The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory

by Brian Greene

  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.