The Fabric of the Cosmos: Space, Time, and the Texture of Reality

by Brian Greene

  12. Quantum mechanics ensures that there is always a nonzero probability that a chance fluctuation will disrupt the cyclic process (e.g., one brane twists relative to the other), causing the model to grind to a halt. Even if the probability is minuscule, sooner or later it will surely come to pass, and hence the cycles cannot continue indefinitely.

  Chapter 14

  1. A. Einstein, "Vierteljahrschrift für gerichtliche Medizin und öffentliches Sanitätswesen" 44 37 (1912). D. Brill and J. Cohen, Phys. Rev. vol. 143, no. 4, 1011 (1966); H. Pfister and K. Braun, Class. Quantum Grav. 2, 909 (1985).

  2. In the four decades since the initial proposal of Schiff and Pugh, other tests of frame dragging have been undertaken. These experiments (carried out by, among others, Bruno Bertotti, Ignazio Ciufolini, and Peter Bender; and I. I. Shapiro, R. D. Reasenberg, J. F. Chandler, and R. W. Babcock) have studied the motion of the moon as well as satellites orbiting the earth, and found some evidence for frame dragging effects. One major advantage of Gravity Probe B is that it is the first fully contained experiment, one that is under complete control of the experimenters, and so should give the most precise and most direct evidence for frame dragging.

  3. Although they are effective at giving a feel for Einstein's discovery, another limitation of the standard images of warped space is that they don't illustrate the warping of time. This is important because general relativity shows that for an ordinary object like the sun, as opposed to something extreme like a black hole, the warping of time (the closer you are to the sun, the slower your clocks will run) is far more pronounced than the warping of space. It's subtler to depict the warping of time graphically and it's harder to convey how warped time contributes to curved spatial trajectories such as the earth's elliptical orbit around the sun, and that's why Figure 3.10 (and just about every attempt to visualize general relativity I've ever seen) focuses solely on warped space. But it's good to bear in mind that in many common astrophysical environments, it's the warping of time that is dominant.

  4. In 1974, Russell Hulse and Joseph Taylor discovered a binary pulsar system—two pulsars (rapidly spinning neutron stars) orbiting one another. Because the pulsars move very quickly and are very close together, Einstein's general relativity predicts that they will emit copious amounts of gravitational radiation. Although it is quite a challenge to detect this radiation directly, general relativity shows that the radiation should reveal itself indirectly through other means: the energy emitted via the radiation should cause the orbital period of the two pulsars to gradually decrease. The pulsars have been observed continuously since their discovery, and indeed, their orbital period has decreased—and in a manner that agrees with the prediction of general relativity to about one part in a thousand. Thus, even without direct detection of the emitted gravitational radiation, this provides strong evidence for its existence. For their discovery, Hulse and Taylor were awarded the 1993 Nobel Prize in Physics.

  5. However, see note 4, above.

  6. From the viewpoint of energetics, therefore, cosmic rays provide a naturally occurring accelerator that is far more powerful than any we have or will construct in the foreseeable future. The drawback is that although the particles in cosmic rays can have extremely high energies, we have no control over what slams into what—when it comes to cosmic ray collisions, we are passive observers. Furthermore, the number of cosmic ray particles with a given energy drops quickly as the energy level increases. While about 10 billion cosmic ray particles with an energy equivalent to the mass of a proton (about one-thousandth of the design capacity of the Large Hadron Collider) strike each square kilometer of earth's surface every second (and quite a few pass through your body every second as well), only about one of the most energetic particles (about 100 billion times the mass of a proton) would strike a given square kilometer of earth's surface each century. Finally, accelerators can slam particles together by making them move quickly, in opposite directions, thereby creating a large center of mass energy. Cosmic ray particles, by contrast, slam into the relatively slow moving particles in the atmosphere. Nevertheless, these drawbacks are not insurmountable. Over the course of many decades, experimenters have learned quite a lot from studying the more plentiful, lower-energy cosmic ray data, and, to deal with the paucity of high-energy collisions, experimenters have built huge arrays of detectors to catch as many particles as possible.

  7. The expert reader will realize that conservation of energy in a theory with dynamic spacetime is a subtle issue. Certainly, the stress tensor of all sources for the Einstein equations is covariantly conserved. But this does not necessarily translate into a global conservation law for energy. And with good reason. The stress tensor does not take account of gravitational energy—a notoriously difficult notion in general relativity. Over short enough distance and time scales—such as occur in accelerator experiments—local energy conservation is valid, but statements about global conservation have to be treated with greater care.

  8. This is true of the simplest inflationary models. Researchers have found that more complicated realizations of inflation can suppress the production of gravitational waves.

  9. A viable dark matter candidate must be a stable, or very long-lived, particle—one that does not disintegrate into other particles. This is expected to be true of the lightest of the supersymmetric partner particles, and hence the more precise statement is that the lightest of the zino, higgsino, or photino is a suitable dark matter candidate.

  10. Not too long ago, a joint Italian-Chinese research group known as the Dark Matter Experiment (DAMA), working out of the Gran Sasso Laboratory in Italy, made the exciting announcement that they had achieved the first direct detection of dark matter. So far, however, no other group has been able to verify the claim. In fact, another experiment, Cryogenic Dark Matter Search (CDMS), based at Stanford and involving researchers from the United States and Russia, has amassed data that many believe rule out the DAMA results to a high degree of confidence. In addition to these dark matter searches, many others are under way. To read about some of these, take a look at http://hepwww.rl.ac.uk/ukdmc/dark_matter/other_searches.htm .

  Chapter 15

  1. This statement ignores hidden-variable approaches, such as Bohm's. But even in such approaches, we'd want to teleport an object's quantum state (its wavefunction), so a mere measurement of position or velocity would be inadequate.

  2. Zeilinger's research group also included Dick Bouwmeester, Jian-Wi Pan, Klaus Mattle, Manfred Eibl, and Harald Weinfurter, and De Martini's has included S. Giacomini, G. Milani, F. Sciarrino, and E. Lombardi.

  3. For the reader who has some familiarity with the formalism of quantum mechanics, here are the essential steps in quantum teleportation. Imagine that the initial state of a photon I have in New York is given by | > 1 = |0> 1 + |1> 1 where |0> and |1> are the two photon polarization states, and we allow for definite, normalized, but arbitrary values of the coefficients. My goal is to give Nicholas enough information so that he can produce a photon in London in exactly the same quantum state. To do so, Nicholas and I first acquire a pair of entangled photons in the state, say | > 23 = (1/sqrt2) |0 2 0 3 > —(1/sqrt2)|1 2 1 3 >. The initial state of the three-photon system is thus | > 123 = ( /sqrt2) {|0 1 0 2 0 3 > — |0 1 1 2 1 3 >} + ( /sqrt2) {|1 1 0 2 0 3 > — |1 1 1 2 1 3 >}. When I perform a Bell-state measurement on Photons 1 and 2, I project this part of the system onto one of four states: | > ± = (1/sqrt2) {|0 1 0 2 > ± |1 1 1 2 >} and | >± = (1/sqrt2) {|0 1 1 2 > ± |1 1 0 2 >}. Now, if we re-express the initial state using this basis of eigenstates for Particles 1 and 2, we find: | > 123 = 1 /2{| > + ( |0 3 > — |1 3 >) + | > — ( |0 3 > + |1 3 >) + | >+ (— |1 3 > + |0 3 >) + | >— (— |1 3 > — |0 3 >}. Thus, after performing my measurement, I will "collapse" the system onto one of these four summands. Once I communicate to Nicholas (via ordinary means), which summand I find, he knows how to manipulate Photon 3 to reproduce the original state of Photon 1. For instance, if I find that my measurement yields state |
> — , then Nicholas does not need to do anything to Photon 3, since, as above, it is already in the original state of Photon 1. If I find any other result, Nicholas will have to perform a suitable rotation (dictated, as you can see, by which result I find), to put Photon 3 into the desired state.

  4. In fact, the mathematically inclined reader will note that it is not hard to prove the so-called no-quantum-cloning theorem. Imagine we have a unitary cloning operator U that takes any given state as input and produces two copies of it as output (U maps | > | >| >, for any input state | >). Note that U acting on a state like (| > + | >) yields (| >| > + | >| >), which is not a two-fold copy of the original state (| > + | >)(| > + | >), and hence no such operator U exists to carry out quantum cloning. (This was first shown by Wootters and Zurek in the early 1980s.)

  5. Many researchers have been involved in developing both the theory and the experimental realization of quantum teleportation. In addition to those discussed in the text, the work of Sandu Popescu while at Cambridge University played an important part in the Rome experiments, and Jeffrey Kimble's group at the California Institute of Technology has pioneered the teleportation of continuous features of a quantum state, to name a few.

  6. For extremely interesting progress on entangling many-particle systems, see, for example, B. Julsgaard, A. Kozhekin, and E. S. Polzik, "Experimental long-lived entanglement of two macroscopic objects," Nature 413 (Sept. 2001), 400-403.

  7. One of the most exciting and active areas of research making use of quantum entanglement and quantum teleportation is the field of quantum computing. For recent general-level presentations of quantum computing, see Tom Siegfried, The Bit and the Pendulum (New York: John Wiley, 2000), and George Johnson, A Shortcut Through Time (New York: Knopf, 2003).

  8. One aspect of the slowing of time at increasing velocity, which we did not discuss in Chapter 3 but will play a role in this chapter, is the so-called twin paradox. The issue is simple to state: if you and I are moving relative to one another at constant velocity, I will think your clock is running slow relative to mine. But since you are as justified as I in claiming to be at rest, you will think that mine is the moving clock and hence is the one that is running slow. That each of us thinks the other's clock is running slow may seem paradoxical, but it's not. At constant velocity, our clocks will continue to get farther apart and hence they don't allow for a direct, face-to-face comparison to determine which is "really" running slow. And all other indirect comparisons (for instance, we compare the times on our clocks by cell phone communication) occur with some elapsed time over some spatial separation, necessarily bringing into play the complications of different observers' notions of now, as in Chapters 3 and 5. I won't go through it here, but when these special relativistic complications are folded into the analysis, there is no contradiction between each of us declaring that the other's clock is running slow (see, e.g., E. Taylor and J. A. Wheeler, Spacetime Physics, for a complete, technical, but elementary discussion) . Where things appear to get more puzzling is if, for example, you slow down, stop, turn around, and head back toward me so that we can compare our clocks face to face, eliminating the complications of different notions of now. Upon our meeting, whose clock will be ahead of whose? This is the so-called twin paradox: if you and I are twins, when we meet again, will we be the same age, or will one of us look older? The answer is that my clock will be ahead of yours—if we are twins, I will look older. There are many ways to explain why, but the simplest to note is that when you change your velocity and experience an acceleration, the symmetry between our perspectives is lost—you can definitively claim that you were moving (since, for example, you felt it— or, using the discussion of Chapter 3, unlike mine, your journey through spacetime has not been along a straight line) and hence that your clock ran slow relative to mine. Less time elapsed for you than for me.

  9. John Wheeler, among others, has suggested a possible central role for observers in a quantum universe, summed up in one of his famous aphorisms: "No elementary phenomenon is a phenomenon until it is an observed phenomenon." You can read more about Wheeler's fascinating life in physics in John Archibald Wheeler and Kenneth Ford, Geons, Black Holes, and Quantum Foam: A Life in Physics (New York: Norton, 1998). Roger Penrose has also studied the relation between quantum physics and the mind in his The Emperor's New Mind, and also in Shadows of the Mind: A Search for the Missing Scienceof Consciousness (Oxford: Oxford University Press, 1994).

  10. See, for example, "Reply to Criticisms" in Albert Einstein, vol. 7 of The Library of Living Philosophers, P. A. Schilpp, ed. (New York: MJF Books, 2001).

  11. W. J. van Stockum, Proc. R. Soc. Edin. A 57 (1937), 135.

  12. The expert reader will recognize that I am simplifying. In 1966, Robert Geroch, who was a student of John Wheeler, showed that it is at least possible, in principle, to construct a wormhole without ripping space. But unlike the more intuitive, space-tearing approach to building wormholes in which the mere existence of the wormhole does not entail time travel, in Geroch's approach the construction phase itself would necessarily require that time become so distorted that one could freely travel backward and forward in time (but no farther back than the initiation of the construction itself).

  13. Roughly speaking, if you passed through a region containing such exotic matter at nearly the speed of light and took the average of all your measurements of the energy density you detected, the answer you'd find would be negative. Physicists say that such exotic matter violates the so-called averaged weak energy condition.

  14. The simplest realization of exotic matter comes from the vacuum fluctuations of the electromagnetic field between the parallel plates in the Casimir experiment, discussed in Chapter 12. Calculations show that the decrease in quantum fluctuations between the plates, relative to empty space, entails negative averaged energy density (as well as negative pressure).

  15. For a pedagogical but technical account of wormholes, see Matt Visser, LorentzianWormholes: From Einstein to Hawking (New York: American Institute of Physics Press, 1996).

  Chapter 16

  1. For the mathematically inclined reader, recall from note 6 of Chapter 6 that entropy is defined as the logarithm of the number of rearrangements (or states), and that's important to get the right answer in this example. When you join two Tupperware containers together, the various states of the air molecules can be described by giving the state of the air molecules in the first container, and then by giving the state of those in the second. Thus, the number of arrangements for the joined containers is the square of the number of arrangements of either separately. After taking the logarithm, this tells us that the entropy has doubled.

  2. You will note that it doesn't really make much sense to compare a volume with an area, as they have different units. What I really mean here, as indicated by the text, is that the rate at which volume grows with radius is much faster than the rate at which surface area grows. Thus, since entropy is proportional to surface area and not volume, it grows more slowly with the size of a region than it would were it proportional to volume.

  3. While this captures the spirit of the entropy bound, the expert reader will recognize that I am simplifying. The more precise bound, as proposed by Raphael Bousso, states that the entropy flux through a null hypersurface (with everywhere non-positive focusing parameter ) is bounded by A/4, where A is the area of a spacelike cross-section of the null hypersurface (the "light-sheet").

  4. More precisely, the entropy of a black hole is the area of its event horizon, expressed in Planck units, divided by 4, and multiplied by Boltzmann's constant.

  5. The mathematically inclined reader may recall from the endnotes to Chapter 8 that there is another notion of horizon—a cosmic horizon—which is the dividing surface between those things with which an observer can and cannot be in causal contact. Such horizons are also believed to support entropy, again proportional to their surface area.

  6. In 1971, the Hungarian-born physicist Dennis Gabor was awarded
the Nobel Prize for the discovery of something called holography. Initially motivated by the goal of improving the resolving power of electron microscopes, Gabor worked in the 1940s on finding ways to capture more of the information encoded in the light waves that bounce off an object. A camera, for example, records the intensity of such light waves; places where the intensity is high yield brighter regions of the photograph, and places where it's low are darker. Gabor and many others realized, though, that intensity is only part of the information that light waves carry. We saw this, for example, in Figure 4.2b: while the interference pattern is affected by the intensity (the amplitude) of the light (higher-amplitude waves yield an overall brighter pattern), the pattern itself arises because the overlapping waves emerging from each of the slits reach their peak, their trough, and various intermediate wave heights at different locations along the detector screen. The latter information is called phase information: two light waves at a given point are said to be in phase if they reinforce each other (they each reach a peak or trough at the same time), out of phase if they cancel each other (one reaches a peak while the other reaches a trough), and, more generally, they have phase relations intermediate between these two extremes at points where they partially reinforce or partially cancel. An interference pattern thus records phase information of the interfering light waves.

  Gabor developed a means for recording, on specially designed film, both the intensity and the phase information of light that bounces off an object. Translated into modern language, his approach is closely akin to the experimental setup of Figure 7.1, except that one of the two laser beams is made to bounce off the object of interest on its way to the detector screen. If the screen is outfitted with film containing appropriate photographic emulsion, it will record an interference pattern—in the form of minute, etched lines on the film's surface—between the unfettered beam and the one that has reflected off the object. The interference pattern will encode both the intensity of the reflected light and phase relations between the two light beams. The ramifications of Gabor's insight for science have been substantial, allowing for vast improvements in a wide range of measurement techniques. But for the public at large, the most prominent impact has been the artistic and commercial development of holograms.