The Fabric of the Cosmos: Space, Time, and the Texture of Reality - Brian Greene (2004)

Endnotes

1 The terms centrifugal and centripetal force are sometimes used when describing spinning motion. But they are merely labels. Our intent is to understand why spinning motion gives rise to force.

2 There is debate concerning Mach’s precise views on the material that follows. Some of his writings are a bit ambiguous and some of the ideas attributed to him arose from subsequent interpretations of his work. Since he seems to have been aware of these interpretations and never offered corrections, some have suggested that he agreed with their conclusions. But historical accuracy might be better served if every time I write “Mach argued” or “Mach’s ideas,” you read it to mean “the prevailing interpretation of an approach initiated by Mach.”

3 While I like human examples because they make an immediate connection between the physics we’re discussing and innate sensations, a drawback is our ability to move, volitionally, one part of our body relative to another—in effect, to use one part of our body as the benchmark for another part’s motion (like someone who spins one of his arms relative to his head). I emphasize uniform spinning motion—spinning motion in which every part of the body spins together—to avoid such irrelevant complications. So, when I talk about your body’s spinning, imagine that, like Newton’s two rocks tied by a rope or a skater in the final moments of an Olympic routine, every part of your body spins at the same rate as every other.

4 Like the pages in any flip book, the pages in Figure 3.3 only show representative moments of time. This may suggest to you the interesting question of whether time is discrete or infinitely divisible. We’ll come back to that question later, but for now imagine that time is infinitely divisible, so our flip book really should have an infinite number of pages interpolating between those shown.

5 It’s easier to picture warped space, but because of their intimate connection, time is also warped by matter and energy. And just as a warp in space means that space is stretched or compressed, as in Figure 3.10, a warp in time means that time is stretched or compressed. That is, clocks experiencing different gravitational pulls—like one on the sun and another in deep, empty space—tick off time at different rates. In fact, it turns out that the warping of space caused by ordinary bodies like the earth and sun (as opposed to black holes) is far less pronounced than the warping they inflict on time.15

6 In special relativity—the special case of general relativity in which the gravitational field is zero—this idea applies unchanged: a zero gravitational field is still a field, one that can be measured and changed, and hence provides a something relative to which acceleration can be defined.

7 To avoid linguistic complications, I’m describing the electron spins as perfectly correlated, even though the more conventional description is one in which they’re perfectly anticorrelated: whatever result one detector finds, the other will find the opposite. To compare with the conventional description, imagine that I’ve interchanged all the clockwise and counterclockwise labels on one of the detectors.

8 Many researchers, including me, believe that Bell’s argument and Aspect’s experiment establish convincingly that the observed correlations between widely separated particles cannot be explained by Scully-type reasoning—reasoning that attributes the correlations to nothing more surprising than the particles’ having acquired definite, correlated properties when they were (previously) together. Others have sought to evade or lessen the stunning nonlocality conclusion to which this has led us. I don’t share their skepticism, but some works for general readers that discuss some of these alternatives are cited in the note section.15

9 Pick any point in the loaf. Draw a slice that includes the point, and which intersects our current now-slice at an angle that is less than 45 degrees. This slice will represent the now-slice—reality—of a distant observer who was initially at rest relative to us, like Chewie, but is now moving relative to us at less than the speed of light. By design, this slice includes the (arbitrary) point in the loaf you happened to pick.4

10 There is an exception to this statement having to do with a certain class of exotic particles. As far as the questions discussed in this chapter are concerned, I consider this likely to be of little relevance and so won’t mention this qualification further. If you are interested, it is briefly discussed in note 2.

11 Note that time-reversal symmetry is not about time itself being reversed or “running” backward. Instead, as we’ve described, time-reversal symmetry is concerned with whether events that happen in time, in one particular temporal order, can also happen in the reverse order. A more appropriate phrase might be event reversal or process reversal or event order reversal, but we’ll stick with the conventional term.

12 Entropy is another example in which terminology complicates ideas. Don’t worry if you have to remind yourself repeatedly that low entropy means high order and that high entropy means low order (equivalently, high disorder). I often have to.

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 1024 H2O 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 this page: 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 gravitational 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 1067 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.