The Fabric of the Cosmos: Space, Time, and the Texture of Reality (74 page)

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.