by Brian Greene
13 Remember, on pages 152-53 we showed the huge difference between the number of ordered and disordered configurations for a mere 693 double-sided sheets of paper. We are now discussing the behavior of roughly 10 24 H 2 O molecules, so the difference between the number of ordered and disordered configurations is breathtakingly monumental. Moreover, the same reasoning holds for all other atoms and molecules within you and within the environment (brains, security cameras, air molecules, and so on). Namely, in the standard explanation in which you can trust your memories, not only would the partially melted ice cubes have begun, at 10 p.m., in a more ordered—less likely—state, but so would everything else: when a video camera records a sequence of events, there is a net increase in entropy (from the heat and noise released by the recording process); similarly, when a brain records a memory, although we understand the microscopic details with less accuracy, there is a net increase in entropy (the brain may gain order but as with any order-producing process, if we take account of heat generated, there is a net increase in entropy). Thus, if we compare the total entropy in the bar between 10 p.m. and 10:30 p.m. in the two scenarios—one in which you trust your memories, and the other in which things spontaneously arrange themselves from an initial state of disorder to be consistent with what you see, now, at 10:30 p.m.—there is an enormous entropy difference. The latter scenario, every step of the way, has hugely more entropy than the former scenario, and so, from the standpoint of probability, is hugely more likely.
14 A closely related point is that should we convince ourselves that the world we see right now just coalesced out of total disorder, the exact same reasoning—invoked anytime later—would require us to abandon our current belief and, instead, attribute the ordered world to a yet more recent fluctuation. Thus, in this way of thinking, every next moment invalidates the beliefs held in each previous moment, a distinctly unconvincing way of explaining the cosmos.
15 That is, a black hole of a given size contains more entropy than anything else of the same size.
16 Even though Feynman's sum over histories approach might seem to make the particle aspect prominent, it is just a particular interpretation of probability waves (since it involves many histories for a single particle, each making its own probabilistic contribution), and so is subsumed by the wavelike side of complementarity. When we speak of something behaving like a particle, we will always mean a conventional particle that travels along one and only one trajectory.
17 If you find this section tough going, you can safely move on to the next section without loss of continuity. But I encourage you to try to get through it, as the results are truly stupendous.
18 Quantum mechanics, rightly, has a reputation as being anything but smooth and gradual; rather, as we will see explicitly in later chapters, it reveals a turbulent and jittery microcosmos. The origin of this jitteriness is the probabilistic nature of the wavefunction—even though things can be one way at one moment, there is a probability that they will be significantly different a moment later— not an ever-present jittery quality of the wavefunction itself.
19 To go beyond the two-dimensional metaphor of a balloon's surface and have a spherical three-dimensional model is easy mathematically but difficult to picture, even for professional mathematicians and physicists. You might be tempted to think of a solid, three-dimensional ball, like a bowling ball without the finger holes. This, however, isn't an acceptable shape. We want all points in the model to be on a completely equal footing, since we believe that every place in the universe is (on average) just like any other. But the bowling ball has all sorts of different points: some are on the outside surface, others are embedded in the interior, one is right in the center. Instead, just as the two-dimensional surface of a balloon surrounds a three- dimensional spherical region (containing the balloon's air), an acceptable round three-dimensional shape would need to surround a four- dimensional spherical region. So the three-dimensional spherical surface of a balloon in a four-dimensional space is an acceptable shape. But if that still leaves you groping for an image, do what just about all professionals do: stick to the easy-to-visualize lower-dimensional analogies. They capture almost all of the essential features. A bit further on, we consider three-dimensional flat space, as opposed to the round shape of a sphere, and that flat space can be visualized.
20 Depending on whether the rate of the universe's expansion is speeding up or slowing down over time, the light emitted from such galaxies may fight a battle that would have made Zeno proud: the light may stream toward us at light speed while the expansion of space makes the distance the light has yet to cover ever larger, preventing the light from ever reaching us. See notes section for details.10
21 Just as the video game screen gives a finite-sized version of flat space that has no edges or boundaries, there are finite-sized versions of the saddle shape that also have no edges or boundaries. I won't discuss this further, save to note that it implies that all three possible curvatures (positive, zero, negative) can be realized by finite-sized shapes without edges or boundaries. (In principle, then, a space-faring Magellan could carry out a cosmic version of his voyage in a universe whose curvature is given by any of the three possibilities.)
22 Today, matter in the universe is more abundant than radiation, so it's convenient to express the critical density in units most relevant for mass—grams per cubic meter. Note too that while 10 -23 grams per cubic meter might not sound like a lot, there are many cubic meters of space out there in the cosmos. Moreover, the farther back in time you look, the smaller the space into which the mass/energy is squeezed, so the denser the universe becomes.
23 Even though a decrease in symmetry means that fewer manipulations go unnoticed, the heat released to the environment during these transformations ensures that overall entropy—including that of the environment—still increases.
24 The terminology isn't particularly important, but briefly, here's where it comes from. The valley in Figure 9.1c and 9.1d has a symmetric shape—it's circular—with every point being on a par with every other (each point denotes a Higgs field value of lowest possible energy). Yet, when the Higgs field's value slides down the bowl, it lands on one particular point on the circular valley, and in so doing "spontaneously" selects one location on the valley as special. In turn, the points on the valley are no longer all on an equal footing, since one has been picked out, and so the Higgs field disrupts or "breaks" the previous symmetry between them. Thus, putting the words together, the process in which the Higgs slides down to one particular nonzero value in the valley is called spontaneous symmetrybreaking. Later in the text, we will describe more tangible aspects of the reduction of symmetry associated with such a formation of a Higgs ocean. 7
25 You might think I've left out an "i" in the last syllable of "inflaton," but I haven't; physicists often give fields names, such as photon and gluon, which end with "on."
26 As the universe expands, the energy loss of photons can be directly observed because their wavelengths stretch—they undergo redshift— and the longer a photon's wavelength, the less energy it has. The microwave background photons have undergone such redshift for nearly 14 billion years, explaining their long—microwave—wavelengths, and their low temperature. Matter undergoes a similar loss of its kinetic energy (energy from particle motion), but the total energy bound up in the mass of particles (their rest energy— the energy equivalent of their mass, when at rest) remains constant.
27 While useful, the rubber-band analogy is not perfect. The inward, negative pressure exerted by the rubber bands impedes the expansion of the box, whereas the inflaton's negative pressure drives the expansion of space. This important difference illustrates the clarification emphasized on page 278: in cosmology, it is not that uniform negative pressure drives expansion (only pressure differences result in forces, so uniform pressure, whether positive or negative, exerts no force). Rather, pressure, like mass, gives rise to a gravitational force. And negative pressure gives rise to a repulsive grav
itational force that drives expansion. This does not affect our conclusions.
28 Some researchers, including Alan Guth and Eddie Farhi, have investigated whether one might, hypothetically, create a new universe in the laboratory by synthesizing a nugget of inflaton field. Beyond the fact that we still don't have direct experimental verification that there is such a thing as an inflaton field, note that the twenty pounds of inflaton field would need to be crammed in a tiny space, roughly 10 -26 or so centimeters on a side, and hence the density would be enormous—some 10 67 times the density of an atomic nucleus—way beyond what we can produce, now or perhaps ever.
29 Don't get confused here: The inflationary stretching of quantum jitters discussed in the last section still produced a minuscule, unavoidable nonuniformity of about 1 part in 100,000. But that tiny nonuniformity overlaid an otherwise smooth universe. We are now describing how the latter—the underlying smooth uniformity—came to be.
30 For ease of writing, we'll consider only fields that reach their lowest energy when their values are zero. The discussion for other fields—Higgs fields—is identical, except the jitters fluctuate about the field's nonzero, lowest-energy value. If you are tempted to say that a region of space is empty only if there is no matter present and all fields are absent, not just that they have the value zero, see notes section. 2
31 The remainder of this chapter recounts the discovery of superstring theory and discusses the theory's essential ideas regarding unification and the structure of spacetime. Readers of The Elegant Universe (especially Chapters 6 through 8) will be familiar with much of this material, and should feel free to skim this chapter and move on to the next.
32 Remember, as noted in Chapter 9, even a puny magnet can overpower the pull of the entire earth's gravity and pick up a paper clip. Numerically, the gravitational force has about 10 -42 times the strength of the electromagnetic force.
33 I might note that the proponents of another approach for merging general relativity and quantum mechanics, loop quantum gravity, to be briefly discussed in Chapter 16, take a viewpoint that is closer to the former conjecture—that spacetime has a discrete structure on the smallest of scales.
34 The relationship to mass arising from a Higgs ocean will be discussed later in the chapter.
35 Were you to count left, right, clockwise, and counterclockwise all separately, you'd conclude that the worm can move in four directions. But when we speak of "independent" directions, we always group those that lie along the same geometrical axis—like left and right, and also clockwise and counterclockwise.
36 Let me prepare you for one relevant development we will encounter in the next chapter. String theorists have known for decades that the equations they generally use to mathematically analyze string theory are approximate (the exact equations have proven difficult to identify and understand). However, most thought that the approximate equations were sufficiently accurate to determine the required number of extra dimensions. More recently (and to the shock of most physicists in the field), some string theorists showed that the approximate equations missed one dimension; it is now accepted that the theory needs seven extra dimensions. As we will see, this does not compromise the material discussed in this chapter, but shows that it fits within a larger, in fact more unified, framework. 20
37 The more precise name for these sticky entities is Dirichlet-p-branes, or D-p-branes for short. We will stick with the shorter p-brane.
38 There is even a proposal, from Lisa Randall, of Harvard, and Raman Sundrum, of Johns Hopkins, in which gravity too can be trapped, not by a sticky brane, but by extra dimensions that curve in just the right way, relaxing even further the constraints on their size.
39 One of these is the planned Laser Interferometer Space Antenna (LISA), a space-based version of LIGO comprising multiple spacecraft, separated by millions of kilometers, playing the role of LIGO's four-kilometer tubes. There are also other detectors that are playing a critical role in the search for gravitational waves, including the German-British detector GEO600, the French-Italian detector VIRGO, and the Japanese detector TAMA300.
40 Since teleportation starts with something here and seeks to make it appear at a distant location, in this section I will often speak as if particles have definite positions. To be more precise, I should always say, "starting with a particle that has a high likelihood of being located here" or "starting with a particle with a 99 percent chance of being located here," with similar language used where the particle is teleported, but for brevity's sake I will use the looser language.
41 For collections of particles—as opposed to individual particles—the quantum state also encodes the relationship of each particle in the collection to every other. So, by exactly reproducing the quantum state of the particles making up the DeLorean, we ensure that they all stand in the same relation to each other; the only change they experience is that their overall location would have been shifted from New York to London.
42 The fragility of the human body is another practical limitation: the acceleration required to reach such high speeds in a reasonable length of time is well beyond what the body can withstand. Note, too, that the slowing of time gives a strategy, in principle, for reaching distant locations in space. If a rocket were to leave earth and head for the Andromeda galaxy, traveling at 99.999999999999999999 percent of light speed, we'd have to wait nearly 6 million years for it to return. But at that speed, time on the rocket slows down relative to time on earth so dramatically that upon returning the astronaut would have aged only eight hours (setting aside the fact that he or she couldn't have survived the accelerations to get up to speed, turn back, and finally stop).
43 Of course, I really should say January 1, 1966, but let's not worry about that.
44 For details on geometrical duality involving both circles and Calabi-Yau shapes, see The Elegant Universe, Chapter 10.
45 If you're reluctant to rewrite Plato, the braneworld scenario gives a version of holography in which shadows are put back in their proper place. Imagine that we live on a three-brane that surrounds a region with four space dimensions (much as the two-dimensional skin of an apple surrounds the apple's three-dimensional interior). The holographic principle in this setting would say that our three-dimensional perceptions would be the shadows of four-dimensional physics taking place in the region surrounded by our brane.
Notes
Chapter 1
1. Lord Kelvin was quoted by the physicist Albert Michelson during his 1894 address at the dedication of the University of Chicago's Ryerson Laboratory (see D. Kleppner, Physics Today, November 1998).
2. Lord Kelvin, "Nineteenth Century Clouds over the Dynamical Theory of Heat and Light," Phil. Mag. II—6th series, 1 (1901).
3. A. Einstein, N. Rosen, and B. Podolsky, Phys. Rev. 47, 777 (1935).
4. Sir Arthur Eddington, The Nature of the Physical World (Cambridge, Eng.: Cambridge University Press, 1928).
5. As described more fully in note 2 of Chapter 6, this is an overstatement because there are examples, involving relatively esoteric particles (such as K-mesons and B-mesons), which show that the so-called weak nuclear force does not treat past and future fully symmetrically. However, in my view and that of many others who have thought about it, since these particles play essentially no role in determining the properties of everyday material objects, they are unlikely to be important in explaining the puzzle of time's arrow (although, I hasten to add, no one knows this for sure). Thus, while it is technically an overstatement, I will assume throughout that the error made in asserting that the laws treat past and future on equal footing is minimal—at least as far as explaining the puzzle of time's arrow is concerned.
6. Timothy Ferris, Coming of Age in the Milky Way (New York: Anchor, 1989).
Chapter 2
1. Isaac Newton, Sir Isaac Newton's Mathematical Principle of Natural Philosophy and His System of the World, trans. A. Motte and Florian Cajori (Berkeley: University of California Press, 1934), vol. 1, p. 1
0.
2. Ibid., p. 6.
3. Ibid.
4. Ibid., p. 12.
5. Albert Einstein, in Foreword to Max Jammer, Concepts of Space: The Histories of Theories of Space in Physics (New York: Dover, 1993).
6. A. Rupert Hall, Isaac Newton, Adventurer in Thought (Cambridge, Eng.: Cambridge University Press, 1992), p. 27.
7. Ibid.
8. H. G. Alexander, ed., The Leibniz-Clarke Correspondence (Manchester: Manchester University Press, 1956).
9. I am focusing on Leibniz as the representative of those who argued against assigning space an existence independent of the objects inhabiting it, but many others also strenuously defended this view, among them Christiaan Huygens and Bishop Berkeley.
10. See, for example, Max Jammer, p. 116.
11. V. I. Lenin, Materialism and Empiriocriticism: Critical Comments on a Reac tionaryPhilosophy (New York: International Publications, 1909). Second English ed. of Materializm' i Empiriokrititsizm': Kriticheskia Zametki ob' Odnoi Reaktsionnoi Filosofii (Moscow: Zveno Press, 1909).
Chapter 3
1. For the mathematically trained reader, these four equations are
denote the electric field, the magnetic field, the electric charge density, the electric current density, the permittivity of free space, and the permeability of free space, respectively. As you can see, Maxwell's equations relate the rate of change of the electromagnetic fields to the presence of electric charges and currents. It is not hard to show that these equations imply a speed for electromagnetic waves given by 1/sqrt 0 , which when evaluated is in fact the speed of light.
2. There is some controversy as to the role such experiments played in Einstein's development of special relativity. In his biography of Einstein, Subtle Is the Lord: The Science and the Life of Albert Einstein (Oxford: Oxford University Press, 1982), pp. 115-19, Abraham Pais has argued, using Einstein's own statements from his later years, that Einstein was aware of the Michelson-Morley results. Albrecht Fölsing in Albert Einstein: A Biography (New York: Viking, 1997), pp. 217-20, also argues that Einstein was aware of the Michelson-Morley result, as well as earlier experimental null results in searching for evidence of the aether, such as the work of Armand Fizeau. But Fölsing and many other historians of science have also argued that such experiments played, at best, a secondary role in Einstein's thinking. Einstein was primarily guided by considerations of mathematical symmetry, simplicity, and an uncanny physical intuition.