Notes

Chapter 1

1. The table below is an elaboration of Table 1.1. It records the masses and force charges of the particles of all three families. Each type of quark can carry three possible strong-force charges that are, somewhat fancifully, labeled as colors—they stand for numerical strong-force charges values. The weak charges recorded are, more precisely, the "third-component" of weak isospin. (We have not listed the "right-handed" components of the particles—they differ by having no weak charge.)

Family 1
Particle | Mass | Electric charge | Weak charge | Strong charge
Electron | .0054 | -1 | -1/2 | 0
Electron-Neutrino | < 10(-8) | 0 | 1/2 | 0
Up Quark | .0047 | 2/3 | 1/2 | red, green, blue
Down Quark | .0074 | -1/3 | -1/2r | red, green, blue
Family 2
Particle | Mass | Electric charge | Weak charge | Strong charge
Muon | .11 | -1 | -1/2 | 0
Muon-Neutrino | < .0003 | 0 | 1/2 | 0
Charm Quark | 1.6 | 2/3 | 1/2 | red, green, blue
Strange Quark | .16 | -1/3 | -1/2 | red, green, blue
Family 3
Particle | Mass | Electric charge | Weak charge | Strong charge
Tau | 1.9 | -1 | -1/2 | 0
Tau-Neutrino | < .033 | 0 | 1/2 | 0
Top Quark | 189 | 2/3 | 1/2 | red, green, blue
Bottom Quark | 5.2 | -1/3 | -1/2 | red, green, blue

2. Strings can also have two freely moving ends (so-called open strings) in addition to the loops (closed strings) illustrated in Figure 1.1. To ease our presentation, for the most part we will focus on closed strings, although essentially all of what we say applies to both.

3. Albert Einstein, in a 1942 letter to a friend, as quoted in Tony Hey and Patrick Walters, Einstein's Mirror (Cambridge, Eng.: Cambridge University Press, 1997).

4. Steven Weinberg, Dreams of a Final Theory (New York: Pantheon, 1992), p.52.

5. Interview with Edward Witten, May 11, 1998.

Chapter 2

1. The presence of massive bodies like the earth does complicate matters by introducing gravitational forces. Since we are now focusing on motion in the horizontal direction—not the vertical direction—we can and will ignore the earth's presence. In the next chapter we will undertake a thorough discussion of gravity.

2. For the mathematically inclined reader, we note that these observations can be turned into quantitative statements. For instance, if the moving light clock has speed v and it takes t seconds for its photon to complete one round-trip journey (as measured by our stationary light clock), then the light clock will have traveled a distance vt when its photon has returned to the lower mirror. We can now use the Pythagorean theorem to calculate that the length of each of the diagonal paths in Figure 2.3 is √((vt/2)2 + h2), where h is the distance between the two mirrors of a light clock (taken to be six inches in the text). The two diagonal paths, taken together, therefore have length 2√((vt/2)2 + h2). Since the speed of light has a constant value, conventionally called c, it takes light 2√(vt/2)2 + h2/c seconds to complete the double diagonal journey. And so, we have the equality t = 2√((vt/2)2 + h2)/c, which can be solved for t, yielding t = 2h/√(c2 - v2). To avoid confusion, let's write this as tmoving = 2h/(√c2 - v2), where the subscript indicates that this is the time we measure for one tick to occur on the moving clock. On the other hand, the time for one tick on our stationary clock is tstationary = 2h/c and as a little algebra reveals, tmoving = tstationary / √(1 - v2/c2), directly showing that one tick on the moving clock takes longer than one tick on the stationary clock. This means that between chosen events, fewer total ticks will take place on the moving clock than on the stationary, ensuring that less time has elapsed for the observer in motion.

3. In case you would be more convinced by an experiment carried out in a less esoteric setting than a particle accelerator, consider the following. During October 1971, J. C. Hafele, then of Washington University in St. Louis, and Richard Keating of the United States Naval Observatory flew cesium-beam atomic clocks on commercial airliners for some 40 hours. After taking into account a number of subtle features having to do with gravitational effects (to be discussed in the next chapter), special relativity claims that the total elapsed time on the moving atomic clocks should be less than the elapsed time on stationary earthbound counterparts by a few hundred billionths of a second. This is just what Hafele and Keating found: Time really does slow down for a clock in motion.

4. Although Figure 2.4 correctly illustrates the shrinking of an object along its direction of motion, the image does not illustrate what we would actually see if an object were somehow to blaze by at nearly light speed (assuming our eyesight or photographic equipment were sharp enough to see anything at all!). To see something, our eyes—or our camera—must receive light that has reflected off the object's surface. But since the reflected light travels to us from various locations on the object, the light we see at any moment traveled to us along paths of different lengths. This results in a kind of relativistic visual illusion in which the object will appear both foreshortened and rotated.

5. For the mathematically inclined reader, we note that from the spacetime position 4-vector x = (ct, x1, x2, x3) = (ct, x→) we can produce the velocity 4-vector u = dx/dτ, where τ is the proper time defined by dτ2 = dt2 - c-2(dx12 + dx22 + dx32). Then, the "speed through spacetime" is the magnitude of the 4-vector u, √(((c2dt2 - dx→2) / (dt2 - c-2dx→2))), which is identically the speed of light, c. Now, we can rearrange the equation c2(dt/dτ)2 - (dx→/dτ)2 = c2, to be c2(dτ/dt)2 + (dx→/dt)2 = c2. This shows that an increase in an object's speed through space, √((dx→/dt)2) must be accompanied by a decrease in dτ/dt, the latter being the object's speed through time (the rate at which time elapses on its own clock, dτ, as compared with that on our stationary clock, dt).

Chapter 3

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, 1962), Vol. I, p. 634.

2. A bit more precisely, Einstein realized that the equivalence principle holds so long as your observations are confined to a small enough region of space—that is, so long as your "compartment" is small enough. The reason is the following. Gravitational fields can vary in strength (and in direction) from place to place. But we are imagining that your whole compartment accelerates as a single unit and therefore your acceleration simulates a single, uniform gravitational force field. As your compartment gets ever smaller, though, there is ever less room over which a gravitational field can vary, and hence the equivalence principle becomes ever more applicable. Technically, the difference between the uniform gravitational field simulated by an accelerated vantage point and a possibly nonuniform "real" gravitational field created by some collection of massive bodies is known as the "tidal" gravitational field (since it accounts for the moon's gravitational effect on tides on earth). This endnote, therefore, can be summarized by saying that tidal gravitational fields become less noticeable as the size of your compartment gets smaller, making accelerated motion and a "real" gravitational field indistinguishable.

3. Albert Einstein, as quoted in Albrecht Fölsing, Albert Einstein (New York: Viking, 1997), p. 315.

4. John Stachel, "Einstein and the Rigidly Rotating Disk," in General Relativity and Gravitation, ed. A. Held (New York: Plenum, 1980), p. 1.

5. Analysis of the Tornado ride, or the "rigidly rotating disk," as it is called in more technical language, easily leads to confusion. In fact, to this day there is not universal agreement on a number of subtle aspects of this example. In the text we have followed the spirit of Einstein's own analysis, and in this endnote we continue to take this viewpoint and seek to clarify a couple of features that you may have found confusing. First, you may be puzzled about why the circumference of the ride is not Lorentz contracted in exactly the same way as the ruler, and hence measured by Slim to have the same length as we originally found. Bear in mind, though, that throughout our discussion the ride was always spinning; we never analyzed the ride when it was at rest. Thus, from our perspective as stationary observers, the only difference between our and Slim's measurement of the ride's circumference is that Slim's ruler is Lorentz contracted; the spinning Tornado ride was spinning when we performed our measurement, and it is spinning as we watch Slim carry out his. Since we see that his ruler is contracted, we realize that he will have to lay it out more times to traverse the entire circumference, thereby measuring a longer length than we did. Lorentz contraction of the ride's circumference would have been relevant only if we compared the properties of the ride when spinning and when at rest, but this is a comparison we did not need.

Second, notwithstanding the fact that we did not need to analyze the ride when it was at rest, you may still be wondering about what would happen when it does slow down and stop. Now, it would seem, we must take account of the changing circumference with changing speed due to different degrees of Lorentz contraction. But how can this be squared with an unchanging radius? This is a subtle problem whose resolution hinges on the fact that there are no fully rigid objects in the real world. Objects can stretch and bend and thereby accommodate the stretching or contracting we have come upon; if not, as Einstein pointed out, a rotating disk that was initially formed by allowing a spinning cast of molten metal to cool while in motion would break apart if its rate of spinning were subsequently changed. For more details on the history of the rigidly rotating disk, see Stachel, "Einstein and the Rigidly Rotating Disk."

6. The expert reader will recognize that in the example of the Tornado ride, that is, in the case of a uniformly rotating frame of reference, the curved three-dimensional spatial sections on which we have focused fit together into a four-dimensional spacetime whose curvature still vanishes.

7. Hermann Minkowski, as quoted in Fölsing, Albert Einstein, p. 189.

8. Interview with John Wheeler, January 27, 1998.

9. Even so, existing atomic clocks are sufficiently accurate to detect such tiny—and even tinier—time warps. For instance, in 1976 Robert Vessot and Martin Levine of the Harvard-Smithsonian Astrophysical Observatory, together with collaboraters at the National Aeronautics and Space Administration (NASA), launched a Scout D rocket from Wallops Island, Virginia, that carried an atomic clock accurate to about a trillionth of a second per hour. They hoped to show that as the rocket gained altitude (thereby decreasing the effect of the earth's gravitational pull), an identical earthbound atomic clock (still subject to the full force of the earth's gravity) would tick more slowly Through a two-way stream of microwave signals, the researchers were able to compare the rate of ticking of the two atomic clocks and, indeed, at the rocket's maximum altitude of 6,000 miles, its atomic clock ran fast by about 4 parts per billion relative to its counterpart on earth, agreeing with theoretical predictions to better than a hundredth of a percent.

10. In the mid-1800s, the French scientist Urbain Jean Joseph Le Verrier discovered that the planet Mercury deviates slightly from the orbit around the sun that is predicted by Newton's law of gravity. For more than half a century explanations for this so-called excess orbital perihelion precession (in plain language, at the end of each orbit, Mercury does not quite wind up where Newton's theory says it should) ran the gamut—the gravitational influence of an undiscovered planet or planetary ring, an undiscovered moon, the effect of interplanetary dust, the oblateness of the sun—but none was sufficiently compelling to win general acceptance. In 1915, Einstein calculated the perihelion precession of Mercury using his newfound equations of general relativity and found an answer that, by his own admission, gave him heart palpitations: The result from general relativity precisely matched observations. This success, certainly, was one significant reason that Einstein had such faith in his theory, but most everyone else awaited confirmation of a prediction, rather than an explanation of a previously known anomaly. For more details, see Abraham Pais, Subtle Is the Lord (New York: Oxford University Press, 1982), p. 253.

11. Robert P. Crease and Charles C. Mann, The Second Creation (New Brunswick, N.J.: Rutgers University Press, 1996), p. 39.

12. Surprisingly, recent research on the detailed rate of cosmic expansion suggests that the universe may in fact incorporate a very small but nonzero cosmological constant.

Chapter 4

1. Richard Feynman, The Character of Physical Law (Cambridge, Mass.: MIT Press, 1965), p. 129.

2. Although Planck's work did solve the infinite energy puzzle, apparently this goal was not what directly motivated his work. Rather, Planck was seeking to understand a closely related issue: the experimental results concerning how energy in a hot oven—a "black body" to be more precise—is distributed over various wavelength ranges. For more details on the history of these developments, the interested reader should consult Thomas S. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912 (Oxford, Eng.: Clarendon, 1978).

3. A little more precisely, Planck showed that waves whose minimum energy content exceeds their purported average energy contribution (according to nineteenthcentury thermodynamics) are exponentially suppressed. This suppression is increasingly sharp as we examine waves of ever larger frequency.

4. Planck's constant is 1.05 x 10-27 grams-centimeters2/second.

5. Timothy Ferris, Coming of Age in the Milky Way (New York: Anchor, 1989), p. 286.

6. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and String Theory, June 21, 1997.

7. It is worthwhile to note that Feynman's approach to quantum mechanics can be used to derive the approach based on wave functions, and vice versa; the two approaches, therefore, are fully equivalent. Nevertheless, the concepts, the language, and the interpretation that each approach emphasizes are rather different, even though the answers each gives are absolutely identical.

8. Richard Feynman, QED: The Strange Theory of Light and Matter (Princeton: Princeton University Press, 1988).

Chapter 5

1. Stephen Hawking, A Brief History of Time (New York: Bantam Books, 1988), p. 175.

2. Richard Feynman, as quoted in Timothy Ferris, The Whole Shebang (New York: Simon & Schuster, 1997), p. 97.

3. In case you are still perplexed about how anything at all can happen within a region of space that is empty, it is important to realize that the uncertainty principle places a limit on how "empty" a region of space can actually be; it modifies what we mean by empty space. For example, when applied to wave disturbances in a field (such as electromagnetic waves traveling in the electromagnetic field) the uncertainty principle shows that the amplitude of a wave and the speed with which its amplitude changes are subject to the same inverse relationship as are the position and speed of a particle: The more precisely the amplitude is specified the less we can possibly know about the speed with which its amplitude changes. Now, when we say that a region of space is empty, we typically mean that, among other things, there are no waves passing through it, and that all fields have value zero. In clumsy but ultimately useful language, we can rephrase this by saying that the amplitudes of all waves that pass through the region are zero, exactly. But if we know the amplitudes exactly, the uncertainty principle implies that the rate of change of the amplitudes is completely uncertain and can take on essentially any value. But if the amplitudes change, this means that in the next moment they will no longer be zero, even though the region of space is still "empty." Again, on average the field will be zero since at some places its value will be positive while at others negative; on average the net energy in the region has not changed. But this is only on average. Quantum uncertainty implies that the energy in the field—even in an empty region of space—fluctuates up and down, with the size of the fluctuations getting larger as the distance and time scales on which the region is examined get smaller. The energy embodied in such momentary field fluctuations can then, through E = mc2, be converted into the momentary creation of pairs of particles and their antiparticles, which annihilate each other in great haste, to keep the energy from changing, on average.

4. Even though the initial equation that Schrödinger wrote down—the one incorporating special relativity—did not accurately describe the quantum-mechanical properties of electrons in hydrogen atoms, it was soon realized to be a valuable equation when appropriately used in other contexts, and, in fact, is still in use today. However, by the time Schrödinger published his equation he had been scooped by Oskar Klein and Walter Gordon, and hence his relativistic equation is called the "Klein-Gordon equation."

5. For the mathematically inclined reader, we note that the symmetry principles used in elementary particle physics are generally based on groups, most notably, Lie groups. Elementary particles are arranged in representations of various groups and the equations governing their time evolution are required to respect the associated symmetry transformations. For the strong force, this symmetry is called SU(3) (the analog of ordinary three-dimensional rotations, but acting on a complex space), and the three colors of a given quark species transform in a three-dimensional representation. The shifting (from red, green, blue to yellow, indigo, violet) mentioned in the text is, more precisely, an SU(3) transformation acting on the "color coordinates" of a quark. A gauge symmetry is one in which the group transformations can have a spacetime dependence: in this case, "rotating" the quark colors differently at different locations in space and moments in time.

6. During the development of the quantum theories of the three nongravitational forces, physicists also came upon calculations that gave infinite results. In time, though, they gradually realized that these infinities could be done away with through a tool known as renormalization. The infinities arising in attempts to merge general relativity and quantum mechanics are far more severe and are not amenable to the renormalization cure. Even more recently, physicists have realized that infinite answers are a signal that a theory is being used to analyze a realm that is beyond the bounds of its applicability. Since the goal of current research is to find a theory whose range of applicability is, in principle, unbounded-the "ultimate" or "final" theory—physicists want to find a theory in which infinite answers do not crop up, regardless of how extreme the physical system being analyzed might be.

7. The size of the Planck length can be understood based upon simple reasoning rooted in what physicists call dimensional analysis. The idea is this. When a theory is formulated as a collection of equations, the abstract symbols must be tied to physical features of the world if the theory is to make contact with reality. In particular, we must introduce a system of units so that if a symbol, say, is meant to refer to a length, we have a scale by which its value can be interpreted. After all, if equations show that the length in question is 5, we need to know if that means 5 centimeters, 5 kilometers, or 5 light years, etc. In a theory that involves general relativity and quantum mechanics, a choice of units emerges naturally, in the following way. There are two constants of nature upon which general relativity depends: the speed of light, c, and Newton's gravitation constant, G. Quantum mechanics depends on one constant of nature ħ. By examining the units of these constants (e.g., c is a velocity, so is expressed as distance divided by time, etc.), one can see that the combination √(ħG/c3) has the units of a length; in fact, it is 1.616 x 10-33 centimeters. This is the Planck length. Since it involves gravitational and spacetime inputs (G and c) and has a quantum mechanical dependence (ħ) as well, it sets the scale for measurements—the natural unit of length—in any theory that attempts to merge general relativity and quantum mechanics. When we use the term "Planck length" in the text, it is often meant in an approximate sense, indicating a length that is within a few orders of magnitude of 10-33 centimeters.

8. Currently, in addition to string theory, two other approaches for merging general relativity and quantum mechanics are being pursued vigorously. One approach is led by Roger Penrose of Oxford University and is known as twistor theory. The other approach—inspired in part by Penrose's work—is led by Abhay Ashtekar of Pennsylvania State University and is known as the new variables method. Although these other approaches will not be discussed further in this book, there is growing speculation that they may have a deep connection to string theory and that possibly, together with string theory, all three approaches are honing in on the same solution for merging general relativity and quantum mechanics.

Chapter 6

1. The expert reader will recognize that this chapter focuses solely on perturbative string theory; nonperturbative aspects are discussed in Chapters 12 and 13.

2. Interview with John Schwarz, December 23, 1997.

3. Similar suggestions were made independently by Tarmaki Yoneya and by Korkut Bardakci and Martin Halpern. The Swedish physicist Lars Brink also contributed significantly to the early development of string theory.

4. Interview with John Schwarz, December 23, 1997,

5. Interview with Michael Green, December 20, 1997.

6. The standard model does suggest a mechanism by which particles acquire mass—the Higgs mechanism, named after the Scottish physicist Peter Higgs. But from the point of view of explaining the particle masses, this merely shifts the burden to explaining properties of a hypothetical "mass-giving particle"—the so-called Higgs boson. Experimental searches for this particle are underway, but once again, if it is found and its properties measured, these will be input data for the standard model, for which the theory offers no explanation.

7. For the mathematically inclined reader, we note that the association between string vibrational patterns and force charges can be described more precisely as follows. When the motion of a string is quantized, its possible vibrational states are represented by vectors in a Hilbert space, much as for any quantum-mechanical system. These vectors can be labeled by their eigenvalues under a set of commuting hermitian operators. Among these operators are the Hamiltonian, whose eigenvalues give the energy and hence the mass of the vibrational state, as well as operators generating various gauge symmetries that the theory respects. The eigenvalues of these latter operators give the force charges carried by the associated vibrational string state.

8. Based upon insights gleaned from the second superstring revolution (discussed in Chapter 12), Witten and, most notably, Joe Lykken of the Fermi National Accelerator Laboratory have identified a subtle, yet possible, loophole in this conclusion. Lykken, exploiting this realization, has suggested that it might be possible for strings to be under far less tension, and therefore be substantially larger in size, than originally thought. So large, in fact, that they might be observable by the next generation of particle accelerators. If this long-shot possibility turns out to be the case, there is the exciting prospect that many of the remarkable implications of string theory discussed in this and the following chapters will be verifiable experimentally within the next decade. But even in the more "conventional" scenario espoused by string theorists, in which strings are typically on the order of 10-33 centimeters in length, there are indirect ways to search for them experimentally, as we will discuss in Chapter 9.

9. The expert reader will recognize that the photon produced in a collision between an electron and a positron is a virtual photon and therefore must shortly relinquish its energy by dissociating into a particle-antiparticle pair.

10. Of course, a camera works by collecting photons that bounce off the object of interest and recording them on a piece of photographic film. Our use of a camera in this example is symbolic, since we are not imagining bouncing photons off of the colliding strings. Rather, we simply want to record in Figure 6.7(c) the whole history of the interaction. Having said that, we should point out one further subtle point that the discussion in the text glosses over. We learned in Chapter 4 that we can formulate quantum mechanics using Feynman's sum-over-paths method, in which we analyze the motion of objects by combining contributions from all possible trajectories that lead from some chosen starting point to some chosen destination (with each trajectory contributing with a statistical weight determined by Feynman). In Figures 6.6 and 6.7 we show one of the infinite number of possible trajectories followed by point particles (Figure 6.6) or by strings (Figure 6.7) taking them from their initial positions to their final destinations. The discussion in this section, however, applies equally well to any of the other possible trajectories and therefore applies to the whole quantum-mechanical process itself. (Feynman's formulation of point-particle quantum mechanics in the sum-over-paths framework was generalized to string theory through the work of Stanley Mandelstarn of the University of California at Berkeley and by the Russian physicist Alexander Polyakov, who is now on the faculty of the physics department of Princeton University.)

Chapter 7

1. Albert Einstein, as quoted in R. Clark, Einstein: The Life and Times (New York: Avon Books, 1984), p. 287.

2. More precisely, spin-½ means that the angular momentum of the electron from its spin is h/2.

3. The discovery and development of supersymmetry has a complicated history In addition to those cited in the text, essential early contributions were made by R. Haag, M. Sohnius, J. T. Lopuszanski, Y. A. Gol'fand, E. P. Lichtman, J. L. Gervais, B. Sakita, V. P. Akulov, D. V. Volkov, and V. A. Soroka, among many others. Some of their work is documented in Rosanne Di Stefano, Notes on the Conceptual Development of Supersymmetry, Institute for Theoretical Physics, State University of New York at Stony Brook, preprint ITP-SB-8878.

4. For the mathematically inclined reader we note that this extension involves augmenting the familiar Cartesian coordinates of spacetime with new quantum coordinates, say u and v, that are anticommuting: u × v = -v × u. Supersymmetry can then be thought of as translations in this quantum-mechanically augmented form of spacetime.

5. For the reader interested in more details of this technical issue we note the following. In note 6 of Chapter 6 we mentioned that the standard model invokes a "mass-giving particle"—the Higgs boson—to endow the particles of Tables 1.1 and 1.2 with their observed masses. For this procedure to work, the Higgs particle itself cannot be too heavy; studies show that its mass should certainly be no greater than about 1,000 times the mass of a proton. But it turns out that quantum fluctuations tend to contribute substantially to the mass of the Higgs particle, potentially driving its mass all the way to the Planck scale. Theorists have found, however, that this outcome, which would uncover a major defect in the standard model, can be avoided if certain parameters in the standard model (most notably, the so-called bare mass of the Higgs particle) are finely tuned to better than 1 part in 1015 to cancel the effects of these quantum fluctuations on the Higgs particle's mass.

6. One subtle point to note about Figure 7.1 is that the strength of the weak force is shown to be between that of the strong and electromagnetic forces, whereas we have previously said that it is weaker than both. The reason for this lies in Table 1.2, in which we see that the messenger particles of the weak force are quite massive, whereas those of the strong and electromagnetic forces are massless. Intrinsically, the strength of the weak force (as measured by its coupling constant—an idea we will come upon in Chapter 12) is as shown in Figure 7.1, but its massive messenger particles are sluggish conveyers of its influence and diminish its effects. In Chapter 14 we will see how the gravitational force fits into Figure 7.1.

7. Edward Witten, lecture at the Heinz Pagels Memorial Lecture Series, Aspen, Colorado, 1997.

8. For an in-depth discussion of these and related ideas, see Steven Weinberg, Dreams of a Final Theory.

Chapter 8

1. This is a simple idea, but since the imprecision of common language can sometimes lead to confusion, two clarifying remarks are in order. First, we are assuming that the ant is constrained to live on the surface of the garden hose. If, on the contrary, the ant could burrow into the interior of the hose—if it could penetrate into the rubber material of the hose—we would need three numbers to specify its position, since we would need to also specify how deeply it had burrowed. But if the ant lives only on the hose's surface, its location can be specified with just two numbers. This leads to our second point. Even with the ant living on the hose's surface, we could, if we so chose, specify its location with three numbers: the ordinary left-right, back-forth, and up-down positions in our familiar three-dimensional space. But once we know that the ant lives on the surface of the hose, the two numbers referred to in the text give the minimal data that uniquely specify the ant's position. This is what we mean by saying that the surface of the hose is two-dimensional.

2. Surprisingly, the physicists Savas Dimopoulos, Nima Arkani-Hamed, and Gia Dvali, building on earlier insights of Ignatios Antomadis and Joseph Lykken, have pointed out that even if an extra curled-up dimension were as large as a millimeter in size, it is possible that it would not yet have been detected experimentally. The reason is that particle accelerators probe the microworld by utilizing the strong, weak, and electromagnetic forces. The gravitational force, being incredibly feeble at technologically accessible energies, is generally ignored. But Dimopoulos and his collaborators note that if the extra curled-up dimension has an impact predominantly on the gravitational force (something, it turns out, that is quite plausible in string theory), all extant experiments could well have overlooked it. New, highly sensitive gravitational experiments will look for such "large" curled-up dimensions in the near future. A positive result would be one of the greatest discoveries of all time.

3. Edwin Abbott, Flatland (Princeton: Princeton University Press, 1991).

4. A. Einstein in letter to T. Kaiuza as quoted in Abraham Pais, "Subtle is the Lord": The Science and the Life of Albert Einstein (Oxford: Oxford University Press, 1982), p. 330.

5. A. Einstein in letter to T. Kaluza as quoted in D. Freedman and P. van Nieuwenhuizen, "The Hidden Dimensions of Spacetime," Scientific American 252 (1985), 62.

6. Ibid.

7. Physicists found that the most difficult feature of the standard model to incorporate through a higher-dimensional formulation is something known as chirality. So as not to overburden the discussion we have not covered this concept in the main text, but for readers who are interested we do so briefly here. Imagine that someone shows you a film of some particular scientific experiment and confronts you with the unusual challenge of determining whether the film shot the experiment directly or whether it shot the experiment by looking at its reflection in a mirror. As the cinematographer was quite expert, there are no telltale signs of a mirror being involved. Is this a challenge you can meet? In the mid-1950s, the theoretical insights of T. D. Lee and C. N. Yang, and the experimental results of C. S. Wu and collaborators, showed that you can meet the challenge, so long as an appropriate experiment had been filmed. Namely, their work established that the laws of the universe are not perfectly mirror symmetric in the sense that the mirror-reflected version of certain processes—those directly dependent on the weak force—cannot happen in our world, even though the original process can. And so, as you watch the film if you see one of these forbidden processes occur, you will know that you are watching a mirror-reflected image of the experiment, as opposed to the experiment itself. Since mirrors interchange left and right, the work of Lee, Yang, and Wu established that the universe is not perfectly left-right symmetric—in the language of the field, the universe is chiral. It is this feature of the standard model (the weak force, in particular) that physicists found nearly impossible to incorporate into a higher-dimensional supergravity framework. To avoid confusion, we note that in Chapter 10 we will discuss a concept in string theory known as "mirror symmetry," but the use of the word "mirror" in that context is completely different from its use here.

8. For the mathematically inclined reader, we note that a Calabi-Yau manifold is a complex Kähler manifold with vanishing first Chern class. In 1957 Calabi conjectured that every such manifold admits a Ricci-flat metric, and in 1977 Yau proved this to be true.

9. This illustration is courtesy of Andrew Hanson of Indiana University, and was made using the Mathematica 3-D graphing package.

10. For the mathematically inclined reader we note that this particular Calabi-Yau space is a real three-dimensional slice through the quintic hypersurface in complex projective four-space.

Chapter 9

1. Edward Witten, "Reflections on the Fate of Spacetime" Physics Today, April 1996, p. 24.

2. Interview with Edward Witten, May 11, 1998.

3. Sheldon Glashow and Paul Ginsparg, "Desperately Seeking Superstrings?" Physics Today, May 1986, p. 7.

4. Sheldon Glashow, in The Superworld I, ed. A. Zichichi (New York: Plenum, 1990), p. 250.

5. Sheldon Glashow, Interactions (New York: Warner Books, 1988), p. 335.

6. Richard Feynman, in Superstrings: A Theory of Everything? ed. Paul Davies and Julian Brown (Cambridge, Eng: Cambridge University Press, 1988).

7. Howard Georgi, in The New Physics, ed. Paul Davies (Cambridge: Cambridge University Press 1989), p. 446.

8. Interview with Edward Witten, March 4, 1998.

9. Interview with Cumrun Vafa, January 12, 1998.

10. Murray Gell-Mann, as quoted in Robert P. Crease and Charles C. Mann, The Second Creation (New Brunswick, N.J.: Rutgers University Press), 1996, p. 414.

11. Interview with Sheldon Glashow, December 28, 1997.

12. Interview with Sheldon Glashow, December 28, 1997.

13. Interview with Howard Georgi, December 28, 1997. During the interview, Georgi also noted that the experimental refutation of the prediction of proton decay that emerged from his and Glashow's first proposed grand unified theory (see Chapter 7) played a significant part in his reluctance to embrace superstring theory. He noted poignantly that his grand unified theory invoked a vastly higher energy realm than any theory previously considered, and when its prediction was proved wrong—when it resulted in his "being slapped down by nature"—his attitude toward studying extremely high energy physics abruptly changed. When I asked him whether experimental confirmation of his grand unified theory might have inspired him to lead the charge to the Planck scale, he responded, "Yes, it likely would have."

14. David Gross, "Superstrings and Unification," in Proceedings of the XXIV International Conference on High Energy Physics, ed. R. Kotthaus and J. Kühn (Berlin: Springer-Verlag, 1988), p. 329.

15. Having said this, it's worth bearing in mind the long-shot possibility, pointed out in endnote 8 of Chapter 6, that strings just might be significantly longer than originally thought and therefore might be subject to direct experimental observation by accelerators within a few decades.

16. For the mathematically inclined reader we note that the more precise mathematical statement is that the number of families is half the absolute value of the Euler number of the Calabi-Yau space. The Euler number itself is the alternating sum of the dimensions of the manifold's homology groups—the latter being what we loosely refer to as multidimensional holes. So, three families emerge from Calabi-Yau spaces whose Euler number is ±6.

17. Interview with John Schwarz, December 23, 1997.

18. For the mathematically inclined reader we note that we are referring to Calabi-Yau manifolds with a finite, nontrivial fundamental group, the order of which, in certain cases, determines the fractional charge denominators.

19. Interview with Edward Witten, March 4, 1998.

20. For the expert we note that some of these processes violate lepton number conservation as well as charge-parity-time (CPT) reversal symmetry.

Chapter 10

1. For completeness, we note that although much of what we have covered to this point in the book applies equally well to open strings (a string with loose ends) or closed-string loops (the strings on which we have focused), the topic discussed here is one in which the two kinds of strings would appear to have different properties. After all, an open string will not get entangled by looping around a circular dimension. Nevertheless, through work that ultimately has played a pivotal part in the second superstring revolution, in 1989 Joe Polchinski from the University of California at Santa Barbara and two of his students, Jian-Hui Dai and Robert Leigh, showed how open strings fit perfectly into the conclusions we find in this chapter.

2. In case you are wondering why the possible uniform vibrational energies are whole number multiples of 1/R, you need only think back to the discussion of quantum mechanics—the warehouse in particular—from Chapter 4. There we learned that quantum mechanics implies that energy, like money, comes in discrete lumps: whole number multiples of various energy denominations. In the case of uniform vibrational string motion in the Garden-hose universe, this energy denomination is precisely 1/R, as we demonstrated in the text using the uncertainty principle. Thus the uniform vibrational energies are whole number multiples of 1/R.

3. Mathematically, the identity between the string energies in a universe with a circular dimension whose radius is either R or 1/R arises from the fact that the energies are of the form v/R + wR, where v is the vibration number and w is the winding number. This equation is invariant under the simultaneous interchange of v and w as well as R and 1/R—i.e., under the interchange of vibration and winding numbers and inversion of the radius. In our discussion we are working in Planck units, but we can work in more conventional units by rewriting the energy formula in terms of √α'—so-called string scale—whose value is about the Planck length, 10-13 centimeter. We can then express string energies as v/R + wR/α', which is invariant under interchange of v and w as well as R and α'/R, where the latter two are now expressed in terms of conventional units of distance.

4. You may be wondering how it's possible for a string that stretches all the way around a circular dimension of radius R to nevertheless measure the radius to be 1/R. Although a thoroughly justifiable concern, its resolution actually lies in the imprecise phrasing of the question itself. You see, when we say that the string is wrapped around a circle of radius R, we are by necessity invoking a definition of distance (so that the phrase "radius R" has meaning). But this definition of distance is the one relevant for the unwound string modes—that is, the vibration modes. From the point of view of this definition of distance—and only this definition—the winding string configurations appear to stretch around the circular part of space. However, from the second definition of distance, the one that caters to the wound-string configurations, they are every bit as localized in space as are the vibration modes from the viewpoint of the first definition of distance, and the radius they "see" is 1/R, as discussed in the text.

This description gives some sense of why wound and unwound strings measure distances that are inversely related. But as the point is quite subtle, it is perhaps worth noting the underlying technical analysis for the mathematically inclined reader. In ordinary point-particle quantum mechanics, distance and momentum (essentially energy) are related by Fourier transform. That is, a position eigenstate |x> on a circle of radius R can be defined by |x>=Σveixp|p> where p = v/R and |p> is a momentum eigenstate (the direct analog of what we have called a uniform-vibration mode of a string—overall motion without change in shape). In string theory, though, there is a second notion of position eigenstate |x~> defined by making use of the winding string states: |x~> = Σweix~p~|p~ > where |p~> is a winding eigenstate with p~ = wR. From these definitions we immediately see that x is periodic with period 2πR while x~ is periodic with period 2π/R, showing that x is a position coordinate on a circle of radius R~ while x~ is the position coordinate on a circle of radius 1/R. Even more explicitly, we can now imagine taking the two wavepackets |x> and |x~> both starting say, at the origin, and allowing them to evolve in time to carry out our operational approach for defining distance. The radius of the circle, as measured by either probe, is then proportional to the required time lapse for the packet to return to its initial configuration. Since a state with energy E evolves with a phase factor involving Et, we see that the time lapse, and hence the radius, is t ~ 1/E ~ R for the vibration modes and t ~ 1/E ~ 1/R for the winding modes.

5. For the mathematically inclined reader, we note that, more precisely, the number of families of string vibrations is one-half the absolute value of the Euler characteristic of the Calabi-Yau space, as mentioned in note 16 of Chapter 9. This is given by the absolute value of difference between h2,1 and h1,1, where hp,q denotes the (p,q) Hodge number. Up to a numerical shift, these count the number of nontrivial homology three-cycles ("three-dimensional holes") and the number of homology twocycles ("two-dimensional holes"). And so, whereas we speak of the total number of holes in the main text, the more precise analysis shows that the number of families depends on the absolute value of difference between the odd- and even-dimensional holes. The conclusion, however, is the same. For instance, if two Calabi-Yau spaces differ by the interchange of their respective h2,1 and h1,1 Hodge numbers, the number of particle families-and the total number of "holes"—will not change.

6. The name comes from the fact that the "Hodge diamonds"—a mathematical summary of the holes of various dimensions in a Calabi-Yau space—for each Calabi-Yau space of a mirror pair are mirror reflections of one another.

7. The term mirror symmetry is also used in other, completely different contexts in physics, such as in the question of chirality—that is, whether the universe is left-right symmetric—as discussed in note 7 of Chapter 8.

Chapter 11

1. The mathematically inclined reader will recognize that we are asking whether the topology of space is dynamical—that is, whether it can change. We note that although we will often use the language of dynamical topology change, in practice we are usually considering a one-parameter family of spacetimes whose topology changes as a function of the parameter. Technically speaking, this parameter is not time, but in certain limits can essentially be identified with time.

2. For the mathematically inclined reader, the procedure involves blowing down rational curves on a Calabi-Yau manifold and then making use of the fact that, under certain circumstances, the resulting singularity can be repaired by distinct small resolutions.

3. K. C. Cole, New York Times Magazine, October 18, 1987, p. 20.

Chapter 12

1. Albert Einstein, as quoted in John D. Barrow, Theories of Everything (New York: Fawcett-Columbine, 1992), p. 13.

2. Let's briefly summarize the differences between the five string theories. To do so, we note that vibrational disturbances along a loop of string can travel clockwise or counterclockwise. The Type IIA and Type 1/R strings differ in that in the latter theory, these clockwise/counterclockwise vibrations are identical, while in the former, they are exactly opposite in form. Opposite has a precise mathematical meaning in this context, but it's easiest to think about in terms of the spins of the resulting vibrational patterns in each theory. In the Type 1/R theory, it turns out that all particles spin in the same direction (they have the same chirality), whereas in the Type IIA theory, they spin in both directions (they have both chiralities). Nevertheless, each theory incorporates supersymmetry. The two heterotic theories differ in a similar but more dramatic way. Each of their clockwise string vibrations looks like those of the Type II string (when focusing on just the clockwise vibrations, the Type IIA and Type 1/R theories are the same), but their counterclockwise vibrations are those of the original bosonic string theory. Although the bosonic string has insurmountable problems when chosen for both clockwise and counterclockwise string vibrations, in 1985 David Gross, Jeffrey Harvey, Emit Martinec, and Ryan Rhom (all then at Princeton University and dubbed the "Princeton String Quartet") showed that a perfectly sensible theory emerges if it is used in combination with the Type II string. The really odd feature of this union is that it has been known since the work of Claude Lovelace of Rutgers University in 1971 and the work of Richard Brower of Boston University, Peter Goddard of Cambridge University, and Charles Thorn of the University of Florida at Gainesville in 1972 that the bosonic string requires a 26-dimensional spacetime, whereas the superstring, as we have discussed, requires a 10-dimensional one. So the heterotic string constructions are a strange hybrid—a heterosis—in which counterclockwise vibrational patterns live in 26 dimensions and clockwise patterns live in 10 dimensions! Before you get caught up in trying to make sense of this perplexing union, Gross and his collaborators showed that the extra 16 dimensions on the bosonic side must be curled up into one of two very special higher-dimensional doughnutlike shapes, giving rise to the Heterotic-O and Heterotic-E theories. Since the extra 16 dimensions on the bosonic side are rigidly curled up, each of these theories behaves as though it really has 10 dimensions, just as in the Type II case. Again, both heterotic theories incorporate a version of supersymmetry. Finally, the Type I theory is a close cousin of the Type 1/R string except that, in addition to the closed loops of string we have discussed in previous chapters, it also has strings with unconnected ends—so-called open strings.

3. When we speak of "exact" answers in this chapter, such as the "exact" motion of the earth, what we really mean is the exact prediction for some physical quantity within some chosen theoretical framework. Until we truly have the final theory—perhaps we now do, perhaps we never will—all of our theories will themselves be approximations to reality. But this notion of approximate has nothing to do with our discussion in this chapter. Here we are concerned with the fact that within a chosen theory, it is often difficult, if not impossible, to extract the exact predictions that the theory makes. Instead, we have to extract such predictions using approximation methods based on a perturbative approach.

4. These diagrams are string theory versions of the so-called Feynman diagrams, invented by Richard Feynman for performing perturbative calculations in point-particle quantum field theory.

5. More precisely, every virtual string pair, that is, every loop in a given diagram, contributes—among other more complicated terms—a multiplicative factor of the string coupling constant. More loops translate into more factors of the string coupling constant. If the string coupling constant is less than 1, repeated multiplications make the overall contribution ever smaller; if it is 1 or larger, repeated multiplications yield a contribution with the same or larger magnitude.

6. For the mathematically inclined reader, we note that the equation states that spacetime must admit a Ricci-flat metric. If we split spacetime into a Cartesian product of four-dimensional Minkowski spacetime and a six-dimensional compact Kähler space, Ricci-flatness is equivalent to the latter being a Calabi-Yau manifold. This is why Calabi-Yau spaces play such a prominent role in string theory

7. Of course, nothing absolutely ensures that these indirect approaches are justified. For example, just as some faces are not left-right symmetric, it might be that the laws of physics are different in other far-flung regions of the universe, as we will discuss briefly in Chapter 14.

8. The expert reader will recognize that these statements require so-called N=2 supersymmetry.

9. To be a little more precise, if we call the Heterotic-O coupling constant gHO and the Type I coupling constant gI, then the relation between the two theories states that they are physically identical so long as gHO = 1/gI, which is equivalent to gI = 1/gHO. When one coupling constant is big the other is small.

10. This is a close analog of the R, 1/R duality discussed previously If we call the Type 1/R string coupling constant gIIB then the statement that appears to be true is that the values gIIB and 1/gIIB describe the same physics. gIIB is big, 1/gIIB is small, and vice versa.

11. If all but four dimensions are curled up, a theory with more than eleven total dimensions necessarily gives rise to massless particles with spin greater than 2, something that both theoretical and experimental considerations rule out.

12. A notable exception is the important 1987 work of Duff, Paul Howe, Takeo Inami, and Kelley Stelle in which they drew on earlier insights of Eric Bergshoeff, Ergin Sezgin, and Townsend to argue that ten-dimensional string theory should have a deep eleven-dimensional connection.

13. More precisely, this diagram should be interpreted as saying that we have a single theory that depends on a number of parameters. The parameters include coupling constants as well as geometrical size and shape parameters. In principle, we should be able to use the theory to calculate particular values for all of these parameters—a particular value for its coupling constant and a particular form for the spacetime geometry—but within our current theoretical understanding, we do not know how to accomplish this. And so, to understand the theory better string theorists study its properties as the values of these parameters are varied over all possibilities. If the parameter values are chosen to lie in any of the six peninsular regions of Figure 1.1, the theory has the properties inherent to one of the five string theories, or to eleven-dimensional supergravity, as marked. If the parameter values are chosen to lie in the central region, the physics is governed by the still mysterious M-theory.

14. We should note, though, that even in the peninsular regions there are some exotic ways in which branes can have an effect on familiar physics. For example, it has been suggested that our three extended spatial dimensions might themselves be a three-brane that is large and unfurled. If so, as we go about our daily business we would be gliding through the interior of a three-dimensional membrane. Investigations of such possibilities are now being undertaken.

15. Interview with Edward Witten, May 11, 1998.

Chapter 13

1. The expert reader will recognize that under mirror symmetry, a collapsing three-dimensional sphere on one Calabi-Yau space gets mapped to a collapsing two-dimensional sphere on the mirror Calabi-Yau space—apparently putting us back in the situation of flops discussed in Chapter 11. The difference, however, is that a mirror rephrasing of this sort results in the antisymmetric tensor field Bμv—the real part of the complexified Kähler form on the mirror Calabi-Yau space—vanishing, and this is a far more drastic sort of singularity than that discussed in Chapter 11.

2. More precisely, these are examples of extremal black holes: black holes that have the minimum mass consistent with the force charges they carry, just like the BPS states in Chapter 12. Similar black holes will also play a pivotal role in the following discussion on black hole entropy.

3. The radiation emitted from a black hole should be just like that emitted from a hot oven—the very problem, discussed at the outset of Chapter 4, that played such a pivotal role in the development of quantum mechanics.

4. It turns out that because the black holes involved in space-tearing conifold transitions are extremal, they do not Hawking radiate, regardless of how light they become.

5. Stephen Hawking, lecture at Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1996.

6. In their initial calculation, Strominger and Vafa found that the mathematics was made easier by working with five—not four—extended spacetime dimensions. Surprisingly, after completing their calculation of the entropy of such a five-dimensional black hole they realized that no theoretician had as yet constructed such hypothetical extremal black holes in the setting of five-dimensional general relativity. Since only by comparing their answer to the area of the event horizon of such a hypothetical black hole could they confirm their results, Strominger and Vafa then set out to mathematically construct such a five-dimensional black hole. They succeeded. It was then a simple matter to show that the microscopic string theory calculation of the entropy was in agreement with what Hawking would have predicted based on the area of the black hole's event horizon. But it is interesting to realize that because the black hole solution was found later, Strominger and Vafa did not know the answer they were shooting for while undertaking their entropy calculation. Since their work, numerous researchers, led most notably by Princeton physicist Curtis Callan, have succeeded in extending the entropy calculations to the more familiar setting of four extended spacetime dimensions, and all are in agreement with Hawking's predictions.

7. Interview with Sheldon Glashow, December 29, 1997.

8. Laplace, Philosophical Essay on Probabilities, trans. Andrew I. Dale (New York: Springer-Verlag, 1995).

9. Stephen Hawking, in Hawking and Roger Penrose, The Nature of Space and Time (Princeton: Princeton University Press, 1995), p. 41.

10. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.

11. Interview with Andrew Strominger, December 29, 1997.

12. Interview with Cumrun Vafa, January 12, 1998.

13. Stephen Hawking, lecture at the Amsterdam Symposium on Gravity, Black Holes, and Strings, June 21, 1997.

14. This issue also has some bearing on the information-loss question, as some physicists have speculated over the years that there might be a central "nugget" embedded in the depths of a black hole that stores all of the information carried by matter that gets trapped within the hole's horizon.

15. In fact, the space-tearing conifold transitions discussed in this chapter involve black holes and hence might seem to be tied up with the question of their singularities. But recall that the conifold tear occurs just as the black hole has shed all its mass, and is therefore not directly related to questions concerning black hole singularities.

Chapter 14

1. More precisely, the universe should be filled with photons conforming to the radiation thermally emitted by a perfectly absorbent body—a "black-body" in the language of thermodynamics—with the stated temperature range. This is the same radiation spectrum emitted quantum mechanically by black holes, as explained by Hawking, and by a hot oven, as explained by Planck.

2. The discussion conveys the spirit of the issues involved although we are glossing over some subtle features having to do with the motion of light in an expanding universe that affect the detailed numerics. In particular, although special relativity declares that nothing can travel faster than the speed of light, this does not preclude two photons carried along on the expanding spatial fabric from receding from one another at a speed exceeding that of light. For example, at the time the universe first became transparent, about 300,000 years ATB, locations in the heavens that were about 900,000 light-years apart would have been able to have influenced each other, even though the distance between them exceeds 300,000 light-years. The extra factor of three comes from the expansion of the spatial fabric. This means that as we run the cosmic film backward in time, by the time we get to 300,000 years ATB, two points in the heavens need only be less than 900,000 light-years apart to have had a chance to influence each other's temperature. These detailed numerics do not change the qualitative features of the issues discussed.

3. For a detailed and lively discussion of the discovery of the inflationary cosmological model and the problems it resolves, see Alan Guth, The Inflationary Universe (Reading, Mass: Addison-Wesley, 1997).

4. For the mathematically inclined reader, we note that the idea underlying this conclusion is the following: If the sum of the spacetime dimensions of the paths swept out by each of two objects is greater than or equal to the spacetime dimension of the arena through which they are moving then they will generically intersect. For instance, point particles sweep out one-dimensional spacetime paths—the sum of the spacetime dimensions for two such particle paths is therefore two. The spacetime dimension of Lineland is also two, and hence their paths will generally intersect (assuming their velocities have not been finely tuned to be exactly equal). Similarly, strings sweep out two-dimensional spacetime paths (their world-sheets); for two strings the sum in question is therefore four. This means that strings moving in four spacetime dimensions (three space and one time) will generally intersect.

5. With the discovery of M-theory and the recognition of an eleventh dimension, string theorists have begun studying ways of curling up all seven extra dimensions in a manner that puts them all on more or less equal footing. The possible choices for such seven-dimensional manifolds are known as Joyce manifolds, after Domenic Joyce of Oxford University, who is credited with finding the first techniques for their mathematical construction.

6. Interview with Cumrun Vafa, January 12, 1998.

7. The expert reader will note that our description is taking place in the so-called string frame of reference, in which increasing curvature during the pre-big bang arises from (a dilaton-driven) increase in the strength of the gravitational force. In the so-called Einstein frame, the evolution would be described as an accelerating contraction phase.

8. Interview with Gabriele Veneziano, May 19, 1998.

9. Smolin's ideas are discussed in his book The Life of the Cosmos (New York: Oxford University Press, 1997).

10. Within string theory, for example, this evolution could be driven by small changes to the shape of the curled-up dimensions from one universe to its offspring. From our results on space-tearing conifold transitions, we know that a sufficiently long sequence of such small changes can take us from one Calabi-Yau to any other, allowing the multiverse to sample the reproductive efficiency of all universes based on strings. After the multiverse has passed through sufficiently many stages of reproduction, Smolin's hypothesis would lead us to expect that the typical universe will have a Calabi-Yau component that is optimized for fertility.

Chapter 15

1. Interview with Edward Witten, March 4, 1998.

2. Some theorists see a hint of this idea in the holographic principle, a concept originated by Susskind and the renowned Dutch physicist Gerard 't Hooft. Just as a hologram can reproduce a three-dimensional visual image from a specially designed two-dimensional film, Susskind and 't Hooft have suggested that all of the physical happenings we encounter may actually be encoded fully through equations defined in a lower-dimensional world. Although this may sound as strange as trying to draw someone's portrait by viewing only their shadow, we can get a sense of what it means, and understand part of Susskind's and 't Hooft's motivation, by thinking about black hole entropy as discussed in Chapter 13. Recall that the entropy of a black hole is determined by the surface area of its event horizon—and not by the total volume of space that the event horizon bounds. Therefore, the disorder of a black hole, and correspondingly the information it can embody, is encoded in the two-dimensional data of surface area. It is almost as if the event horizon of the black hole acts like a hologram by capturing all the information content of the black hole's three-dimensional interior. Susskind and 't Hooft have generalized this idea to the whole universe by suggesting that everything that occurs in the "interior" of the universe is merely a reflection of data and equations defined on a distant, bounding surface. Recently, work by the Harvard physicist Juan Maldacena, together with important subsequent work by Witten and of Princeton physicists Steven Gubser, Igor Klebanov, and Alexander Polyakov, has shown that, at least in certain cases, string theory embodies the holographic principle. In a manner that is currently being investigated vigorously, it appears that the physics of a universe governed by string theory has an equivalent description that involves only physics that takes place on such a bounding surface—a surface necessarily of lower dimensionality than the interior. Some string theorists have suggested that fully understanding the holographic principle and its role in string theory may well lead to the third superstring revolution.

3. Sir Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World, trans. Motte and Cajori (Berkeley: University of California Press, 1962), Vol. 1, p. 6.

4. If you are familiar with linear algebra, one simple and relevant way of thinking about noncommutative geometry is to replace conventional Cartesian coordinates, which commute under multiplication, with matrices, which do not.

5. Interview with Cumrun Vafa, January 12, 1998.

6. Interview with Edward Witten, May 11, 1998.

7. Quoted in Banesh Hoffman with Helen Dukas, Albert Einstein, Creator and Rebel (New York: Viking, 1972), p. 18.

8. Martin J. Klein, "Einstein: The Life and Times, by R. W. Clark," (book review) Science 174, pp. 1315-16.

9. Jacob Bronkowski, The Ascent of Man (Boston: Little, Brown, 1973), p. 20.

Glossary of Scientific Terms

Absolute zero. The lowest possible temperature, about -273 degrees Celsius, or 0 on the Kelvin scale.

Acceleration. A change in an object's speed or direction. See also velocity.

Accelerator. See particle accelerator.

Amplitude. The maximum height of a wave peak or the maximum depth of a wave trough.

Anthropic principle. Doctrine that one explanation for why the universe has the properties we observe is that, were the properties different, it is likely that life would not form and therefore we would not be here to observe the changes.

Antimatter. Matter that has the same gravitational properties as ordinary matter, but that has an opposite electric charge as well as opposite nuclear force charges.

Antiparticle. A particle of antimatter.

ATB. Acronym for "after the bang"; usually used in reference to time elapsed since the big bang.

Atom. Fundamental building block of matter, consisting of a nucleus (comprising protons and neutrons) and an orbiting swarm of electrons.

Big bang. Currently accepted theory that the expanding universe began some 15 billion years ago from a state of enormous energy, density, and compression.

Big crunch. One hypothesized future for the universe in which the current expansion stops, reverses, and results in all space and all matter collapsing together; a reversal of the big bang.

Black hole. An object whose immense gravitational field entraps anything, even light, that gets too close (closer than the black hole's event horizon).

Black-hole entropy. The entropy embodied within a black hole.

Boson. A particle, or pattern of string vibration, with a whole number amount of spin; typically a messenger particle.

Bosonic string theory. First known string theory; contains vibrational patterns that are all bosons.

BPS states. Configurations in a supersymmetric theory whose properties can be determined exactly by arguments rooted in symmetry.

Brane. Any of the extended objects that arise in string theory. A one-brane is a string, a two-brane is a membrane, a three-brane has three extended dimensions, etc, More generally, a p-brane has p spatial dimensions.

Calabi-Yau space, Calabi-Yau shape. A space (shape) into which the extra spatial dimensions required by string theory can be curled up, consistent with the equations of the theory

Charge. See force charge.

Chiral, Chirality. Feature of fundamental particle physics that distinguishes left- from right-handed, showing that the universe is not fully left-right symmetric.

Closed string. A type of string that is in the shape of a loop.

Conifold transition. Evolution of the Calabi-Yau portion of space in which its fabric rips and repairs itself, yet with mild and acceptable physical consequences in the context of string theory. The tears involved are more severe than those in a flop transition.

Cosmic microwave background radiation. Microwave radiation suffusing the universe, produced during the big bang and subsequently thinned and cooled as the universe expanded.

Cosmological constant. A modification of general relativity's original equations, allowing for a static universe; interpretable as a constant energy density of the vacuum.

Coupling constant. See string coupling constant.

Curled-up dimension. A spatial dimension that does not have an observably large spatial extent; a spatial dimension that is crumpled, wrapped, or curled up into a tiny size, thereby evading direct detection.

Curvature. The deviation of an object or of space or of spacetime from a flat form and therefore from the rules of geometry codified by Euclid.

Dimension. An independent axis or direction in space or spacetime. The familiar space around us has three dimensions (left-right, back-forth, up-down) and the familiar spacetime has four (the previous three axes plus the past-future axis). Superstring theory requires the universe to have additional spatial dimensions.

Dual, Duality, Duality symmetries. Situation in which two or more theories appear to be completely different, yet actually give rise to identical physical consequences.

Electromagnetic field. Force field of the electromagnetic force, consisting of electric and magnetic lines of force at each point in space.

Electromagnetic force. One of the four fundamental forces, a union of the electric and magnetic forces.

Electromagnetic gauge symmetry. Gauge symmetry underlying quantum electrodynamics.

Electromagnetic radiation. The energy carried by an electromagnetic wave.

Electromagnetic wave. A wavelike disturbance in an electromagnetic field; all such waves travel at the speed of light. Visible light, X rays, microwaves, and infrared radiation are examples.

Electron. Negatively charged particle, typically found orbiting the nucleus of an atom.

Electroweak theory. Relativistic quantum field theory describing the weak force and the electromagnetic force in one unified framework.

Eleven-dimensional supergravity. Promising higher-dimensional supergravity theory developed in the 1970s, subsequently ignored, and more recently shown to be an important part of string theory.

Entropy. A measure of the disorder of a physical system; the number of rearrangements of the ingredients of a system that leave its overall appearance intact.

Equivalence principle. See principle of equivalence.

Event horizon. The one-way surface of a black hole; once penetrated, the laws of gravity ensure that there is no turning back, no escaping the powerful gravitational grip of the black hole.

Extended dimension. A space (and spacetime) dimension that is large and directly apparent; a dimension with which we are ordinarily familiar, as opposed to a curled-up dimension.

Extremal black holes. Black holes endowed with the maximal amount of force charge possible for a given total mass.

Families. Organization of matter particles into three groups, with each group being known as a family The particles in each successive family differ from those in the previous by being heavier, but carry the same electric and nuclear force charges.

Fermion. A particle, or pattern of string vibration, with half a whole odd number amount of spin; typically a matter particle.

Feynman sum-over-paths. See sum-over-paths.

Field, Force field. From a macroscopic perspective, the means by which a force communicates its influence; described by a collection of numbers at each point in space that reflect the strength and direction of the force at that point.

Flat. Subject to the rules of geometry codified by Euclid; a shape, like the surface of a perfectly smooth tabletop, and its higher-dimensional generalizations.

Flop transition. Evolution of the Calabi-Yau portion of space in which its fabric rips and repairs itself, yet with mild and acceptable physical consequences in the context of string theory.

Foam. See spacetime foam.

Force charge. A property of a particle that determines how it responds to a particular force. For instance, the electric charge of a particle determines how it responds to the electromagnetic force.

Frequency. The number of complete wave cycles a wave completes each second.

Gauge symmetry. Symmetry principle underlying the quantum-mechanical description of the three nongravitational forces; the symmetry involves the invariance of a physical system under various shifts in the values of force charges, shifts that can change from place to place and from moment to moment.

General relativity. Einstein's formulation of gravity, which shows that space and time communicate the gravitational force through their curvature.

Gluon. Smallest bundle of the strong force field; messenger particle of the strong force.

Grand unification. Class of theories that merge all three nongravitational forces into a single theoretical framework.

Gravitational force. The weakest of the four fundamental forces of nature. Described by Newton's universal theory of gravity, and subsequently by Einstein's general relativity.

Graviton. Smallest bundle of the gravitational force field; messenger particle for the gravitational force.

Heterotic-E string theory (Heterotic E8 × E8 string theory). One of the five superstring theories; involves closed strings whose right-moving vibrations resemble those of the Type II string and whose left-moving vibrations involve those of the bosonic string. Differs in important but subtle ways from the Heterotic-O string theory.

Heterotic-O string theory (Heterotic O(32) string theory). One of the five superstring theories; involves closed strings whose right-moving vibrations resemble those of the Type II string and whose left-moving vibrations involve those of the bosonic string. Differs in important but subtle ways from the Heterotic-E string theory.

Higher-dimensional supergravity. Class of supergravity theories in more than four spacetime dimensions.

Horizon problem. Cosmological puzzle associated with the fact that regions of the universe that are separated by vast distances nevertheless have nearly identical properties such as temperature. Inflationary cosmology offers a solution.

Infinities. Typical nonsensical answer emerging from calculations that involve general relativity and quantum mechanics in a point-particle framework.

Inflation, Inflationary cosmology. Modification to the earliest moments of the standard big bang cosmology in which universe undergoes a brief burst of enormous expansion.

Initial conditions. Data describing the beginning state of a physical system.

Interference pattern. Wave pattern that emerges from the overlap and the intermingling of waves emitted from different locations.

Kaluza-Klein theory. Class of theories incorporating extra curled-up dimensions, together with quantum mechanics.

Kelvin. A temperature scale in which temperatures are quoted relative to absolute zero.

Klein-Gordon equation. A fundamental equation of relativistic quantum field theory.

Laplacian determinism. Clockwork conception of the universe in which complete knowledge of the state of the universe at one moment completely determines its state at all future and past moments.

Light clock. A hypothetical clock that measures elapsed time by counting the number of round-trip journeys completed by a single photon between two mirrors.

Lorentz contraction. Feature emerging from special relativity, in which a moving object appears shortened along its direction of motion.

Macroscopic. Refers to scales typically encountered in the everyday world and larger; roughly the opposite of microscopic.

Massless black hole. In string theory, a particular kind of black hole that may have large mass initially, but that becomes ever lighter as a piece of the Calabi-Yau portion of space shrinks. When the portion of space has shrunk down to a point, the initially massive black hole has no remaining mass—it is massless. In this state, it no longer manifests such usual black hole properties as an event horizon.

Maxwell's theory, Maxwell's electromagnetic theory. Theory uniting electricity and magnetism, based on the concept of the electromagnetic field, devised by Maxwell in the 1880s; shows that visible light is an example of an electromagnetic wave.

Messenger particle. Smallest bundle of a force field; microscopic conveyer of a force.

Mirror symmetry. In the context of string theory, a symmetry showing that two different Calabi-Yau shapes, known as a mirror pair, give rise to identical physics when chosen for the curled-up dimensions of string theory.

M-theory. Theory emerging from the second superstring revolution that unites the previous five superstring theories within a single overarching framework. M-theory appears to be a theory involving eleven spacetime dimensions, although many of its detailed properties have yet to be understood.

Multidimensional hole. A generalization of the hole found in a doughnut to higher-dimensional versions.

Multi-doughnut, Multi-handled doughnut. A generalization of a doughnut shape (a torus) that has more than one hole.

Multiverse. Hypothetical enlargement of the cosmos in which our universe is but one of an enormous number of separate and distinct universes.

Neutrino. Chargeless species of particle, subject only to the weak force.

Neutron. Chargeless particle, typically found in the nucleus of an atom, consisting of three quarks (two down-quarks, one up-quark).

Newton's laws of motion. Laws describing the motion of bodies based on the conception of an absolute and immutable space and time; these laws held sway until Einstein's discovery of special relativity.

Newton's universal theory of gravity. Theory of gravity declaring that the force of attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. Subsequently supplanted by Einstein's general relativity.

Nonperturbative. Feature of a theory whose validity is not dependent on approximate, perturbative calculations; an exact feature of a theory.

Nucleus. The core of an atom, consisting of protons and neutrons.

Observer. Idealized person or piece of equipment, often hypothetical, that measures relevant properties of a physical system.

One-loop process. Contribution to a calculation in perturbation theory in which one virtual pair of strings (or particles in a point-particle theory) is involved.

Open string. A type of string with two free ends.

Oscillatory pattern. See vibrational pattern.

Particle accelerator. Machine for boosting particles to nearly light speed and slamming them together in order to probe the structure of matter.

Perturbation theory. Framework for simplifying a difficult problem by finding an approximate solution that is subsequently refined as more details, initially ignored, are systematically included.

Perturbative approach, Perturbative method. See perturbation theory.

Phase. When used in reference to matter, describes its possible states: solid phase, liquid phase, gas phase. More generally, refers to the possible descriptions of a physical system as features on which it depends (temperature, string coupling constant values, form of spacetime, etc.) are varied.

Phase transition. Evolution of a physical system from one phase to another.

Photoelectric effect. Phenomenon in which electrons are ejected from a metallic surface when light is shone upon it.

Photon. Smallest packet of the electromagnetic force field; messenger particle of the electromagnetic force; smallest bundle of light.

Planck energy. About 1,000 kilowatt hours. The energy necessary to probe to distances as small as the Planck length. The typical energy of a vibrating string in string theory.

Planck length. About 10-33 centimeters. The scale below which quantum fluctuations in the fabric of spacetime would become enormous. The size of a typical string in string theory.

Planck mass. About ten billion billion times the mass of a proton; about one-hundredth of a thousandth of a gram; about the mass of a small grain of dust. The typical mass equivalent of a vibrating string in string theory.

Planck's constant. Denoted by the symbol h, Planck's constant is a fundamental parameter in quantum mechanics. It determines the size of the discrete units of energy, mass, spin, etc. into which the microscopic world is partitioned. Its value is 1.05 × 10-17 grams-cm/sec.

Planck tension. About 1039 tons. The tension on a typical string in string theory.

Planck time. About 10-43 seconds. Time at which the size of the universe was roughly the Planck length; more precisely, time it takes light to travel the Planck length.

Primordial nucleosynthesis. Production of atomic nuclei occurring during the first three minutes after the big bang.

Principle of equivalence. Core principle of general relativity declaring the indistinguishability of accelerated motion and immersion in a gravitational field (over small enough regions of observation). Generalizes the principle of relativity by showing that all observers, regardless of their state of motion, can claim to be at rest, so long as they acknowledge the presence of a suitable gravitational field.

Principle of relativity. Core principle of special relativity declaring that all constant-velocity observers are subject to an identical set of physical laws and that, therefore, every constant-velocity observer is justified in claiming that he or she is at rest. This principle is generalized by the principle of equivalence.

Product. The result of multiplying two numbers.

Proton. Positively charged particle, typically found in the nucleus of an atom, consisting of three quarks (two up-quarks and one down-quark).

Quanta. The smallest physical units into which something can be partitioned, according to the laws of quantum mechanics. For instance, photons are the quanta of the electromagnetic field.

Quantum chromodynamics (QCD). Relativistic quantum field theory of the strong force and quarks, incorporating special relativity.

Quantum claustrophobia. See quantum fluctuations.

Quantum determinism. Property of quantum mechanics that knowledge of the quantum state of a system at one moment completely determines its quantum state at future and past moments. Knowledge of the quantum state, however, determines only the probability that one or another future will actually ensue.

Quantum electrodynamics (QED). Relativistic quantum field theory of the electromagnetic force and electrons, incorporating special relativity.

Quantum electroweak theory. See electroweak theory.

Quantum field theory. See relativistic quantum field theory.

Quantum fluctuation. Turbulent behavior of a system on microscopic scales due to the uncertainty principle.

Quantum foam. See spacetime foam.

Quantum geometry. Modification of Riemannian geometry required to describe accurately the physics of space on ultramicroscopic scales, where quantum effects become important.

Quantum gravity. A theory that successfully mergers quantum mechanics and general relativity, possibly involving modifications of one or both. String theory is an example of a theory of quantum gravity.

Quantum mechanics. Framework of laws governing the universe whose unfamiliar features such as uncertainty, quantum fluctuations, and wave-particle duality become most apparent on the microscopic scales of atoms and subnuclear particles.

Quantum tunneling. Feature of quantum mechanics showing that objects can pass through barriers that should be impenetrable according to Newton's classical laws of physics.

Quark. A particle that is acted upon by the strong force. Quarks exist in six varieties (up, down, charm, strange, top, bottom) and three "colors" (red, green, blue).

Radiation. The energy carried by waves or particles.

Reciprocal. The inverse of a number; for example, the reciprocal of 3 is 1/3, the reciprocal of ½ is 2.

Relativistic quantum field theory. Quantum-mechanical theory of fields, such as the electromagnetic field, that incorporates special relativity.

Resonance. One of the natural states of oscillation of a physical system.

Riemannian geometry. Mathematical framework for describing curved shapes of any dimension. Plays a central role in Einstein's description of spacetime in general relativity.

Schrödinger equation. Equation governing the evolution of probability waves in quantum mechanics.

Schwarzschild solution. Solution to the equations of general relativity for a spherical distribution of matter; one implication of this solution is the possible existence of black holes.

Second law of thermodynamics. Law stating that total entropy always increases.

Second superstring revolution. Period in the development of string theory beginning around 1995 in which some nonperturbative aspects of the theory began to be understood.

Singularity. Location where the fabric of space or spacetime suffers a devastating rupture.

Smooth, Smooth space. A spatial region in which the fabric of space is flat or gently curved, with no pinches, ruptures, or creases of any kind.

Space-tearing flop transition. See flop transition.

Spacetime. A union of space and time originally emerging from special relativity. Can be viewed as the "fabric" out of which the universe is fashioned; it constitutes the dynamical arena within which the events of the universe take place.

Spacetime foam. Frothy, writhing, tumultuous character of the spacetime fabric on ultramicroscopic scales, according to a conventional point-particle perspective. An essential reason for the incompatibility of quantum mechanics and general relativity prior to string theory.

Special relativity. Einstein's laws of space and time in the absence of gravity (see also general relativity).

Sphere. The outer surface of a ball. The surface of a familiar three-dimensional ball has two dimensions (which can be labeled by two numbers such as "latitude" and "longitude," as on the surface of the earth). The concept of a sphere, though, applies more generally to balls and hence their surfaces, in any number of dimensions. A one-dimensional sphere is a fancy name for a circle; a zero-dimensional sphere is two points (as explained in the text). A three-dimensional sphere is harder to picture; it is the surface of a four-dimensional ball.

Spin. A quantum-mechanical version of the familiar notion of the same name; particles have an intrinsic amount of spin that is either a whole number or half a whole number (in multiples of Planck's constant), and which never changes.

Standard model of cosmology. Big bang theory together with an understanding of the three nongravitational forces as summarized by the standard model of particle physics.

Standard model of particle physics, Standard model, Standard theory. An enormously successful theory of the three nongravitational forces and their action on matter. Effectively the union of quantum chromodynamics and the electroweak theory.

String. Fundamental one-dimensional object that is the essential ingredient in string theory

String coupling constant. A (positive) number that governs how likely it is for a given string to split apart into two strings or for two strings to join together into one—the basic processes in string theory. Each string theory has its own string coupling constant, the value of which should be determined by an equation; currently such equations are not understood well enough to yield any useful information. Coupling constants less than 1 imply that perturbative methods are valid.

String mode. A possible configuration (vibrational pattern, winding configuration) that a string can assume.

String theory. Unified theory of the universe postulating that fundamental ingredients of nature are not zero-dimensional point particles but tiny one-dimensional filaments called strings. String theory harmoniously unites quantum mechanics and general relativity, the previously known laws of the small and the large, that are otherwise incompatible. Often short for superstring theory.

Strong force, Strong nuclear force. Strongest of the four fundamental forces, responsible for keeping quarks locked inside protons and neutrons and for keeping protons and neutrons crammed inside of atomic nuclei.

Strong force symmetry. Gauge symmetry underlying the strong force, associated with invariance of a physical system under shifts in the color charges of quarks.

Strongly coupled. Theory whose string coupling constant is larger than 1.

Strong-weak duality. Situation in which a strongly coupled theory is dual—physically identical—to a different, weakly coupled theory.

Sum-over-paths. Formulation of quantum mechanics in which particles are envisioned to travel from one point to another along all possible paths between them.

Supergravity. Class of point-particle theories combining general relativity and supersymmetry.

Superpartners. Particles whose spins differ by ½ unit and that are paired by supersymmetry.

Superstring theory. String theory that incorporates supersymmetry.

Supersymmetric quantum field theory. Quantum field theory incorporating supersymmetry.

Supersymmetric standard model. Generalization of the standard model of particle physics to incorporate supersymmetry. Entails a doubling of the known elementary particle species.

Supersymmetry. A symmetry principle that relates the properties of particles with a whole number amount of spin (bosons) to those with half a whole (odd) number amount of spin (fermions).

Symmetry. A property of a physical system that does not change when the system is transformed in some manner. For instance, a sphere is rotationally symmetrical since its appearance does not change if it is rotated.

Symmetry breaking. A reduction in the amount of symmetry a system appears to have, usually associated with a phase transition.

Tachyon. Particle whose mass (squared) is negative; its presence in a theory generally yields inconsistencies.

Thermodynamics. Laws developed in the nineteenth century to describe aspects of heat, work, energy, entropy, and their mutual evolution in a physical system.

Three-brane. See brane.

Three-dimensional sphere. See sphere.

Time dilation. Feature emerging from special relativity, in which the flow of time slows down for an observer in motion.

T.O.E. (Theory of Everything). A quantum-mechanical theory that encompasses all forces and all matter.

Topologically distinct. Two shapes that cannot be deformed into one another without tearing their structure in some manner.

Topology. Classification of shapes into groups that can be deformed into one another without ripping or tearing their structure in any way

Topology-changing transition. Evolution of spatial fabric that involves rips or tears, thereby changing the topology of space.

Torus. The two-dimensional surface of a doughnut.

Two-brane. See brane.

Two-dimensional sphere. See sphere.

Type I string theory. One of the five superstring theories; involves both open and closed strings.

Type IIA string theory. One of the five superstring theories; involves closed strings with left-right symmetric vibrational patterns.

Type IIB string theory. One of the five superstring theories; involves closed strings with left-right asymmetric vibrational patterns.

Ultramicroscopic. Length scales shorter than the Planck length (and also time scales shorter than the Planck time).

Uncertainty principle. Principle of quantum mechanics, discovered by Heisenberg, that there are features of the universe, like the position and velocity of a particle, that cannot be known with complete precision. Such uncertain aspects of the microscopic world become ever more severe as the distance and time scales on which they are considered become ever smaller. Particles and fields undulate and jump between all possible values consistent with the quantum uncertainty. This implies that the microscopic realm is a roiling frenzy, awash in a violent sea of quantum fluctuations.

Unified theory, Unified field theory. Any theory that describes all four forces and all of matter within a single, all-encompassing framework.

Uniform vibration. The overall motion of a string in which it moves without changes in shape.

Velocity. The speed and the direction of an object's motion.

Vibrational mode. See vibrational pattern.

Vibrational pattern. The precise number of peaks and troughs as well as their amplitude as a string oscillates.

Vibration number. Whole number describing the energy in the uniform vibrational motion of a string; the energy in its overall motion as opposed to that associated with changes in its shape.

Virtual particles. Particles that erupt from the vacuum momentarily; they exist on borrowed energy, consistent with the uncertainty principle, and rapidly annihilate, thereby repaying the energy loan.

Wave function. Probability waves upon which quantum mechanics is founded.

Wavelength. The distance between successive peaks or troughs of a wave.

Wave-particle duality. Basic feature of quantum mechanics that objects manifest both wavelike and particle-like properties.

W bosons. See weak gauge boson.

Weak force, Weak nuclear force. One of the four fundamental forces, best known for mediating radioactive decay.

Weak gauge boson. Smallest bundle of the weak force field; messenger particle of the weak force; called W or Z boson.

Weak gauge symmetry. Gauge symmetry underlying the weak force.

Weakly coupled. Theory whose string coupling constant is less than 1.

Winding energy. The energy embodied by a string wound around a circular dimension of space.

Winding mode. A string configuration that wraps around a circular spatial dimension.

Winding number. The number of times a string is wound around a circular spatial dimension.

World-sheet. Two-dimensional surface swept out by a string as it moves.

Wormhole. A tube-like region of space connecting one region of the universe to another.

Z boson. See weak gauge boson.

Zero-dimensional sphere. See sphere.

References and Suggestions for Further Reading

Abbot, Edwin A. Flatland: A Romance of Many Dimensions. Princeton: Princeton University Press, 1991.

Barrow, John D. Theories of Everything. New York: Fawcett-Columbine, 1992.

Bronowski, Jacob. The Ascent of Man. Boston: Little, Brown, 1973.

Clark, Ronald W. Einstein, The Life and Times. New York: Avon, 1984.

Crease, Robert P., and Charles C. Mann. The Second Creation. New Brunswick, N.J.: Rutgers University Press, 1996.

Davies, P. C. W. Superforce. New York: Simon & Schuster, 1984.

Davies, P. C. W., and J. Brown, eds. Superstrings: A Theory of Everything? Cambridge, Eng.: Cambridge University Press, 1988.

Deutsch, David. The Fabric of Reality. New York: Allen Lane, 199 7.

Einstein, Albert. The Meaning of Relativity. Princeton: Princeton University Press, 1988.

——. Relativity. New York: Crown, 1961.

Ferris, Timothy Coming of Age in the Milky Way New York: Anchor, 1989.

——. The Whole Shebang. New York: Simon & Schuster, 1997.

Fölsing, Albrecht. Albert Einstein. New York: Viking, 1997.

Feynman, Richard, The Character of Physical Law. Cambridge, Mass.: MIT Press, 1995.

Gamow, George. Mr. Tompkins in Paperback. Cambridge, Eng.: Cambridge University Press, 1993.

Gell-Mann, Murray. The Quark and the Jaguar. New York: Freeman, 1994.

Glashow, Sheldon. Interactions. New York: Time-Warner Books, 1988.

Guth, Alan H. The Inflationary Universe. Reading, Mass.: Addison-Wesley, 1997.

Hawking, Stephen. A Brief History of Time. New York: Bantam Books, 1988.

Hawking, Stephen, and Roger Penrose. The Nature of Space and Time. Princeton: Princeton University Press, 1996.

Hey, Tony, and Patrick Walters. Einstein's Mirror. Cambridge, Eng.: Cambridge University Press, 1997.

Kaku, Michio. Beyond Einstein. New York: Anchor, 1987.

——. Hyperspace. New York: Oxford University Press, 1994.

Lederman, Leon, with Dick Teresi. The God Particle. Boston: Houghton Mifflin, 1993.

Lindley, David. The End of Physics. New York: Basic Books, 1993.

——. Where Does the Weirdness Go? New York: Basic Books, 1996.

Overbye, Dennis, Lonely Hearts of the Cosmos. New York: HarperCollins, 199 1.

Pais, Abraham. Subtle Is the Lord: The Science and the Life of Albert Einstein. New York: Oxford University Press, 1982.

Penrose, Roger. The Emperor's New Mind. Oxford, Eng.: Oxford University Press, 1989.

Rees, Martin J. Before the Beginning. Reading, Mass.: Addison-Wesley, 1997.

Smolin, Lee. The Life of the Cosmos. New York: Oxford University Press, 1997.

Thorne, Kip. Black Holes and Time Warps. New York: Norton, 1994.

Weinberg, Steven. The First Three Minutes. New York: Basic Books, 1993.

——. Dreams of a Final Theory. New York: Pantheon, 1992.

Wheeler, John A. A Journey into Gravity and Spacetime. New York: Scientific American Library, 1990.