Tweet
Notes: Chapter 12
1. The circumstantial evidence I have in mind here relies on the fact that the strengths of all three nongravitational forces depend on the energy and temperature of the environment in which the forces act. At low energies and temperatures, such as those of our everyday environment, the strengths of all three forces are different. But there is indirect theoretical and experimental evidence that at very high temperatures, such as occurred in the earliest moments of the universe, the strengths of all three forces converge, indicating, albeit indirectly, that all three forces themselves may fundamentally be unified, and appear distinct only at low energies and temperatures. For a more detailed discussion see, for example, The Elegant Universe, Chapter 7. Return to Text
2. Once we know that a field, like any of the known force fields, is an ingredient in the makeup of the cosmos, then we know that it exists everywhere — it is stitched into the fabric of the cosmos. It is impossible to excise the field, much as it is impossible to excise space itself. The nearest we can come to eliminating a field's presence, therefore, is to have it take on a value that minimizes its energy. For force fields, like the electromagnetic force, that value is zero, as discussed in the text. For fields like the inflaton or the standard-model Higgs field (which, for simplicity, we do not consider here), that value can be some nonzero number that depends on the field's precise potential energy shape, as we discussed in Chapters 9 and 10. As mentioned in the text, to keep the discussion streamlined we are only explicitly discussing quantum fluctuations of fields whose lowest energy state is achieved when their value is zero, although fluctuations associated with Higgs or inflaton fields require no modification of our conclusions. Return to Text
3. Actually, the mathematically inclined reader should note that the uncertainty principle dictates that energy fluctuations are inversely proportional to the time resolution of our measurements, so the finer the time resolution with which we examine a field's energy, the more wildly the field will undulate. Return to Text
4. In this experiment, Lamoreaux verified the Casimir force in a modified setup involving the attraction between a spherical lens and a quartz plate. More recently, Gianni Carugno, Roberto Onofrio, and their collaborators at the University of Padova have undertaken the more difficult experiment involving the original Casimir framework of two parallel plates. (Keeping the plates perfectly parallel is quite an experimental challenge.) So far, they have confirmed Casimir's predictions to a level of 15 percent. Return to Text
5. In retrospect, these insights also show that if Einstein had not introduced the cosmological constant in 1917, quantum physicists would have introduced their own version a few decades later. As you will recall, the cosmological constant was an energy Einstein envisioned suffusing all of space, but whose origin he — and modern-day proponents of a cosmological constant — left unspecified. We now realize that quantum physics suffuses empty space with uttering fields, and as we directly see through Casimir's discovery, the resulting microscopic field frenzy fills space with energy. In fact, a major challenge facing theoretical physics is to show that the combined contribution of all field jitters yields a total energy in empty space — a total cosmological constant — that is within the observational limit currently determined by the supernova observations discussed in Chapter 10. So far, no one has been able to do this; carrying out the analysis exactly has proven to be beyond the capacity of current theoretical methods, and approximate calculations have gotten answers wildly larger than observations allow, strongly suggesting that the approximations are way off. Many view explaining the value of the cosmological constant (whether it is zero, as long thought, or small and nonzero as suggested by the inflation and the supernova data) as one of the most important open problems in theoretical physics. Return to Text
6. In this section, I describe one way of seeing the conflict between general relativity and quantum mechanics. But I should note, in keeping with our theme of seeking the true nature of space and time, that other, somewhat less tangible but potentially important puzzles arise in attempting to merge general relativity and quantum mechanics. One that's particularly tantalizing arises when the straightforward application of the procedure for transforming classical nongravitational theories (like Maxwell's electrodynamics) into a quantum theory is extended to classical general relativity (as shown by Bryce DeWitt in what is now called the Wheeler-DeWitt equation). In the central equation that emerges, it turns out that the time variable does not appear. So, rather than having an explicit mathematical embodiment of time — as is the case in every other fundamental theory — in this approach to quantizing gravity, temporal evolution must be kept track of by a physical feature of the universe (such as its density) that we expect to change in a regular manner. As yet, no one knows if this procedure for quantizing gravity is appropriate (although much progress in an offshoot of this formalism, called loop quantum gravity, has been recently achieved; see Chapter 16), so it is not clear whether the absence of an explicit time variable is hinting at something deep (time as an emergent concept?) or not. In this chapter we focus on a different approach for merging general relativity and quantum mechanics, superstring theory.Return to Text
7. It is somewhat of a misnomer to speak of the "center" of a black hole as if it were a place in space. The reason, roughly speaking, is that when one crosses a black hole's event horizon — its outer edge — the roles of space and time are interchanged. In fact, just as you can't resist going from one second to the next in time, so you can't resist being pulled to the black hole's "center" once you've crossed the event horizon. It turns out that this analogy between heading forward in time and heading toward a black hole's center is strongly motivated by the mathematical description of black holes. Thus, rather than thinking of the black hole's center as a location in space, it is better to think of it as a location in time. Furthermore, since you can't go beyond the black hole's center, you might be tempted to think of it as a location in spacetime where time comes to an end. This may well be true. But since the standard general relativity equations break down under such extremes of huge mass density, our ability to make.definite statements of this sort is compromised. Clearly, this suggests that if we had equations that don't break down deep inside a black hole, we might gain important insights into the nature of time. That is one of the goals of superstring theory. Return to Text
8. As in earlier chapters, by "observable universe" I mean that part of the universe with which we could have had, at least in principle, communication during the time since the bang. In a universe that is infinite in spatial extent, as discussed in Chapter 8, all of space does not shrink to a point at the moment of the bang. Certainly, everything in the observable part of the universe will be squeezed into an ever smaller space as we head back to the beginning, but, although hard to picture, there are things — infinitely far away — that will forever remain separate from us, even as the density of matter and energy grows ever higher. Return to Text
9. Leonard Susskind, in "The Elegant Universe," NOVA, three-hour PBS series first higher. aired October 28 and November 4, 2003. Return to Text
10. Indeed, the difficulty of designing experimental tests for superstring theory has been a crucial stumbling block, one that has substantially hindered the theory's acceptance. However, as we will see in later chapters, there has been much progress in this direction; string theorists have high hopes that upcoming accelerator and space-based experiments will provide at least circumstantial evidence in support of the theory, and with luck, maybe even more. Return to Text
11. Although I haven't covered it explicitly in the text, note that every known particle has an antiparticle — a particle with the same mass but opposite force charges (like the opposite sign of electric charge). The electron's antiparticle is the positron; the up-quark's antiparticle is, not surprisingly, the anti-up-quark; and so on. Return to Text
12. As we will see in Chapter 13, recent work in string theory has suggested that strings may be much larger than the Planck length, and this has a number of potentially critical implications — including the possibility of making the theory experimentally testable. Return to Text
13. The existence of atoms was initially argued through indirect means (as an explanation of the particular ratios in which various chemical substances would combine, and later, through Brownian motion); the existence of the first black holes was confirmed (to many physicists' satisfaction) by seeing their effect on gas that falls toward them from nearby stars, instead of "seeing" them directly. Return to Text
14. Since even a placidly vibrating string has some amount of energy, you might wonder how it's possible for a string vibrational pattern to yield a massless particle. The answer, once again, has to do with quantum uncertainty. No matter how placid a string is, quantum uncertainty implies that it has a minimal amount of jitter and jiggle. And, through the weirdness of quantum mechanics, these uncertainty-induced jitters have negative energy. When this is combined with the positive energy from the most gentle of ordinary string vibrations, the total mass/energy is zero. Return to Text
15. For the mathematically inclined reader, the more precise statement is that the square of the masses of string vibrational modes are given by integer multiples of the square of the Planck mass. Even more precisely (and of relevance to recent developments covered in Chapter 13), the square of these masses are integer multiples of the string scale (which is proportional to the inverse square of the string length). In conventional formulations of string theory, the string scale and the Planck mass are close, which is why I've simplified the main text and only introduced the Planck mass. However, in Chapter 13 we will consider situations in which the string scale can be different from the Planck mass. Return to Text
16. It's not too hard to understand, in rough terms, how the Planck length crept into Klein's analysis. General relativity and quantum mechanics invoke three fundamental constants of nature: c (the velocity of light), G (the basic strength of the gravitational force) and h (Planck's constant describing the size of quantum effects). These three constants can be combined to produce a quantity with units of length: (hG/c^3)^1/2, which, by definition, is the Planck length. After substituting the numerical values of the three constants, one finds the Planck length to be about 1.616 x 10^-33 centimeters. Thus, unless a dimensionless number with value differing substantially from 1 should emerge from the the theory — something that doesn't often happen in a simple, well-formulated physical theory — we expect the Planck length to be the characteristic size of lengths, such as the length of the curled-up spatial dimension. Nevertheless, do note that this does not rule out the possibility that dimensions can be larger than the Planck length, and in Chapter 13 we will see interesting recent work that has investigated this possibility vigorously. Return to Text
17. Incorporating a particle with the electron's charge, and with its relatively tiny mass, proved a formidable challenge. Return to Text
18. Note that the uniform symmetry requirement that we used in Chapter 8 to narrow down the shape of the universe was motivated by astronomical observations (such as those of the microwave background radiation) within the three large dimensions. These symmetry constraints have no bearing on the shape of the possible six tiny extra space dimensions. Figure 12.9a is based on an image created by Andrew Hanson. Return to Text
19. You might wonder about whether there might not only be extra space dimensions, but also extra time dimensions. Researchers (such as Itzhak Bars at the University of Southern California) have investigated this possibility, and shown that it is at least possible to formulate theories with a second time dimension that seem to be physically reasonable. But whether this second time dimension is really on a par with the ordinary time dimension or is just a mathematical device has never been settled fully; the general feeling is more toward the latter than the former. By contrast, the most straightforward reading of string theory says that the extra space dimensions are every bit as real as the three we know about. Return to Text
20. String theory experts (and those who have read The Elegant Universe, Chapter 12) will recognize that the more precise statement is that certain formulations of string theory (discussed in Chapter.13 of this book) admit limits involving eleven spacetime dimensions. There is still debate as to whether string theory is best thought of as fundamentally being an eleven spacetime dimensional theory, or whether the eleven dimensional formulation should be viewed as a particular limit (e.g., when the string coupling constant is taken large in the Type IIA formulation), on a par with other limits. As this distinction does not have much impact on our general-level discussion, I have chosen the former viewpoint, largely for the linguistic ease of having a fixed and uniform total number of
dimensions. Return to Text
2. Once we know that a field, like any of the known force fields, is an ingredient in the makeup of the cosmos, then we know that it exists everywhere — it is stitched into the fabric of the cosmos. It is impossible to excise the field, much as it is impossible to excise space itself. The nearest we can come to eliminating a field's presence, therefore, is to have it take on a value that minimizes its energy. For force fields, like the electromagnetic force, that value is zero, as discussed in the text. For fields like the inflaton or the standard-model Higgs field (which, for simplicity, we do not consider here), that value can be some nonzero number that depends on the field's precise potential energy shape, as we discussed in Chapters 9 and 10. As mentioned in the text, to keep the discussion streamlined we are only explicitly discussing quantum fluctuations of fields whose lowest energy state is achieved when their value is zero, although fluctuations associated with Higgs or inflaton fields require no modification of our conclusions. Return to Text
3. Actually, the mathematically inclined reader should note that the uncertainty principle dictates that energy fluctuations are inversely proportional to the time resolution of our measurements, so the finer the time resolution with which we examine a field's energy, the more wildly the field will undulate. Return to Text
4. In this experiment, Lamoreaux verified the Casimir force in a modified setup involving the attraction between a spherical lens and a quartz plate. More recently, Gianni Carugno, Roberto Onofrio, and their collaborators at the University of Padova have undertaken the more difficult experiment involving the original Casimir framework of two parallel plates. (Keeping the plates perfectly parallel is quite an experimental challenge.) So far, they have confirmed Casimir's predictions to a level of 15 percent. Return to Text
5. In retrospect, these insights also show that if Einstein had not introduced the cosmological constant in 1917, quantum physicists would have introduced their own version a few decades later. As you will recall, the cosmological constant was an energy Einstein envisioned suffusing all of space, but whose origin he — and modern-day proponents of a cosmological constant — left unspecified. We now realize that quantum physics suffuses empty space with uttering fields, and as we directly see through Casimir's discovery, the resulting microscopic field frenzy fills space with energy. In fact, a major challenge facing theoretical physics is to show that the combined contribution of all field jitters yields a total energy in empty space — a total cosmological constant — that is within the observational limit currently determined by the supernova observations discussed in Chapter 10. So far, no one has been able to do this; carrying out the analysis exactly has proven to be beyond the capacity of current theoretical methods, and approximate calculations have gotten answers wildly larger than observations allow, strongly suggesting that the approximations are way off. Many view explaining the value of the cosmological constant (whether it is zero, as long thought, or small and nonzero as suggested by the inflation and the supernova data) as one of the most important open problems in theoretical physics. Return to Text
6. In this section, I describe one way of seeing the conflict between general relativity and quantum mechanics. But I should note, in keeping with our theme of seeking the true nature of space and time, that other, somewhat less tangible but potentially important puzzles arise in attempting to merge general relativity and quantum mechanics. One that's particularly tantalizing arises when the straightforward application of the procedure for transforming classical nongravitational theories (like Maxwell's electrodynamics) into a quantum theory is extended to classical general relativity (as shown by Bryce DeWitt in what is now called the Wheeler-DeWitt equation). In the central equation that emerges, it turns out that the time variable does not appear. So, rather than having an explicit mathematical embodiment of time — as is the case in every other fundamental theory — in this approach to quantizing gravity, temporal evolution must be kept track of by a physical feature of the universe (such as its density) that we expect to change in a regular manner. As yet, no one knows if this procedure for quantizing gravity is appropriate (although much progress in an offshoot of this formalism, called loop quantum gravity, has been recently achieved; see Chapter 16), so it is not clear whether the absence of an explicit time variable is hinting at something deep (time as an emergent concept?) or not. In this chapter we focus on a different approach for merging general relativity and quantum mechanics, superstring theory.Return to Text
7. It is somewhat of a misnomer to speak of the "center" of a black hole as if it were a place in space. The reason, roughly speaking, is that when one crosses a black hole's event horizon — its outer edge — the roles of space and time are interchanged. In fact, just as you can't resist going from one second to the next in time, so you can't resist being pulled to the black hole's "center" once you've crossed the event horizon. It turns out that this analogy between heading forward in time and heading toward a black hole's center is strongly motivated by the mathematical description of black holes. Thus, rather than thinking of the black hole's center as a location in space, it is better to think of it as a location in time. Furthermore, since you can't go beyond the black hole's center, you might be tempted to think of it as a location in spacetime where time comes to an end. This may well be true. But since the standard general relativity equations break down under such extremes of huge mass density, our ability to make.definite statements of this sort is compromised. Clearly, this suggests that if we had equations that don't break down deep inside a black hole, we might gain important insights into the nature of time. That is one of the goals of superstring theory. Return to Text
8. As in earlier chapters, by "observable universe" I mean that part of the universe with which we could have had, at least in principle, communication during the time since the bang. In a universe that is infinite in spatial extent, as discussed in Chapter 8, all of space does not shrink to a point at the moment of the bang. Certainly, everything in the observable part of the universe will be squeezed into an ever smaller space as we head back to the beginning, but, although hard to picture, there are things — infinitely far away — that will forever remain separate from us, even as the density of matter and energy grows ever higher. Return to Text
9. Leonard Susskind, in "The Elegant Universe," NOVA, three-hour PBS series first higher. aired October 28 and November 4, 2003. Return to Text
10. Indeed, the difficulty of designing experimental tests for superstring theory has been a crucial stumbling block, one that has substantially hindered the theory's acceptance. However, as we will see in later chapters, there has been much progress in this direction; string theorists have high hopes that upcoming accelerator and space-based experiments will provide at least circumstantial evidence in support of the theory, and with luck, maybe even more. Return to Text
11. Although I haven't covered it explicitly in the text, note that every known particle has an antiparticle — a particle with the same mass but opposite force charges (like the opposite sign of electric charge). The electron's antiparticle is the positron; the up-quark's antiparticle is, not surprisingly, the anti-up-quark; and so on. Return to Text
12. As we will see in Chapter 13, recent work in string theory has suggested that strings may be much larger than the Planck length, and this has a number of potentially critical implications — including the possibility of making the theory experimentally testable. Return to Text
13. The existence of atoms was initially argued through indirect means (as an explanation of the particular ratios in which various chemical substances would combine, and later, through Brownian motion); the existence of the first black holes was confirmed (to many physicists' satisfaction) by seeing their effect on gas that falls toward them from nearby stars, instead of "seeing" them directly. Return to Text
14. Since even a placidly vibrating string has some amount of energy, you might wonder how it's possible for a string vibrational pattern to yield a massless particle. The answer, once again, has to do with quantum uncertainty. No matter how placid a string is, quantum uncertainty implies that it has a minimal amount of jitter and jiggle. And, through the weirdness of quantum mechanics, these uncertainty-induced jitters have negative energy. When this is combined with the positive energy from the most gentle of ordinary string vibrations, the total mass/energy is zero. Return to Text
15. For the mathematically inclined reader, the more precise statement is that the square of the masses of string vibrational modes are given by integer multiples of the square of the Planck mass. Even more precisely (and of relevance to recent developments covered in Chapter 13), the square of these masses are integer multiples of the string scale (which is proportional to the inverse square of the string length). In conventional formulations of string theory, the string scale and the Planck mass are close, which is why I've simplified the main text and only introduced the Planck mass. However, in Chapter 13 we will consider situations in which the string scale can be different from the Planck mass. Return to Text
16. It's not too hard to understand, in rough terms, how the Planck length crept into Klein's analysis. General relativity and quantum mechanics invoke three fundamental constants of nature: c (the velocity of light), G (the basic strength of the gravitational force) and h (Planck's constant describing the size of quantum effects). These three constants can be combined to produce a quantity with units of length: (hG/c^3)^1/2, which, by definition, is the Planck length. After substituting the numerical values of the three constants, one finds the Planck length to be about 1.616 x 10^-33 centimeters. Thus, unless a dimensionless number with value differing substantially from 1 should emerge from the the theory — something that doesn't often happen in a simple, well-formulated physical theory — we expect the Planck length to be the characteristic size of lengths, such as the length of the curled-up spatial dimension. Nevertheless, do note that this does not rule out the possibility that dimensions can be larger than the Planck length, and in Chapter 13 we will see interesting recent work that has investigated this possibility vigorously. Return to Text
17. Incorporating a particle with the electron's charge, and with its relatively tiny mass, proved a formidable challenge. Return to Text
18. Note that the uniform symmetry requirement that we used in Chapter 8 to narrow down the shape of the universe was motivated by astronomical observations (such as those of the microwave background radiation) within the three large dimensions. These symmetry constraints have no bearing on the shape of the possible six tiny extra space dimensions. Figure 12.9a is based on an image created by Andrew Hanson. Return to Text
19. You might wonder about whether there might not only be extra space dimensions, but also extra time dimensions. Researchers (such as Itzhak Bars at the University of Southern California) have investigated this possibility, and shown that it is at least possible to formulate theories with a second time dimension that seem to be physically reasonable. But whether this second time dimension is really on a par with the ordinary time dimension or is just a mathematical device has never been settled fully; the general feeling is more toward the latter than the former. By contrast, the most straightforward reading of string theory says that the extra space dimensions are every bit as real as the three we know about. Return to Text
20. String theory experts (and those who have read The Elegant Universe, Chapter 12) will recognize that the more precise statement is that certain formulations of string theory (discussed in Chapter.13 of this book) admit limits involving eleven spacetime dimensions. There is still debate as to whether string theory is best thought of as fundamentally being an eleven spacetime dimensional theory, or whether the eleven dimensional formulation should be viewed as a particular limit (e.g., when the string coupling constant is taken large in the Type IIA formulation), on a par with other limits. As this distinction does not have much impact on our general-level discussion, I have chosen the former viewpoint, largely for the linguistic ease of having a fixed and uniform total number of
dimensions. Return to Text