The Fabric of the Cosmos: Space, Time, and the Texture of Reality - Brian Greene (2004)
Part III. SPACETIME AND COSMOLOGY
Chapter 8. Of Snowflakes and Spacetime
SYMMETRY AND THE EVOLUTION OF THE COSMOS
Richard Feynman once said that if he had to summarize the most important finding of modern science in one sentence he would choose “The world is made of atoms.” When we recognize that so much of our understanding of the universe relies on the properties and interactions of atoms—from the reason that stars shine and the sky is blue to the explanation for why you feel this book in your hand and see these words with your eyes—we can well appreciate Feynman’s choice for encapsulating our scientific legacy. Many of today’s leading scientists agree that if they were offered a second sentence, they’d choose “Symmetry underlies the laws of the universe.” During the last few hundred years there have been many upheavals in science, but the most lasting discoveries have a common characteristic: they’ve identified features of the natural world that remain unchanged even when subjected to a wide range of manipulations. These unchanging attributes reflect what physicists call symmetries, and they have played an increasingly vital role in many major advances. This has provided ample evidence that symmetry—in all its mysterious and subtle guises—shines a powerful light into the darkness where truth awaits discovery.
In fact, we will see that the history of the universe is, to a large extent, the history of symmetry. The most pivotal moments in the evolution of the universe are those in which balance and order suddenly change, yielding cosmic arenas qualitatively different from those of preceding eras. Current theory holds that the universe went through a number of these transitions during its earliest moments and that everything we’ve ever encountered is a tangible remnant of an earlier, more symmetric cosmic epoch. But there is an even grander sense, a metasense, in which symmetry lies at the core of an evolving cosmos. Time itself is intimately entwined with symmetry. As will become clear, the practical connotation of time as a measure of change, as well as the very existence of a kind of cosmic time that allows us to speak sensibly of things like “the age and evolution of the universe as a whole,” rely sensitively on aspects of symmetry. And as scientists have examined that evolution, looking back toward the beginning in search of the true nature of space and time, symmetry has established itself as the most sure-footed of guides, providing insights and answers that would otherwise have been completely out of reach.
Symmetry and the Laws of Physics
Symmetry abounds. Hold a cue ball in your hand and rotate it this way or that—spin it around any axis—and it looks exactly the same. Put a plain, round dinner plate on a placemat and rotate it about its center: it looks completely unchanged. Gently catch a newly formed snowflake and rotate it so that each tip is moved into the position previously held by its neighbor, and you’d be hard pressed to notice that you’d done anything at all. Take the letter “A,” flip it about a vertical axis passing through its apex, and it will provide you with a perfect replica of the original.
As these examples make clear, the symmetries of an object are the manipulations, real or imagined, to which it can be subjected with no effect on its appearance. The more kinds of manipulations an object can sustain with no discernible effect, the more symmetric it is. A perfect sphere is highly symmetric, since any rotation about its center—using an up-down axis, a left-right axis, or any axis in fact—leaves it looking exactly the same. A cube is less symmetric, since only rotations in units of 90 degrees about axes that pass through the center of its faces (and combinations thereof) leave it looking unchanged. Of course, should someone perform any other rotation, such as in Figure 8.1c, you obviously can still recognize the cube, but you also can see clearly that someone has tampered with it. By contrast, symmetries are like the deftest of prowlers; they are manipulations that leave no evidence whatsoever.
Figure 8.1 If a cube, as in (a), is rotated by 90 degrees, or multiples thereof, around axes passing through any of its faces, it looks unchanged, as in (b). But any other rotations can be detected, as in (c).
All these are examples of symmetries of objects in space. The symmetries underlying the known laws of physics are closely related to these, but zero in on a more abstract question: what manipulations—once again, real or imagined—can be performed on you or on the environment that will have absolutely no effect on the laws that explain the physical phenomena you observe? Notice that to be a symmetry, manipulations of this sort are not required to leave your observations unchanged. Instead, we are concerned with whether the laws governing those observations—the laws that explain what you see before, and then what you see after, some manipulation—are unchanged. As this is a central idea, let’s see it at work in some examples.
Imagine that you’re an Olympic gymnast and for the last four years you’ve been training diligently in your Connecticut gymnastics center. Through seemingly endless repetition, you’ve got every move in your various routines down perfectly—you know just how hard to push off the balance beam to execute an aerial walkover, how high to jump in the floor exercise for a double-twisting layout, how fast to swing on the parallel bars to launch your body on a perfect double-somersault dismount. In effect, your body has taken on an innate sense of Newton’s laws, since it is these very laws that govern your body’s motion. Now, when you finally do your routines in front of a packed audience in New York City, the site of the Olympic competition itself, you’re banking on the same laws holding, since you intend to perform your routines exactly as you have in practice. Everything we know about Newton’s laws lends credence to your strategy. Newton’s laws are not specific to one location or another. They don’t work one way in Connecticut and another way in New York. Rather, we believe his laws work in exactly the same way regardless of where you are. Even though you have changed location, the laws that govern your body’s motion remain as unaffected as the appearance of a cue ball that has been rotated.
This symmetry is known as translational symmetry or translational invariance. It applies not only to Newton’s laws but also to Maxwell’s laws of electromagnetism, to Einstein’s special and general relativities, to quantum mechanics, and to just about any proposal in modern physics that anyone has taken seriously.
Notice one important thing, though. The details of your observations and experiences can and sometimes will vary from place to place. Were you to perform your gymnastics routines on the moon, you’d find that the path your body took in response to the same upward jumping force of your legs would be very different. But we fully understand this particular difference and it is already integrated into the laws themselves. The moon is less massive than the earth, so it exerts less gravitational pull; as a result, your body travels along different trajectories. And this fact—that the gravitational pull of a body depends on its mass—is an integral part of Newton’s law of gravity (as well as of Einstein’s more refined general relativity). The difference between your earth and moon experiences doesn’t imply that the law of gravity has changed from place to place. Instead, it merely reflects an environmental difference that the law of gravity already accommodates. So when we said that the known laws of physics apply equally well in Connecticut or New York—or, let’s now add, on the moon—that was true, but bear in mind that you may need to specify environmental differences on which the laws depend. Nevertheless, and this is the key conclusion, the explanatory framework the laws provide is not at all changed by a change in location. A change in location does not require physicists to go back to the drawing board and come up with new laws.
The laws of physics didn’t have to operate this way. We can imagine a universe in which physical laws are as variable as those of local and national governments; we can imagine a universe in which the laws of physics with which we are familiar tell us nothing about the laws of physics on the moon, in the Andromeda galaxy, in the Crab nebula, or on the other side of the universe. In fact, we don’t know with absolute certainty that the laws that work here are the same ones that work in far-flung corners of the cosmos. But we do know that should the laws somehow change way out there, it must be way out there, because ever more precise astronomical observations have provided ever more convincing evidence that the laws are uniform throughout space, at least the space we can see. This highlights the amazing power of symmetry. We are bound to planet earth and its vicinity. And yet, because of translational symmetry, we can learn about fundamental laws at work in the entire universe without straying from home, since the laws we discover here are those laws.
Rotational symmetry or rotational invariance is a close cousin of translational invariance. It is based on the idea that every spatial direction is on an equal footing with every other. The view from earth certainly doesn’t lead you to this conclusion. When you look up, you see very different things than you do when you look down. But, again, this reflects details of the environment; it is not a characteristic of the underlying laws themselves. If you leave earth and float in deep space, far from any stars, galaxies, or other heavenly bodies, the symmetry becomes evident: there is nothing that distinguishes one particular direction in the black void from another. They are all on a par. You wouldn’t have to give a moment’s thought to whether a deep-space laboratory you’re setting up to investigate properties of matter or forces should be oriented this way or that, since the underlying laws are insensitive to this choice. If one night a prankster were to change the laboratory’s gyroscopic settings, causing it to rotate some number of degrees about some particular axis, you’d expect this to have no consequences whatsoever for the laws of physics probed by your experiments. Every measurement ever done fully confirms this expectation. Thus, we believe that the laws that govern the experiments you carry out and explain the results you find are insensitive both to where you are—this is translational symmetry—and to how you happen to be oriented in space—this is rotational symmetry.1
As we discussed in Chapter 3, Galileo and others were well aware of another symmetry that the laws of physics should respect. If your deep-space laboratory is moving with constant velocity—regardless of whether you’re moving 5 miles per hour this way or 100,000 miles per hour that way—the motion should have absolutely no effect on the laws that explain your observations, because you are as justified as the next guy in claiming that you are at rest and it’s everything else that is moving. Einstein, as we have seen, extended this symmetry in a thoroughly unanticipated way by including the speed of light among the observations that would be unaffected by either your motion or the motion of the light’s source. This was a stunning move because we ordinarily throw the particulars of an object’s speed into the environmental details bin, recognizing that the speed observed generally depends upon the motion of the observer. But Einstein, seeing light’s symmetry stream through the cracks in nature’s Newtonian façade, elevated light’s speed to an inviolable law of nature, declaring it to be as unaffected by motion as the cue ball is unaffected by rotations.
General relativity, Einstein’s next major discovery, fits squarely within this march toward theories with ever greater symmetry. Just as you can think of special relativity as establishing symmetry among all observers moving relative to one another with constant velocity, you can think of general relativity as going one step farther and establishing symmetry among all accelerated vantage points as well. This is extraordinary because, as we’ve emphasized, although you can’t feel constant velocity motion, you can feel accelerated motion. So it would seem that the laws of physics describing your observations must surely be different when you are accelerating, to account for the additional force you feel. Such is the case with Newton’s approach; his laws, the ones that appear in all first-year physics textbooks, must be modified if utilized by an accelerating observer. But through the principle of equivalence, discussed in Chapter 3, Einstein realized that the force you feel from accelerating is indistinguishable from the force you feel in a gravitational field of suitable strength (the greater the acceleration, the greater the gravitational field). Thus, according to Einstein’s more refined perspective, the laws of physics do not change when you accelerate, as long as you include an appropriate gravitational field in your description of the environment. General relativity treats all observers, even those moving at arbitrary non-constant velocities, equally—they are completely symmetric—since each can claim to be at rest by attributing the different forces felt to the effect of different gravitational fields. The differences in the observations between one accelerating observer and another are therefore no more surprising and provide no greater evidence of a change in nature’s laws than do the differences you find when performing your gymnastics routine on earth or the moon.2
These examples give some sense of why many consider, and I suspect Feynman would have agreed, that the copious symmetries underlying natural law present a close runner-up to the atomic hypothesis as a summary of our deepest scientific insights. But there is more to the story. Over the last few decades, physicists have elevated symmetry principles to the highest rung on the explanatory ladder. When you encounter a proposed law of nature, a natural question to ask is: Why this law? Why special relativity? Why general relativity? Why Maxwell’s theory of electromagnetism? Why the Yang-Mills theories of the strong and weak nuclear forces (which we’ll look at shortly)? One important answer is that these theories make predictions that have been repeatedly confirmed by precision experiments. This is essential to the confidence physicists have in the theories, certainly, but it leaves out something important.
Physicists also believe these theories are on the right track because, in some hard-to-describe way, they feel right, and ideas of symmetry are essential to this feeling. It feels right that no location in the universe is somehow special compared with any other, so physicists have confidence that translational symmetry should be among the symmetries of nature’s laws. It feels right that no particular constant-velocity motion is somehow special compared with any other, so physicists have confidence that special relativity, by fully embracing symmetry among all constant-velocity observers, is an essential part of nature’s laws. It feels right, moreover, that any observational vantage point—regardless of the possibly accelerated motion involved—should be as valid as any other, and so physicists believe that general relativity, the simplest theory incorporating this symmetry, is among the deep truths governing natural phenomena. And, as we shall shortly see, the theories of the three forces other than gravity— electromagnetism and the strong and weak nuclear forces—are founded on other, somewhat more abstract but equally compelling principles of symmetry. So the symmetries of nature are not merely consequences of nature’s laws. From our modern perspective, symmetries are the foundation from which laws spring.
Symmetry and Time
Beyond their role in fashioning the laws governing nature’s forces, ideas of symmetry are vital to the concept of time itself. No one has as yet found the definitive, fundamental definition of time, but, undoubtedly, part of time’s role in the makeup of the cosmos is that it is the bookkeeper of change. We recognize that time has elapsed by noticing that things now are different from how they were then. The hour hand on your watch points to a different number, the sun is in a different position in the sky, the pages in your unbound copy of War and Peace are more disordered, the carbon dioxide gas that rushed from your bottle of Coke is more spread out—all this makes plain that things have changed, and time is what provides the potential for such change to be realized. To paraphrase John Wheeler, time is nature’s way of keeping everything—all change, that is—from happening all at once.
The existence of time thus relies on the absence of a particular symmetry: things in the universe must change from moment to moment for us even to define a notion of moment to moment that bears any resemblance to our intuitive conception. If there were perfect symmetry between how things are now and how they were then, if the change from moment to moment were of no more consequence than the change from rotating a cue ball, time as we normally conceive it wouldn’t exist.3 That’s not to say the spacetime expanse, schematically illustrated in Figure 5.1, wouldn’t exist; it could. But since everything would be completely uniform along the time axis, there’d be no sense in which the universe evolves or changes. Time would be an abstract feature of this reality’s arena—the fourth dimension of the spacetime continuum—but otherwise, it would be unrecognizable.
Nevertheless, even though the existence of time coincides with the lack of one particular symmetry, its application on a cosmic scale requires the universe to be highly respectful of a different symmetry. The idea is simple and answers a question that may have occurred to you while reading Chapter 3. If relativity teaches us that the passage of time depends on how fast you move and on the gravitational field in which you happen to be immersed, what does it mean when astronomers and physicists speak of the entire universe’s being a particular definite age—an age which these days is taken to be about 14 billion years? Fourteen billion years according to whom? Fourteen billion years on which clock? Would beings living in the distant Tadpole galaxy also conclude that the universe is 14 billion years old, and if so, what would have ensured that their clocks have been ticking away in synch with ours? The answer relies on symmetry—symmetry in space.
If your eyes could see light whose wavelength is much longer than that of orange or red, you would not only be able to see the interior of your microwave oven burst into activity when you push the start button, but you would also see a faint and nearly uniform glow spread throughout what the rest of us perceive as a dark night sky. More than four decades ago, scientists discovered that the universe is suffused with microwave radiation—long-wavelength light—that is a cool relic of the sweltering conditions just after the big bang.4 This cosmic microwave background radiation is perfectly harmless. Early on, it was stupendously hot, but as the universe evolved and expanded, the radiation steadily diluted and cooled. Today it is just about 2.7 degrees above absolute zero, and its greatest claim to mischief is its contribution of a small fraction of the snow you see on your television set when you disconnect the cable and turn to a station that isn’t broadcasting.
But this faint static gives astronomers what tyrannosaurus bones give paleontologists: a window onto earlier epochs that is crucial to reconstructing what happened in the distant past. An essential property of the radiation, revealed by precision satellite measurements over the last decade, is that it is extremely uniform. The temperature of the radiation in one part of the sky differs from that in another part by less than a thousandth of a degree. On earth, such symmetry would make the Weather Channel of little interest. If it were 85 degrees in Jakarta, you would immediately know that it was between 84.999 degrees and 85.001 degrees in Adelaide, Shanghai, Cleveland, Anchorage, and everywhere else for that matter. On a cosmic scale, by contrast, the uniformity of the radiation’s temperature is fantastically interesting, as it supplies two critical insights.
First, it provides observational evidence that in its earliest stages the universe was not populated by large, clumpy, high-entropy agglomerations of matter, such as black holes, since such a heterogeneous environment would have left a heterogeneous imprint on the radiation. Instead, the uniformity of the radiation’s temperature attests to the young universe being homogeneous; and, as we saw in Chapter 6, when gravity matters— as it did in the dense early universe—homogeneity implies low entropy. That’s a good thing, because our discussion of time’s arrow relied heavily on the universe’s starting out with low entropy. One of our goals in this part of the book is to go as far as we can toward explaining this observation—we want to understand how the homogeneous, low-entropy, highly unlikely environment of the early universe came to be. This would take us a big step closer to grasping the origin of time’s arrow.
Second, although the universe has been evolving since the big bang, on average the evolution must have been nearly identical across the cosmos. For the temperature here and in the Whirlpool galaxy, and in the Coma cluster, and everywhere else to agree to four decimal places, the physical conditions in every region of space must have evolved in essentially the same way since the big bang. This is an important deduction, but you must interpret it properly. A glance at the night sky certainly reveals a varied cosmos: planets and stars of various sorts sprinkled here and there throughout space. The point, though, is that when we analyze the evolution of the entire universe we take a macro perspective that averages over these “small”-scale variations, and large-scale averages do appear to be almost completely uniform. Think of a glass of water. On the scale of molecules, the water is extremely heterogeneous: there is an H2O molecule over here, an expanse of empty space, another H2O molecule over there, and so on. But if we average over the small-scale molecular lumpiness and examine the water on the “large” everyday scales we can see with the naked eye, the water in the glass looks perfectly uniform. The nonuniformity we see when gazing skyward is like the microscopic view from a single H 2O molecule. But as with the glass of water, when the universe is examined on large enough scales—scales on the order of hundreds of millions of light-years—it appears extraordinarily homogeneous. The uniformity of the radiation is thus a fossilized testament to the uniformity of both the laws of physics and the details of the environment across the cosmos.
This conclusion is of great consequence because the universe’s uniformity is what allows us to define a concept of time applicable to the universe as a whole. If we take the measure of change to be a working definition of elapsed time, the uniformity of conditions throughout space is evidence of the uniformity of change throughout the cosmos, and thus implies the uniformity of elapsed time as well. Just as the uniformity of earth’s geological structure allows a geologist in America, and one in Africa, and another in Asia to agree on earth’s history and age, the uniformity of cosmic evolution throughout all of space allows a physicist in the Milky Way galaxy, and one in the Andromeda galaxy, and another in the Tadpole galaxy to all agree on the universe’s history and age. Concretely, the homogeneous evolution of the universe means that a clock here, a clock in the Andromeda galaxy, and a clock in the Tadpole galaxy will, on average, have been subject to nearly identical physical conditions and hence will have ticked off time in nearly the same way. The homogeneity of space thus provides a universal synchrony.
While I have so far left out important details (such as the expansion of space, covered in the next section) the discussion highlights the core of the issue: time stands at the crossroads of symmetry. If the universe had perfect temporal symmetry—if it were completely unchanging—it would be hard to define what time even means. On the other hand, if the universe did not have symmetry in space—if, for example, the background radiation were thoroughly haphazard, having wildly different temperatures in different regions—time in a cosmological sense would have little meaning. Clocks in different locations would tick off time at different rates, and so if you asked what things were like when the universe was 3 billion years old, the answer would depend on whose clock you were looking at to see that those 3 billion years had elapsed. That would be complicated. Fortunately, our universe does not have so much symmetry as to render time meaningless, but does have enough symmetry that we can avoid such complexities, allowing us to speak of its overall age and its overall evolution through time.
So, let’s now turn our attention to that evolution and consider the history of the universe.
Stretching the Fabric
The history of the universe sounds like a big subject, but in broad-brush outline it is surprisingly simple and relies in large part on one essential fact: The universe is expanding. As this is the central element in the unfolding of cosmic history, and, surely, is one of humanity’s most profound discoveries, let’s briefly examine how we know it is so.
In 1929, Edwin Hubble, using the 100-inch telescope at the Mount Wilson observatory in Pasadena, California, found that the couple of dozen galaxies he could detect were all rushing away.5 In fact, Hubble found that the more distant a galaxy is, the faster its recession. To give a sense of scale, more refined versions of Hubble’s original observations (that have studied thousands of galaxies using, among other equipment, the Hubble Space Telescope) show that galaxies that are 100 million light-years from us are moving away at about 5.5 million miles per hour, those at 200 million light-years are moving away twice as fast, at about 11 million miles per hour, those at 300 million light-years’ distance are moving away three times as fast, at about 16.5 million miles per hour, and so on. Hubble’s was a shocking discovery because the prevailing scientific and philosophical prejudice held that the universe was, on its largest scales, static, eternal, fixed, and unchanging. But in one stroke, Hubble shattered that view. And in a wonderful confluence of experiment and theory, Einstein’s general relativity was able to provide a beautiful explanation for Hubble’s discovery.
Actually, you might not think that coming up with an explanation would be particularly difficult. After all, if you were to pass by a factory and see all sorts of material violently flying outward in all directions, you would likely think that there had been an explosion. And if you traveled backward along the paths taken by the scraps of metal and chunks of concrete, you’d find them all converging on a location that would be a likely contender for where the explosion occurred. By the same reasoning, since the view from earth—as attested to by Hubble’s and subsequent observations—shows that galaxies are rushing outward, you might think our position in space was the location of an ancient explosion that uniformly spewed out the raw material of stars and galaxies. The problem with this theory, though, is that it singles out one region of space—our region—as unique by making it the universe’s birthplace. And were that the case, it would entail a deep-seated asymmetry: the physical conditions in regions far from the primordial explosion—far from us—would be very different from those here. As there is no evidence for such asymmetry in astronomical data, and furthermore, as we are highly suspect of anthropocentric explanations laced with pre-Copernican thinking, a more sophisticated interpretation of Hubble’s discovery is called for, one in which our location does not occupy some special place in the cosmic order.
General relativity provides such an interpretation. With general relativity, Einstein found that space and time are flexible, not fixed, rubbery, not rigid; and he provided equations that tell us precisely how space and time respond to the presence of matter and energy. In the 1920s, the Russian mathematician and meteorologist Alexander Friedmann and the Belgian priest and astronomer Georges Lemaître independently analyzed Einstein’s equations as they apply to the entire universe, and the two found something striking. Just as the gravitational pull of the earth implies that a baseball popped high above the catcher must either be heading farther upward or must be heading downward but certainly cannot be staying put (except for the single moment when it reaches its highest point), Friedmann and Lemaître realized that the gravitational pull of the matter and radiation spread throughout the entire cosmos implies that the fabric of space must either be stretching or contracting, but that it could not be staying fixed in size. In fact, this is one of the rare examples in which the metaphor not only captures the essence of the physics but also its mathematical content since, it turns out, the equations governing the baseball’s height above the ground are nearly identical to Einstein’s equations governing the size of the universe.6
The flexibility of space in general relativity provides a profound way to interpret Hubble’s discovery. Rather than explaining the outward motion of galaxies by a cosmic version of the factory explosion, general relativity says that for billions of years space has been stretching. And as it has swelled, space has dragged the galaxies away from each other much as the black specks in a poppy seed muffin are dragged apart as the dough rises in baking. Thus, the origin of the outward motion is not an explosion that took place within space. Instead, the outward motion arises from the relentless outward swelling of space itself.
To grasp this key idea more fully, think also of the superbly useful balloon model of the expanding universe that physicists often invoke (an analogy that can be traced at least as far back as a playful cartoon, which you can see in the endnotes, that appeared in a Dutch newspaper in 1930 following an interview with Willem de Sitter, a scientist who made substantial contributions to cosmology7). This analogy likens our three-dimensional space to the easier-to-visualize two-dimensional surface of a spherical balloon, as in Figure 8.2a, that is being blown up to larger and larger size. The galaxies are represented by numerous evenly spaced pennies glued to the balloon’s surface. Notice that as the balloon expands, the pennies all move away from one another, providing a simple analogy for how expanding space drives all galaxies to separate.
An important feature of this model is that there is complete symmetry among the pennies, since the view any particular Lincoln sees is the same as the view any other Lincoln sees. To picture it, imagine shrinking yourself, lying down on a penny, and looking out in all directions across the balloon’s surface (remember, in this analogy the balloon’s surface represents all of space, so looking off the balloon’s surface has no meaning). What will you observe? Well, you will see pennies rushing away from you in all directions as the balloon expands. And if you lie down on a different penny what will you observe? The symmetry ensures you’ll see the same thing: pennies rushing away in all directions. This tangible image captures well our belief—supported by increasingly precise astronomical surveys—that an observer in any one of the universe’s more than 100 billion galaxies, gazing across his or her night sky with a powerful telescope, would, on average, see an image similar to the one we see: surrounding galaxies rushing away in all directions.
And so, unlike a factory explosion within a fixed, preexisting space, if outward motion arises because space itself is stretching, there need be no special point—no special penny, no special galaxy—that is the center of the outward motion. Every point—every penny, every galaxy—is completely on a par with every other. The view from any location seems like the view from the center of an explosion: each Lincoln sees all other Lincolns rushing away; an observer, like us, in any galaxy sees all other galaxies rushing away. But since this is true for all locations, there is no special or unique location that is the center from which the outward motion is emanating.
Moreover, not only does this explanation account qualitatively for the outward motion of galaxies in a manner that is spatially homogeneous, it also explains the quantitative details found by Hubble and confirmed with greater precision by subsequent observations. As illustrated in Figure 8.2b, if the balloon swells during some time interval, doubling in size for example, all spatial separations will double in size as well: pennies that were 1 inch apart will now be 2 inches apart, pennies that were 2 inches apart will now be 4 inches apart, pennies that were 3 inches apart will now be 6 inches apart, and so on. Thus, in any given time interval, the increase in separation between two pennies is proportional to the initial distance between them. And since a greater increase in separation during a given time interval means a greater speed, pennies that are farther away from one another separate more quickly. In essence, the farther away from each other two pennies are, the more of the balloon’s surface there is between them, and so the faster they’re pushed apart when it swells. Applying exactly the same reasoning to expanding space and the galaxies it contains, we get an explanation for Hubble’s observations. The farther away two galaxies are, the more space there is between them, so the faster they’re pushed away from one another as space swells.
Figure 8.2 (a) If evenly spaced pennies are glued to the surface of a sphere, the view seen by any Lincoln is the same as that seen by any other. This aligns with the belief that the view from any galaxy in the universe, on average, is the same as that seen from any other. (b ) If the sphere expands, the distances between all pennies increase. Moreover, the farther apart two pennies are in 8.2a, the greater the separation they experience from the expansion in 8.2b. This aligns well with measurements showing that the farther away from a given vantage point a galaxy is, the faster it moves away from that point. Note that no one penny is singled out as special, also in keeping with our belief that no galaxy in the universe is special or the center of the expansion of space.By attributing the observed motion of galaxies to the swelling of space, general relativity provides an explanation that not only treats all locations in space symmetrically, but also accounts for all of Hubble’s data in one fell swoop. It is this kind of explanation, one that elegantly steps outside the box (in this case, one that actually uses the “box”—space, that is) to explain observations with quantitative precision and artful symmetry, that physicists describe as almost being too beautiful to be wrong. There is essentially universal agreement that the fabric of the space is stretching.
Time in an Expanding Universe
Using a slight variation on the balloon model, we can now understand more precisely how symmetry in space, even though space is expanding, yields a notion of time that applies uniformly across the cosmos. Imagine replacing each penny by an identical clock, as in Figure 8.3. We know from relativity that identical clocks will tick off time at different rates if they are subject to different physical influences—different motions, or different gravitational fields. But the simple yet key observation is that the complete symmetry among all Lincolns on the inflating balloon translates to complete symmetry among all the clocks. All the clocks experience identical physical conditions, so all tick at exactly the same rate and record identical amounts of elapsed time. Similarly, in an expanding universe in which there is a high degree of symmetry among all the galaxies, clocks that move along with one or another galaxy must also tick at the same rate and hence record an identical amount of elapsed time. How could it be otherwise? Each clock is on a par with every other, having experienced, on average, nearly identical physical conditions. This again shows the stunning power of symmetry. Without any calculation or detailed analysis, we realize that the uniformity of the physical environment, as evidenced by the uniformity of the microwave background radiation and the uniform distribution of galaxies throughout space,8 allows us to infer uniformity of time.
Although the reasoning here is straightforward, the conclusion may nevertheless be confusing. Since the galaxies are all rushing apart as space expands, clocks that move along with one or another galaxy are also rushing apart. What’s more, they’re moving relative to each other at an enormous variety of speeds determined by the enormous variety of distances between them. Won’t this motion cause the clocks to fall out of synchronization, as Einstein taught us with special relativity? For a number of reasons, the answer is no; here is one particularly useful way to think about it.
Recall from Chapter 3 that Einstein discovered that clocks that move through space in different ways tick off time at different rates (because they divert different amounts of their motion through time into motion through space; remember the analogy with Bart on his skateboard, first heading north and then diverting some of his motion to the east). But the clocks we are now discussing are not moving through space at all. Just as each penny is glued to one point on the balloon and only moves relative to other pennies because of the swelling of the balloon’s surface, each galaxy occupies one region of space and, for the most part, only moves relative to other galaxies because of the expansion of space. And this means that, with respect to space itself, all the clocks are actually stationary, so they tick off time identically. It is precisely these clocks—clocks whose only motion comes from the expansion of space—that provide the synchronized cosmic clocks used to measure the age of the universe.
Figure 8.3 Clocks that move along with galaxies—whose motion, on average, arises only from the expansion of space—provide universal cosmic timepieces. They stay synchronized even though they separate from one another, since they move with space but not through space.
Notice, of course, that you are free to take your clock, hop aboard a rocket, and zip this way and that across space at enormous speeds, undergoing motion significantly in excess of the cosmic flow from spatial expansion. If you do this, your clock will tick at a different rate and you will find a different length of elapsed time since the bang. This is a perfectly valid point of view, but it is completely individualistic: the elapsed time measured is tied to the history of your particular whereabouts and states of motion. When astronomers speak of the universe’s age, though, they are seeking something universal—they are seeking a measure that has the same meaning everywhere. The uniformity of change throughout space provides a way of doing that.9
In fact, the uniformity of the microwave background radiation provides a ready-made test of whether you actually are moving with the cosmic flow of space. You see, although the microwave radiation is homogeneous across space, if you undertake additional motion beyond that from the cosmic flow of spatial expansion, you will not observe the radiation to be homogeneous. Just as the horn on a speeding car has a higher pitch when approaching and a lower pitch when receding, if you are zipping around in a spaceship, the crests and troughs of the microwaves heading toward the front of your ship will hit at a higher frequency than those traveling toward the back of your ship. Higher-frequency microwaves translate into higher temperatures, so you’d find the radiation in the direction you are heading to be a bit warmer than the radiation reaching you from behind. As it turns out, here on “spaceship” earth, astronomers do find the microwave background to be a little warmer in one direction in space and a little colder in the opposite direction. The reason is that not only does the earth move around the sun, and the sun move around the galactic center, but the entire Milky Way galaxy has a small velocity, in excess of cosmic expansion, toward the constellation Hydra. Only when astronomers correct for the effect these relatively slight additional motions have on the microwaves we receive does the radiation exhibit the exquisite uniformity of temperature between one part of the sky and another. It is this uniformity, this overall symmetry between one location and another, that allows us to speak sensibly of time when describing the entire universe.
Subtle Features of an Expanding Universe
A few subtle points in our explanation of cosmic expansion are worthy of emphasis. First, remember that in the balloon metaphor, it is only the balloon’s surface that plays any role—a surface that is only two-dimensional (each location can be specified by giving two numbers analogous to latitude and longitude on earth), whereas the space we see when we look around has three dimensions. We make use of this lower-dimensional model because it retains the concepts essential to the true, three-dimensional story but is far easier to visualize. It’s important to bear this in mind, especially if you have been tempted to say that there is a special point in the balloon model: the center point in the interior of the balloon away from which the whole rubber surface is moving. While this observation is true, it is meaningless in the balloon analogy because any point not on the balloon’s surface plays no role. The surface of the balloon represents all of space; points that do not lie on the surface of the balloon are merely irrelevant by-products of the analogy and do not correspond to any location in the universe.19
Second, if the speed of recession is larger and larger for galaxies that are farther and farther away, doesn’t that mean that galaxies that are sufficiently distant will rush away from us at a speed greater than the speed of light? The answer is a resounding, definite yes. Yet there is no conflict with special relativity. Why? Well, it’s closely related to the reason clocks moving apart due to the cosmic flow of space stay synchronized. As we emphasized in Chapter 3, Einstein showed that nothing can move through space faster than light. But galaxies, on average, hardly move through space at all. Their motion is due almost completely to the stretching of space itself. And Einstein’s theory does not prohibit space from expanding in a way that drives two points—two galaxies—away from each other at greater than light speed. His results only constrain speeds for which motion from spatial expansion has been subtracted out, motion in excess of that arising from spatial expansion. Observations confirm that for typical galaxies zipping along with the cosmic flow, such excess motion is minimal, fully in keeping with special relativity, even though their motion relative to each other, arising from the swelling of space itself, may exceed the speed of light.20
Third, if space is expanding, wouldn’t that mean that in addition to galaxies being driven away from each other, the swelling space within each galaxy would drive all its stars to move farther apart, and the swelling space within each star, and within each planet, and within you and me and everything else, would drive all the constituent atoms to move farther apart, and the swelling of space within each atom would drive all the subatomic constituents to move farther apart? In short, wouldn’t swelling space cause everything to grow in size, including our meter sticks, and in that way make it impossible to discern that any expansion had actually happened? The answer: no. Think again about the balloon-and-penny model. As the surface of the balloon swells, all the pennies are driven apart, but the pennies themselves surely do not expand. Of course, had we represented the galaxies by little circles drawn on the balloon with a black marker, then indeed, as the balloon grew in size the little circles would grow as well. But pennies, not blackened circles, capture what really happens. Each penny stays fixed in size because the forces holding its zinc and copper atoms together are far stronger than the outward pull of the expanding balloon to which it is glued. Similarly, the nuclear force holding individual atoms together, and the electromagnetic force holding your bones and skin together, and the gravitational force holding planets and stars intact and bound together in galaxies, are stronger than the outward swelling of space, and so none of these objects expands. Only on the largest of scales, on scales much larger than individual galaxies, does the swelling of space meet little or no resistance (the gravitational pull between widely separated galaxies is comparatively small, because of the large separations involved) and so only on such super galactic scales does the swelling of space drive objects apart.
Cosmology, Symmetry, and the Shape of Space
If someone were to wake you in the middle of the night from a deep sleep and demand you tell them the shape of the universe—the overall shape of space—you might be hard pressed to answer. Even in your groggy state, you know that Einstein showed space to be kind of like Silly Putty and so, in principle, it can take on practically any shape. How, then, can you possibly answer your interrogator’s question? We live on a small planet orbiting an average star on the outskirts of a galaxy that is but one of hundreds of billions dispersed throughout space, so how in the world can you be expected to know anything at all about the shape of the entire universe? Well, as the fog of sleep begins to lift, you gradually realize that the power of symmetry once again comes to the rescue.
If you take account of scientists’ widely held belief that, over large-scale averages, all locations and all directions in the universe are symmetrically related to one another, then you’re well on your way to answering the interrogator’s question. The reason is that almost all shapes fail to meet this symmetry criterion, because one part or region of the shape fundamentally differs from another. A pear bulges significantly at the bottom but less so at the top; an egg is flatter in the middle but pointier at its ends. These shapes, although exhibiting some degree of symmetry, do not possess complete symmetry. By ruling out such shapes, and limiting yourself only to those in which every region and direction is like every other, you are able to narrow down the possibilities fantastically.
We’ve already encountered one shape that fits the bill. The balloon’s spherical shape was the key ingredient in establishing the symmetry between all the Lincolns on its swelling surface, and so the three-dimensional version of this shape, the so-called three-sphere, is one candidate for the shape of space. But this is not the only shape that yields complete symmetry. Continuing to reason with the more easily visualized two-dimensional models, imagine an infinitely wide and infinitely long rubber sheet—one that is completely uncurved—with evenly spaced pennies glued to its surface. As the entire sheet expands, there once again is complete spatial symmetry and complete consistency with Hubble’s discovery: every Lincoln sees every other Lincoln rush away with a speed proportional to its distance, as in Figure 8.4. Hence, a three-dimensional version of this shape, like an infinite expanding cube of transparent rubber with galaxies evenly sprinkled throughout its interior, is another possible shape for space. (If you prefer culinary metaphors, think of an infinitely large version of the poppy seed muffin mentioned earlier, one that is shaped like a cube but goes on forever, with poppy seeds playing the role of galaxies. As the muffin bakes, the dough expands, causing each poppy seed to rush away from the others.) This shape is called flat space because, unlike the spherical example, it has no curvature (a meaning of “flat” that mathematicians and physicists use, but that differs from the colloquial meaning of “pancake-shaped.”)11
One nice thing about both the spherical and the infinite flat shapes is that you can walk endlessly and never reach an edge or a boundary. This is appealing because it allows us to avoid thorny questions: What is beyond the edge of space? What happens if you walk into a boundary of space? If space has no edges or boundaries, the question has no meaning. But notice that the two shapes realize this attractive feature in different ways. If you walk straight ahead in a spherically shaped space, you’ll find, like Magellan, that sooner or later you return to your starting point, never having encountered an edge. By contrast, if you walk straight ahead in infinite flat space, you’ll find that, like the Energizer Bunny, you can keep going and going, again never encountering an edge, but also never returning to where your journey began. While this might seem like a fundamental difference between the geometry of a curved and a flat shape, there is a simple variation on flat space that strikingly resembles the sphere in this regard.
Figure 8.4 (a) The view from any penny on an infinite flat plane is the same as the view from any other. (b) The farther apart two pennies are in Figure 8.4a, the greater the increase in their separation when the plane expands.
Figure 8.5 (a) A video game screen is flat (in the sense of “uncurved”) and has a finite size, but contains no edges or boundaries since it “wraps around.” Mathematically, such a shape is called a two-dimensional torus. (b) A three-dimensional version of the same shape, called a three-dimensional torus, is also flat (in the sense of uncurved) and has a finite volume, and also has no edges or boundaries, because it wraps around. If you pass through one face, you enter the opposite face.
To picture it, think of one of those video games in which the screen appears to have edges but in reality doesn’t, since you can’t actually fall off: if you move off the right edge, you reappear on the left; if you move off the top edge, you reappear on the bottom. The screen “wraps around,” identifying top with bottom and left with right, and in that way the shape is flat (uncurved) and has finite size, but has no edges. Mathematically, this shape is called a two-dimensional torus; it is illustrated in Figure 8.5a.12 The three-dimensional version of this shape—a three-dimensional torus—provides another possible shape for the fabric of space. You can think of this shape as an enormous cube that wraps around along all three axes: when you walk through the top you reappear at the bottom, when you walk through the back, you reappear at the front, when you walk through the left side, you reappear at the right, as in Figure 8.5b. Such a shape is flat—again, in the sense of being uncurved, not in the sense of being like a pancake—three-dimensional, finite in all directions, and yet has no edges or boundaries.
Beyond these possibilities, there is still another shape consistent with the symmetric expanding space explanation for Hubble’s discovery. Although it’s hard to picture in three dimensions, as with the spherical example there is a good two-dimensional stand-in: an infinite version of a Pringle’s potato chip. This shape, often referred to as a saddle, is a kind of inverse of the sphere: Whereas a sphere is symmetrically bloated outward, the saddle is symmetrically shrunken inward, as illustrated in Figure 8.6. Using a bit of mathematical terminology, we say that the sphere is positively curved (bloats outward), the saddle is negatively curved (shrinks inward), and flat space—whether infinite or finite—has no curvature (no bloating or shrinking).21
Researchers have proven that this list—uniformly positive, negative, or zero—exhausts the possible curvatures for space that are consistent with the requirement of symmetry between all locations and in all directions. And that is truly stunning. We are talking about the shape of the entire universe, something for which there are endless possibilities. Yet, by invoking the immense power of symmetry, researchers have been able to narrow the possibilities sharply. And so, if you allow symmetry to guide your answer, and your late-night interrogator grants you a mere handful of guesses, you’ll be able to meet his challenge.13
All the same, you might wonder why we’ve come upon a variety of possible shapes for the fabric of space. We inhabit a single universe, so why can’t we specify a unique shape? Well, the shapes we’ve listed are the only ones consistent with our belief that every observer, regardless of where in the universe they’re located, should see on the largest of scales an identical cosmos. But such considerations of symmetry, while highly selective, are not able to go all the way and pick out a unique answer. For that we need Einstein’s equations from general relativity.
Figure 8.6 Using the two-dimensional analogy for space, there are three types of curvature that are completely symmetric—that is, curvatures in which the view from any location is the same as that from any other. They are (a)positive curvature, which uniformly bloats outward, as on a sphere; (b) zero curvature, which does not bloat at all, as on an infinite plane or finite video game screen; (c) negative curvature, which uniformly shrinks inward, as on a saddle.
As input, Einstein’s equations take the amount of matter and energy in the universe (assumed, again by consideration of symmetry, to be distributed uniformly) and as output, they give the curvature of space. The difficulty is that for many decades astronomers have been unable to agree on how much matter and energy there actually is. If all the matter and energy in the universe were to be smeared uniformly throughout space, and if, after this was done, there turned out to be more than the so-called critical density of about .00000000000000000000001 (10−23) grams in every cubic meter22—about five hydrogen atoms per cubic meter—Einstein’s equations would yield a positive curvature for space; if there were less than the critical density, the equations would imply negative curvature; if there were exactly the critical density, the equations would tell us that space has no overall curvature. Although this observational issue is yet to be settled definitively, the most refined data are tipping the scales on the side of no curvature—the flat shape. But the question of whether the Energizer Bunny could move forever in one direction and vanish into the darkness, or would one day circle around and catch you from behind—whether space goes on forever or wraps back like a video screen—is still completely open.14
Even so, even without a final answer to the shape of the cosmic fabric, what’s abundantly clear is that symmetry is the essential consideration allowing us to comprehend space and time when applied to the universe as a whole. Without invoking the power of symmetry, we’d be stuck at square one.
Cosmology and Spacetime
We can now illustrate cosmic history by combining the concept of expanding space with the loaf-of-bread description of spacetime from Chapter 3. Remember, in the loaf-of-bread portrayal, each slice—even though two-dimensional—represents all of three-dimensional space at a single moment of time from the perspective of one particular observer. Different observers slice up the loaf at different angles, depending on details of their relative motion. In the examples encountered previously, we did not take account of expanding space and, instead, imagined that the fabric of the cosmos was fixed and unchanging over time. We can now refine those examples by including cosmological evolution.
To do so, we will take the perspective of observers who are at rest with respect to space—that is, observers whose only motion arises from cosmic expansion, just like the Lincolns glued to the balloon. Again, even though they are moving relative to one another, there is symmetry among all such observers—their watches all agree—and so they slice up the spacetime loaf in exactly the same way. Only relative motion in excess of that coming from spatial expansion, only relative motion through space as opposed to motion from swelling space, would result in their watches falling out of synch and their slices of the spacetime loaf being at different angles. We also need to specify the shape of space, and for purposes of comparison we will consider some of the possibilities discussed above.
The easiest example to draw is the flat and finite shape, the video game shape. In Figure 8.7a, we show one slice in such a universe, a schematic image you should think of as representing all of space right now. For simplicity, imagine that our galaxy, the Milky Way, is in the middle of the figure, but bear in mind that no location is in any way special compared with any other. Even the edges are illusory. The top side is not a place where space ends, since you can pass through and reappear at the bottom; similarly, the left side is not a place where space ends, since you can pass through and reappear on the right side. To accommodate astronomical observations, each side should extend at least 14 billion light-years (about 85 billion trillion miles) from its midpoint, but each could be much longer.
Note that right now we can’t literally see the stars and galaxies as drawn on this now slice since, as we discussed in Chapter 5, it takes time for the light emitted by any object right now to reach us. Instead, the light we see when we look up on a clear, dark night was emitted long ago—millions and even billions of years ago—and only now has completed the long journey to earth, entered our telescopes, and allowed us to marvel at the wonders of deep space. Since space is expanding, eons ago, when this light was emitted, the universe was a lot smaller. We illustrate this in Figure 8.7b in which we have put our current now slice on the right-hand side of the loaf and included a sequence of slices to the left that depict our universe at ever earlier moments of time. As you can see, the overall size of space and the separations between individual galaxies both decrease as we look at the universe at ever earlier moments.
Figure 8.7 (a) A schematic image depicting all of space right now, assuming space is flat and finite in extent, i.e. shaped like a video game screen. Note that the galaxy on the upper right wraps around on the left. (b) A schematic image depicting all of space as it evolves through time, with a few time slices emphasized for clarity. Note that the overall size of space and the separation between galaxies decrease as we look farther back in time.
In Figure 8.8, you can also see the history of light, emitted by a distant galaxy perhaps a billion years ago, as it has traveled toward us here in the Milky Way. On the initial slice in Figure 8.8a, the light is first emitted, and on subsequent slices you can see the light getting closer and closer even as the universe gets larger and larger, and finally you can see it reaching us on the rightmost time slice. In Figure 8.8b, by connecting the locations on each slice that the light’s leading edge passed through during its journey, we show the light’s path through spacetime. Since we receive light from many directions, Figure 8.8c shows a sample of trajectories through space and time that various light beams take to reach us now.
Figure 8.8 (a) Light emitted long ago from a distant galaxy gets closer and closer to the Milky Way on subsequent time slices. (b ) When we finally see the distant galaxy, we are looking at it across both space and time, since the light we see was emitted long ago. The path through spacetime followed by the light is highlighted. (c) The paths through spacetime taken by light emitted from various astronomical bodies that we see today.
The figures dramatically show how light from space can be used as a cosmic time capsule. When we look at the Andromeda galaxy, the light we receive was emitted some 3 million years ago, so we are seeing Andromeda as it was in the distant past. When we look at the Coma cluster, the light we receive was emitted some 300 million years ago and hence we are seeing the Coma cluster as it was in an even earlier epoch. If right now all the stars in all the galaxies in this cluster were to go supernova, we would still see the same undisturbed image of the Coma cluster and would do so for another 300 million years; only then would light from the exploding stars have had enough time to reach us. Similarly, should an astronomer in the Coma cluster who is on our current now-slice turn a superpowerful telescope toward earth, she will see an abundance of ferns, arthropods, and early reptiles; she won’t see the Great Wall of China or the Eiffel Tower for almost another 300 million years. Of course, this astronomer, well trained in basic cosmology, realizes that she is seeing light emitted in earth’s distant past, and in laying out her own cosmic spacetime loaf will assign earth’s early bacteria to their appropriate epoch, their appropriate set of time slices.
All of this assumes that both we and the Coma cluster astronomer are moving only with the cosmic flow from spatial expansion, since this ensures that her slicing of the spacetime loaf coincides with ours—it ensures that her now-lists agree with ours. However, should she break ranks and move through space substantially in excess of the cosmic flow, her slices will tilt relative to ours, as in Figure 8.9. In this case, as we found with Chewie in Chapter 5, this astronomer’s now will coincide with what we consider to be our future or our past (depending on whether the additional motion is toward or away from us). Notice, though, that her slices will no longer be spatially homogeneous. Each angled slice in Figure 8.9 intersects the universe in a range of different epochs and so the slices are far from uniform. This significantly complicates the description of cosmic history, which is why physicists and astronomers generally don’t contemplate such perspectives. Instead, they usually consider only the perspective of observers moving solely with the cosmic flow, since this yields slices that are homogeneous—but fundamentally speaking, each viewpoint is as valid as any other.
Figure 8.9 The time slice of an observer moving significantly in excess of the cosmic flow from spatial expansion.
As we look farther to the left on the cosmic spacetime loaf, the universe gets ever smaller and ever denser. And just as a bicycle tire gets hotter and hotter as you squeeze more and more air into it, the universe gets hotter and hotter as matter and radiation are compressed together more and more tightly by the shrinking of space. If we head back to a mere ten millionths of a second after the beginning, the universe gets so dense and so hot that ordinary matter disintegrates into a primordial plasma of nature’s elementary constituents. And if we continue our journey, right back to nearly time zero itself—the time of the big bang—the entire known universe is compressed to a size that makes the dot at the end of this sentence look gargantuan. The densities at such an early epoch were so great, and the conditions were so extreme, that the most refined physical theories we currently have are unable to give us insight into what happened. For reasons that will become increasingly clear, the highly successful laws of physics developed in the twentieth century break down under such intense conditions, leaving us rudderless in our quest to understand the beginning of time. We will see shortly that recent developments are providing a hopeful beacon, but for now we acknowledge our incomplete understanding of what happened at the beginning by putting a fuzzy patch on the far left of the cosmic spacetime loaf—our verson of the terra incognita on maps of old. With this finishing touch, we present Figure 8.10 as a broad-brush illustration of cosmic history.
Figure 8.10 Cosmic history—the spacetime “loaf”—for a universe that is flat and of finite spatial extent. The fuzziness at the top denotes our lack of understanding near the beginning of the universe.
We’ve so far assumed that space is shaped like a video game screen, but the story has many of the same features for the other possibilities. For example, if the data ultimately show that the shape of space is spherical, then, as we go ever farther back in time, the size of the sphere gets ever smaller, the universe gets ever hotter and denser, and at time zero we encounter some kind of big bang beginning. Drawing an illustration analogous to Figure 8.10 is challenging since spheres don’t neatly stack one next to the other (you can, for example, imagine a “spherical loaf” with each slice being a sphere that surrounds the previous), but aside from the graphic complications, the physics is largely the same.
The cases of infinite flat space and of infinite saddle-shaped space also share many features with the two shapes already discussed, but they do differ in one essential way. Take a look at Figure 8.11, in which the slices represent flat space that goes on forever (of which we can show only a portion, of course). As you look at ever earlier times, space shrinks; galaxies get closer and closer together the farther back you look in Figure 8.11b. However, the overall size of space stays the same. Why? Well, infinity is a funny thing. If space is infinite and you shrink all distances by a factor of two, the size of space becomes half of infinity, and that is still infinite. So although everything gets closer together and the densities get ever higher as you head further back in time, the overall size of the universe stays infinite; things get dense everywhere on an infinite spatial expanse. This yields a rather different image of the big bang.
Normally, we imagine the universe began as a dot, roughly as in Figure 8.10, in which there is no exterior space or time. Then, from some kind of eruption, space and time unfurled from their compressed form and the expanding universe took flight. But if the universe is spatially infinite, there was already an infinite spatial expanse at the moment of the big bang. At this initial moment, the energy density soared and an incomparably large temperature was reached, but these extreme conditions existed everywhere, not just at one single point. In this setting, the big bang did not take place at one point; instead, the big bang eruption took place everywhere on the infinite expanse. Comparing this to the conventional single-dot beginning, it is as though there were many big bangs, one at each point on the infinite spatial expanse. After the bang, space swelled, but its overall size didn’t increase since something already infinite can’t get any bigger. What did increase are the separations between objects like galaxies (once they formed), as you can see by looking from left to right in Figure 8.11b. An observer like you or me, looking out from one galaxy or another, would see surrounding galaxies all rushing away, just as Hubble discovered.
Bear in mind that this example of infinite flat space is far more than academic. We will see that there is mounting evidence that the overall shape of space is not curved, and since there is no evidence as yet that space has a video game shape, the flat, infinitely large spatial shape is the front-running contender for the large-scale structure of spacetime.
Figure 8.11 (a) Schematic depiction of infinite space, populated by galaxies. (b) Space shrinks at ever earlier times—so galaxies are closer and more densely packed at earlier times—but the overall size of infinite space stays infinite. Our ignorance of what happens at the earliest times is again denoted by a fuzzy patch, but here the patch extends through the infinite spatial expanse.
Cosmology and Symmetry
Considerations of symmetry have clearly been indispensable in the development of modern cosmological theory. The meaning of time, its applicability to the universe as a whole, the overall shape of space, and even the underlying framework of general relativity, all rest on foundations of symmetry. Even so, there is yet another way in which ideas of symmetry have informed the evolving cosmos. Through the course of its history, the temperature of the universe has swept across an enormous range, from the ferociously hot moments just after the bang to the few degrees above absolute zero you’d find today if you took a thermometer into deep space. And, as I will explain in the next chapter, because of a critical interdependence between heat and symmetry, what we see today is likely but a cool remnant of the far richer symmetry that molded the early universe and determined some of the most familiar and essential features of the cosmos.