moving at high speed relative to the earth and consequently her life slows down relative to her twin on Earth. But we have just spent a significant fraction of this book arguing that there is no such thing as absolute motion. In other words, the answer to the question “Who is doing the moving?” is “Whoever you want.” Anybody and everybody is free to decide that they are standing still, and the other guy is whizzing around the universe at high speed relative to them. And so it is for the astronaut twin, who is free to say that she is standing perfectly still in her space rocket, watching the earth fly away at high speed. For her, it is therefore the earthbound twin who ages more slowly. Who is right? Can it really be that each of the twins ages more slowly relative to the other? Well it has to be like that—that is what the theory says. There is no paradox yet, because any problems you might be having in believing that each twin observes the other to be aging more slowly are not real problems. They are due to the fact that you are clinging to the idea of universal time. But time is not universal; that much we have learned, and that means there is no contradiction at all. Now comes the apparent paradox: What happens if the astronaut twin returns back to Earth sometime in the future and meets up with her earthbound twin? Obviously they cannot both be younger than the other. What is going on? Is one of them actually older than the other? If so, who?
The answer can be found in our understanding of spacetime. In Figure 8 we show the paths through spacetime taken by the twins, as measured using clocks and rulers at rest relative to the earth. The earthbound twin stays on the earth and consequently her path snakes along the time axis. In other words, almost all of her allocated speed through spacetime is expended traveling through time. Her astronaut twin, on the other hand, heads off at close to light speed. Returning to the motorcyclist analogy, that means she charges off in a “northeasterly” direction, using up as much of her spacetime speed as she can to push through space at close to the cosmic speed limit. On the spacetime diagram shown in Figure 8, that means she travels close to 45 degrees. At some point, however, she needs to turn around and come back to the earth. The picture shows that we are supposing that she heads back again at close to light speed but this time in a “northwesterly” direction. Obviously the twins take different paths through spacetime, even though they started and finished at the same point.
Now just like distances in space, the length of two different paths in spacetime can be different. To reiterate, although everyone must agree on the length of any particular path through spacetime, the lengths of different paths need not be the same. This is really no different from saying that the distance from Chamonix to Courmayeur depends upon whether you went through the Mont Blanc tunnel or hiked over the Alps. Of course, walking over a mountain means you travel a longer distance than tunneling through it. In our discussion of the motorcyclist speeding over the spacetime landscape, we established that the time measured on the motorcyclist’s wristwatch provides a direct way to measure the spacetime distance he traveled: we just need to multiply the elapsed time by c to get the spacetime distance. We can turn this statement on its head and say that once we know the spacetime distance traveled by each of the twins, then we can figure out the time that passes according to each. That is, we can think of each twin as a voyager through spacetime with their wristwatches measuring the spacetime distance that they travel.
Now comes the key idea. Look again at the formula for distances in spacetime, s2 = (ct)2—x2. The spacetime distance is biggest if we can follow a path that has x = 0. Any other path must be shorter because we have to subtract the (always positive) x2 contribution. But the earthbound twin snakes along the time direction with x close to zero, so her path must be the longest possible path. Actually, that is just another way of saying what we already know: that the earthbound twin is traveling as fast as possible through time and so it is she who ages the most.
Our explanation so far has been presented from the viewpoint of the earthbound twin. To fully satisfy ourselves that there is no paradox, we should see how things look from the viewpoint of the astronaut twin. For her, the earthbound twin is the one doing the traveling while she snakes along her own time axis. It looks like the paradox is back again; since the astronaut twin is at rest relative to her spaceship, it seems that she should speed maximally through time and hence age the most. But there is a very subtle point here. The distance equation does not apply if we set out to use the astronaut twin’s clocks and rulers to measure distances and times. More precisely, it fails when the astronaut twin undergoes the acceleration that turns the spaceship around. Why does it fail? The arguments we presented when we figured it out seemed pretty watertight. But if one uses an accelerating system of clocks and rulers to make measurements, as the astronaut twin must, then the assumption that spacetime is unchanging and the same everywhere that we used to write down the distance equation is wrong. Over the time of the acceleration, the astronaut twin will be pushed back into her seat, in much the same way that you are pushed back into your seat when you press the accelerator pedal on a car. For a start, that immediately picks out a special direction in space: the direction of the acceleration. The existence of that force must be accounted for in the distance equation, and that is where the loophole resides. It is a little too complicated for us to go into the mathematical details, but the upshot is that when the spaceship fires its rockets to turn around, the earthbound twin ages rapidly relative to the astronaut twin and that more than makes up for the fact that she ages more slowly during the nonaccelerating phases of the expedition. There is no paradox.
We can’t resist quoting some numbers, because the effect can be startling. Space travel is most comfortable for those onboard the spaceship if the rockets are firing in order to sustain an acceleration equal to “one g.” That means that the space travelers feel their own weight inside the rocket. So let’s imagine a journey of 10 years at that acceleration, followed by 10 more years decelerating at the same rate, at which point we turn the spaceship around and head back to Earth, accelerating for 10 more years and decelerating for a further 10 before finally arriving back. In total the travelers onboard the spaceship will have been journeying for a total of 40 years. The question is how many years have passed on Earth? We’ll just quote the result because the mathematics is (only a little) beyond the level of this book. The result is that a breathtaking 59,000 years will have passed on Earth!
This has been a remarkable journey, and we hope the reader has followed us into the world of spacetime. We are now ready to head directly to E = mc2. Armed with spacetime and our invariant definition of distance, we ask a simple but very important question: Are there other invariant quantities that also describe the properties of real objects in the real world? Of course, distances aren’t the only things that are important. Objects have mass, they can be hard or soft, hot or cold, solid, liquid, or gas. Since all objects live in spacetime, is it possible to describe everything about the world in an invariant way? We will discover in the next chapter that it is, and the consequences are profound, for this is the road that leads directly to E = mc2.