The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  Heat and Symmetry

  When things get very hot or very cold, they sometimes change. And sometimes the change is so pronounced that you can't even recognize the things with which you began. Because of the torrid conditions just after the bang, and the subsequent rapid drop in temperature as space expanded and cooled, understanding the effects of temperature change is crucial in grappling with the early history of the universe. But let's start simpler. Let's start with ice.

  If you heat a very cold piece of ice, at first not much happens. Although its temperature rises, its appearance remains pretty much unchanged. But if you raise its temperature all the way to 0 degrees Celsius and you keep the heat on, suddenly something dramatic does happen. The solid ice starts to melt and turns into liquid water. Don't let the familiarity of this transformation dull the spectacle. Without previous experiences involving ice and water, it would be a challenge to realize the intimate connection between them. One is a rock-hard solid while the other is a viscous liquid. Simple observation reveals no direct evidence that their molecular makeup, H 2 O, is identical. If you'd never before seen ice or water and were presented with a vat of each, at first you would likely think they were unrelated. And yet, as either crosses through 0 degrees Celsius, you'd witness a wondrous alchemy as each transmutes into the other.

  If you continue to heat liquid water, you again find that for a while not much happens beyond a steady rise in temperature. But then, when you reach 100 degrees Celsius, there is another sharp change: the liquid water starts to boil and transmute into steam, a hot gas that again is not obviously connected to liquid water or to solid ice. Yet, of course, all three share the same molecular composition. The changes from solid to liquid and liquid to gas are known as phase transitions. Most substances go through a similar sequence of changes if their temperatures are varied through a wide enough range. 1

  Symmetry plays a central role in phase transitions. In almost all cases, if we compare a suitable measure of something's symmetry before and after it goes through a phase transition, we find a significant change. On a molecular scale, for instance, ice has a crystalline form with H 2 O molecules arranged in an ordered, hexagonal lattice. Like the symmetries of the box in Figure 8.1, the overall pattern of the ice molecules is left unchanged only by certain special manipulations, such as rotations in units of 60 degrees about particular axes of the hexagonal arrangement. By contrast, when we heat ice, the crystalline arrangement melts into a jumbled, uniform clump of molecules—liquid water—that remains unchanged under rotations by any angle, about any axis. So, by heating ice and causing it to go through a solid-to-liquid phase transition, we have made it more symmetric. (Remember, although you might intuitively think that something more ordered, like ice, is more symmetric, quite the opposite is true; something is more symmetric if it can be subjected to more transformations, such as rotations, while its appearance remains unchanged.)

  Similarly, if we heat liquid water and it turns into gaseous steam, the phase transition also results in an increase of symmetry. In a clump of water, the individual H 2 O molecules are, on average, packed together with the hydrogen side of one molecule next to the oxygen side of its neighbor. If you were to rotate one or another molecule in a clump it would noticeably disrupt the molecular pattern. But when the water boils and turns into steam, the molecules flit here and there freely; there is no longer any pattern to the orientations of the H 2 O molecules and hence, were you to rotate a molecule or group of molecules, the gas would look the same. Thus, just as the ice-to-water transition results in an increase in symmetry, the water-to-steam transition does so as well. Most (but not all 2 ) substances behave in a similar way, experiencing an increase of symmetry when they undergo solid-to-liquid and liquid-to-gas phase transitions.

  The story is much the same when you cool water or almost any other substance; it just takes place in reverse. For example, when you cool gaseous steam, at first not much happens, but as its temperature drops to 100 degrees Celsius, it suddenly starts to condense into liquid water; when you cool liquid water, not much happens until you reach 0 degrees Celsius, at which point it suddenly starts to freeze into solid ice. And, following the same reasoning regarding symmetries—but in reverse—we conclude that both of these phase transitions are accompanied by a decrease in symmetry. 23

  So much for ice, water, steam, and their symmetries. What does all this have to do with cosmology? Well, in the 1970s, physicists realized that not only can objects in the universe undergo phase transitions, but the cosmosas a whole can do so as well. Over the last 14 billion years, the universe has steadily expanded and decompressed. And just as a decompressing bicycle tire cools off, the temperature of the expanding universe has steadily dropped. During much of this decrease in temperature, not much happened. But there is reason to believe that when the universe passed through particular critical temperatures—the analogs of 100 degrees Celsius for steam and 0 degrees Celsius for water—it underwent radical change and experienced a drastic reduction in symmetry. Many physicists believe that we are now living in a "condensed" or "frozen" phase of the universe, one that is very different from earlier epochs. The cosmological phase transitions did not literally involve a gas condensing into a liquid, or a liquid freezing into a solid, although there are many qualitative similarities with these more familiar examples. Rather, the "substance" that condensed or froze when the universe cooled through particular temperatures is a field—more precisely, a Higgs field. Let's see what this means.

  Force, Matter, and Higgs Fields

  Fields provide the framework for much of modern physics. The electromagnetic field, discussed in Chapter 3, is perhaps the simplest and most widely appreciated of nature's fields. Living among radio and television broadcasts, cell phone communications, the sun's heat and light, we are all constantly awash in a sea of electromagnetic fields. Photons are the elementary constituents of electromagnetic fields and can be thought of as the microscopic transmitters of the electromagnetic force. When you see something, you can think of it in terms of a waving electromagnetic field entering your eye and stimulating your retina, or in terms of photon particles entering your eye and doing the same thing. For this reason, the photon is sometimes described as the messenger particle of the electromagnetic force.

  The gravitational field is also familiar since it constantly and consistently anchors us, and everything around us, to the earth's surface. As with electromagnetic fields, we are all immersed in a sea of gravitational fields; the earth's is dominant, but we also feel the gravitational fields of the sun, the moon, and the other planets. Just as photons are particles that constitute an electromagnetic field, physicists believe that gravitons are particles that constitute a gravitational field. Graviton particles have yet to be discovered experimentally, but that's not surprising. Gravity is by far the weakest of all forces (for example, an ordinary refrigerator magnet can pick up a paper clip, thereby overcoming the pull of the entire earth's gravity) and so it's understandable that experimenters have yet to detect the smallest constituents of the feeblest force. Even without experimental confirmation, though, most physicists believe that just as photons transmit the electromagnetic force (they are the electromagnetic force's messenger particles), gravitons transmit the gravitational force (they are the gravitational force's messenger particles). When you drop a glass, you can think of the event in terms of the earth's gravitational field pulling on the glass, or, using Einstein's more refined geometrical description, you can think of it in terms of the glass's sliding along an indentation in the spacetime fabric caused by the earth's presence, or—if gravitons do indeed exist—you can also think of it in terms of graviton particles firing back and forth between the earth and the glass, communicating a gravitational "message" that "tells" the glass to fall toward the earth.

  Beyond these well-known force fields, there are two other forces of nature, the strong nuclear force and the weak nuclear force, and they also exert their influence via fields. The nuclear forc
es are less familiar than electromagnetism and gravity because they operate only on atomic and subatomic scales. Even so, their impact on daily life, through nuclear fusion that causes the sun to shine, nuclear fission at work in atomic reactors, and radioactive decay of elements like uranium and plutonium, is no less significant. The strong and weak nuclear force fields are called Yang- Mills fields after C. N. Yang and Robert Mills, who worked out their theoretical underpinnings in the 1950s. And just as electromagnetic fields are composed of photons, and gravitational fields are believed to be composed of gravitons, the strong and weak fields also have particulate constituents. The particles of the strong force are called gluons and those of the weak force are called W and Z particles. The existence of these force particles was confirmed by accelerator experiments carried out in Germany and Switzerland in the late 1970s and early 1980s.

  The field framework also applies to matter. Roughly speaking, the probability waves of quantum mechanics may themselves be thought of as space-filling fields that provide the probability that some or other particle of matter is at some or other location. An electron, for instance, can be thought of as a particle—one that can leave a dot on a phosphor screen, as in Figure 4.4—but it can (and must) also be thought of in terms of a waving field, one that can contribute to an interference pattern on a phosphor screen as in Figure 4.3b. 3 In fact, although I won't go into it in greater detail here, 4 an electron's probability wave is closely associated with something called an electron field— a field that in many ways is similar to an electromagnetic field but in which the electron plays a role analogous to the photon's, being the electron field's smallest constituent. The same kind of field description holds true for all other species of matter particles as well.

  Having discussed both matter fields and force fields, you might think we've covered everything. But there is general agreement that the story told thus far is not quite complete. Many physicists strongly believe that there is yet a third kind of field, one that has never been experimentally detected but that over the last couple of decades has played a pivotal role both in modern cosmological thought and in elementary particle physics. It is called a Higgs field, after the Scottish physicist Peter Higgs. 5 And if the ideas in the next section are right, the entire universe is permeated by an ocean of Higgs field—a cold relic of the big bang—that is responsible for many of the properties of the particles that make up you and me and everything else we've ever encountered.

  Fields in a Cooling Universe

  Fields respond to temperature much as ordinary matter does. The higher the temperature, the more ferociously the value of a field will—like the surface of a rapidly boiling pot of water—undulate up and down. At the chilling temperature characteristic of deep space today (2.7 degrees above absolute zero, or 2.7 Kelvin, as it is usually denoted), or even at the warmer temperatures here on earth, field undulations are minuscule. But the temperature just after the big bang was so enormous—at 10 -43 seconds after the bang, the temperature is believed to have been about 10 32 Kelvin—that all fields violently heaved to and fro.

  As the universe expanded and cooled, the initially huge density of matter and radiation steadily dropped, the vast expanse of the universe became ever emptier, and field undulations became ever more subdued. For most fields this meant that their values, on average, got closer to zero. At some moment, the value of a particular field might jitter slightly above zero (a peak) and a moment later it might dip slightly below zero (a trough), but on average the value of most fields closed in on zero—the value we intuitively associate with absence or emptiness.

  Here's where the Higgs field comes in. It's a variety of field, researchers have come to realize, that had properties similar to other fields' at the scorchingly high temperatures just after the big bang: it fluctuated wildly up and down. But researchers believe that (just as steam condenses into liquid water when its temperature drops sufficiently) when the temperature of the universe dropped sufficiently, the Higgs field condensed into a particular nonzero value throughout all of space. Physicists refer to this as the formation of a nonzero Higgs field vacuum expectation value— but to ease the technical jargon, I'll refer to this as the formation of a Higgs ocean.

  It's kind of like what would happen if you were to drop a frog into a hot metal bowl, as in Figure 9.1a, with a pile of worms lying in the center. At first, the frog would jump this way and that—high up, low down, left, right—in a desperate attempt to avoid burning its legs, and on average would stay so far from the worms that it wouldn't even know they were there. But as the bowl cooled, the frog would calm itself, would hardly jump at all, and, instead, would gently slide down to the most restful spot at the bowl's bottom. There, having closed in on the bowl's center, it would finally rendezvous with its dinner, as in Figure 9.1b.

  But if the bowl were shaped differently, as in Figure 9.1c, things would turn out differently. Imagine again that the bowl starts out very hot and that the worm pile still lies at the bowl's center, now high up on the central bump. Were you to drop the frog in, it would again wildly jump this way and that, remaining oblivious to the prize perched on the central plateau. Then, as the bowl cooled, the frog would again settle itself, reduce its jumping, and slide down the bowl's smooth sides. But because of the new shape, the frog would never make it to the bowl's center. Instead, it would slide down into the bowl's valley and remain at a distance from the worm pile, as in Figure 9.1d.

  Figure 9.1 ( a ) A frog dropped into a hot metal bowl incessantly jumps around. ( b ) When the bowl cools, the frog calms down, jumps much less, and slides down to the bowl's middle.

  If we imagine that the distance between the frog and the worm pile represents the value of a field—the farther the frog is from the worms, the larger the value of the field—and the height of the frog represents the energy contained in that field value—the higher up on the bowl the frog happens to be, the more energy the field contains—then these examples convey well the behavior of fields as the universe cools. When the universe is hot, fields jump wildly from value to value, much as the frog jumps from place to place in the bowl. As the universe cools, fields "calm down," they jump less often and less frantically, and their values slide downward to lower energy.

  Figure 9.1 ( c ) As in (a), but with a hot bowl of a different shape. ( d ) As in (b), but now when the bowl cools, the frog slides down to the valley, which is some distance from the bowl's center (where the worms are located).

  But here's the thing. As with the frog example, there's a possibility of two qualitatively different outcomes. If the shape of the field's energy bowl—its so-called potential energy— is similar to that in Figure 9.1a, the field's value throughout space will slide all the way down to zero, the bowl's center, just as the frog slides all the way down to the worm pile. However, if the field's potential energy looks like that in Figure 9.1c, the field's value will not make it all the way to zero, to the energy bowl's center. Instead, just as the frog will slide down to the valley, which is a nonzero distance from the worm pile, the field's value will also slide down to the valley—a nonzero distance from the bowl's center—and that means the field will have a nonzero value. 6 The latter behavior is characteristic of Higgs fields. As the universe cools, a Higgs field's value gets caught in the valley and never makes it to zero. And since what we're describing would happen uniformly throughout space, the universe would be permeated by a uniform and nonzero Higgs field—a Higgs ocean.

  The reason this happens sheds light on the fundamental peculiarity of Higgs fields. As a region of space becomes ever cooler and emptier—as matter and radiation get ever more sparse—the energy in the region gets ever lower. Taking this to the limit, you know you've reached the emptiest a region of space can be when you've lowered its energy as far as possible. For ordinary fields suffusing a region of space, their energy contribution is lowest when their value has slid all the way down to the center of the bowl as in Figure 9.1b; they have zero energy when their value is zero. That makes
good, intuitive sense since we associate emptying a region of space with setting everything, including field values, to zero.

  But for a Higgs field, things work differently. Just as a frog can reach the central plateau in Figure 9.1c and be zero distance from the worm pile only if it has enough energy to jump up from the surrounding valley, a Higgs field can reach the bowl's center, and have value zero, only if it too embodies enough energy to surmount the bowl's central bump. If, to the contrary, the frog has little or no energy, it will slide to the valley in Figure 9.1d—a nonzero distance from the worm pile. Similarly, a Higgs field with little or no energy will also slide to the bowl's valley—a nonzero distance from the bowl's center—and hence it will have a nonzero value.

  To force a Higgs field to have a value of zero—the value that would seem to be the closest you can come to completely removing the field from the region, the value that would seem to be the closest you can come to a state of nothingness—you would have to raise its energy and, energetically speaking, the region of space would not be as empty as it possibly could. Even though it sounds contradictory, removing the Higgs field— reducing its value to zero, that is—is tantamount to adding energy to the region. As a rough analogy, think of one of those fancy noise reduction headphones that produce sound waves to cancel those coming from the environment that would otherwise impinge on your eardrums. If the headphones work perfectly, you hear silence when they produce their sounds, but you hear the ambient noise if you shut them off. Researchers have come to believe that just as you hear less when the headphones are suffused with the sounds they are programmed to produce, so cold, empty space harbors as little energy as it possibly can—it is as empty as it can be—when it is suffused with an ocean of Higgs field. Researchers refer to the emptiest space can be as the vacuum, and so we learn that the vacuum may actually be permeated by a uniform Higgs field.