The Fabric of the Cosmos: Space, Time, and the Texture of Reality (75 page)

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.

Ordinary photographs look flat because they record only light intensity. To get depth, you need phase information. The reason is that as a light wave travels, it cycles from peak to trough to peak again, and so phase information—or, more precisely, phase differences between light beams that reflect off nearby parts of an object—encodes differences in how far the light rays have traveled. For example, if you look at a cat straight on, its eyes are a little farther away than its nose and this depth difference is encoded in the phase difference between the light beams' reflecting off each facial element. By shining a laser through a hologram, we are able to exploit the phase information the hologram records, and thereby add depth to the image. We've all seen the results: stunning three-dimensional projections generated from two-dimensional pieces of plastic. Note, though, that your eyes do not use phase information to see depth. Instead, your eyes use parallax: the slight difference in the angles at which light from a given point travels to reach your left eye and your right eye supplies information that your brain decodes into the point's distance. That's why, for example, if you lose sight in one eye (or just keep it closed for a while), your depth perception is compromised.

7. For the mathematically inclined reader, the statement here is that a beam of light, or massless particles more generally, can travel from any point in the interior of antideSitter space to spatial infinity and back, in finite time.

8. For the mathematically inclined reader, Maldacena worked in the context of AdS
₅
× S
⁵
, with the boundary theory arising from the boundary of AdS
₅
.

9. This statement is more one of sociology than of physics. String theory grew out of the tradition of quantum particle physics, while loop quantum gravity grew out of the tradition of general relativity. However, it is important to note that, as of today, only string theory can make contact with the successful predictions of general relativity, since only string theory convincingly reduces to general relativity on large distance scales. Loop quantum gravity is understood well in the quantum domain, but bridging the gap to large-scale phenomena has proven difficult.

10. More precisely, as discussed further in Chapter 13 of
The Elegant Universe,
we have known how much entropy black holes contain since the work of Bekenstein and Hawking in the 1970s. However, the approach those researchers used was rather indirect, and never identified microscopic rearrangements—as in Chapter 6—that would account for the entropy they found. In the mid-1990s, this gap was filled by two string theorists, Andrew Strominger and Cumrun Vafa, who cleverly found a relation between black holes and certain configurations of branes in string/M-theory. Roughly, they were able to establish that certain special black holes would admit exactly the same number of rearrangements of their basic ingredients (whatever those ingredients might be) as do particular, special combinations of branes. When they counted the number of such brane rearrangements (and took the logarithm) the answer they found was the area of the corresponding black hole, in Planck units, divided by 4—exactly the answer for black hole entropy that had been found years before. In loop quantum gravity, researchers have also been able to show that the entropy of a black hole is proportional to its surface area, but getting the exact answer (surface area in Planck units divided by 4) has proven more of a challenge. If a particular parameter, known as the Immirzi parameter, is chosen appropriately, then indeed the exact black hole entropy emerges from the mathematics of loop quantum gravity, but as yet there is no universally accepted fundamental explanation, within the theory itself, of what sets the correct value of this parameter.

11. As I have throughout the chapter, I am suppressing quantitatively important but conceptually irrelevant numerical parameters.

absolute space: Newton's view of space; envisions space as unchanging and independent of its contents.

absolute spacetime: View of space emerging from special relativity; envisions space through the entirety of time, from any perspective, as unchanging and independent of its contents.

absolutist: Perspective holding that space is absolute.

acceleration: Motion that involves a change in speed and/or direction.

accelerator, atom smasher: Research tool of particle physics that collides particles together at high speed.

aether, luminiferous aether: Hypothetical substance filling space that provides the medium for light to propogate; discredited.

arrow of time: Direction in which time seems to point—from past to future.

background independence: Property of a physical theory in which space and time emerge from a more fundamental concept, rather than being inserted axiomatically.

big bang theory/standard big bang theory: Theory describing a hot, expanding universe from a moment after its birth.

big crunch: One possible end to the universe, analogous to a reverse of the big bang in which space collapses in on itself.

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

braneworld scenario: Possibility within string/M-theory that our familiar three-spatial dimensions are a three-brane.

Casimir force: Quantum mechanical force exerted by an imbalance of vacuum field fluctuations.

classical physics: As used in this book, the physical laws of Newton and Maxwell. More generally, often used to refer to all nonquantum laws of physics, including special and general relativity.

closed strings: Filaments of energy in string theory, in the shape of loops.

collapse of probability wave, collapse of wavefunction: Hypothetical development in which a probability wave (a wavefunction) goes from a spread-out to a spiked shape.

Copenhagen interpretation: Interpretation of quantum mechanics that envisions large objects as being subject to classical laws and small objects as being subject to quantum laws.

cosmic microwave background radiation: Remnant electromagnetic radiation (photons) from the early universe, which permeates space.

cosmic horizon, horizon: Locations in space beyond which light has not had time to reach us, since the beginning of the universe.

cosmological constant: A hypothetical energy and pressure, uniformly filling space; origin and composition unknown.

cosmology: Study of origin and evolution of the universe. critical density: Amount of mass/energy density required for space to be flat; about 10
^-23
grams per cubic meter.

D-branes, Dirichlet
-p-
branes: A
p-
brane that is "sticky"; a
p-
brane to which open string endpoints are attached.

dark energy: A hypothetical energy and pressure, uniformly filling space; more general notion than a cosmological constant as its energy/pressure can vary with time.

dark matter: Matter suffused through space, exerting gravity but not emitting light.

electromagnetic field: The field which exerts the electromagnetic force.

electromagnetic force: One of nature's four forces; acts on particles that have electric charge.

electron field: The field for which the electron particle is the smallest bundle or constituent.

electroweak theory: The theory unifying the electromagnetic and the weak nuclear forces into the electroweak force.

electroweak Higgs field: Field that acquires a nonzero value in cold, empty space; gives rise to masses for fundamental particles.

energy bowl: See
potential energy bowl.

entropy: A measure of the disorder of a physical system; the number of rearrangements of a system's fundamental constituents that leave its gross, overall appearance unchanged.

entanglement, quantum entanglement: Quantum phenomenon in which spatially distant particles have correlated properties.

event horizon: Imaginary sphere surrounding a black hole delineating the points of no return; anything crossing the event horizon cannot escape the black hole's gravity. field: A "mist" or "essence" permeating space; can convey a force or describe the presence/motion of particles. Mathematically, involves a number or collection of numbers at each point in space, signifying the field's value.

flat space: Possible shape of the spatial universe having no curvature.

flatness problem: Challenge for cosmological theories to explain observed flatness of space.

general relativity: Einstein's theory of gravity; invokes curvature of space and time.

gluons: Messenger particles of the strong nuclear force.

gravitons: Hypothetical messenger particles of the gravitational force.

grand unification: Theory attempting to unify the strong, weak, and electromagnetic forces.

Higgs field: See
electroweak Higgs field.

Higgs field vacuum expectation value: Situation in which a Higgs field acquires a nonzero value in empty space; a Higgs ocean.

Higgs ocean: Shorthand, peculiar to this book, for a Higgs field vacuum expectation value.

Higgs particles: Finest quantum constituents of a Higgs field.

horizon problem: Challenge for cosmological theories to explain how regions of space, beyond each other's cosmological horizon, have nearly identical properties.

inertia: Property of an object that resists its being accelerated.

inflationary cosmology: Cosmological theory incorporating a brief but enormous burst of spatial expansion in the early universe.

inflaton field: The field whose energy and negative pressure drives inflationary expansion.

interference: Phenomenon in which overlapping waves create a distinctive pattern; in quantum mechanics, involves seemingly exclusive alternatives combining together.

Kaluza-Klein theory: Theory of universe involving more than three spatial dimensions.

Kelvin: Scale in which temperatures are quoted relative to absolute zero (the lowest possible temperature, -273° on the Celsius scale). luminiferous aether: See
aether.

M-theory: Currently incomplete theory unifying all five versions of string theory; a fully quantum mechanical theory of all forces and all matter.

Mach's principle: Principle that all motion is relative and that the standard of rest is provided by average mass distribution in the universe.

Many Worlds interpretation: Interpretation of quantum mechanics in which all potentialities embodied by a probability wave are realized in separate universes.

messenger particle: Smallest "packet" or "bundle" of a force, which communicates the forces' influence.

microwave background radiation: See
cosmic microwave background radiation.

negative curvature: Shape of space containing less than the critical density; saddle-shaped.

observable universe: Part of universe within our cosmic horizon; part of universe close enough so that light it emitted can have reached us by today; part of universe we can see.

open strings: Filaments of energy in string theory, in the shape of snippets.
p-
brane: Ingredient of string/M-theory with p-spatial dimensions. See also D-brane.

Planck length: Size (10
^-33
centimeters) below which the conflict between quantum mechanics and general relativity becomes manifest; size below which conventional notion of space breaks down.

Planck mass: Mass (10
^-5
grams, mass of a grain of dust; ten billion billion times the proton mass); typical mass of a vibrating string.

Planck time: Time (10
^-43
seconds) it takes light to traverse one Planck length; time interval below which conventional notion of time breaks down.

phase transition: Qualitative change in a physical system when its temperature is varied through a sufficiently wide range.

photon: Messenger particle of the electromagnetic force; a "bundle" of light.

potential energy: Energy stored in a field or object.

potential energy bowl: Shape describing the energy a field contains for a given field value; technically called the field's potential energy.

probability wave: Wave in quantum mechanics that encodes the probability that a particle will be found at a given location.

quantum chromodynamics: Quantum mechanical theory of the strong nuclear force.

quantum fluctuations, quantum jitters: The unavoidable, rapid variations in the value of a field on small scales, arising from quantum uncertainty.

quantum measurement problem: Problem of explaining how the myriad possibilities encoded in a probability wave give way to a single outcome when measured.

quantum mechanics: Theory, developed in the 1920s and 1930s, for describing the realm of atoms and subatomic particles.

quarks: Elementary particles subject to the strong nuclear force; there are six varieties (up, down, strange, charm, top, bottom).

relationist: Perspective holding that all motion is relative and space is not absolute.

rotational invariance, rotational symmetry: Characteristic of a physical system, or of a theoretical law, of being unaffected by a rotation.

second law of thermodynamics: Law that says that, on average, the entropy of a physical system will tend to rise from any given moment.

spacetime: The union of space and time first articulated by special relativity.

special relativity: Einstein's theory in which space and time are not individually absolute, but instead depend upon the relative motion between distinct observers. spin: Quantum mechanical property of elementary particles in which, somewhat like a top, they undergo rotational motion (they have intrinsic angular momentum).

spontaneous symmetry breaking: Technical name for the formation of a Higgs ocean; process by which a previously manifest symmetry is hidden or spoiled.

standard candles: Objects of a known intrinsic brightness that are useful for measuring astronomical distances.

standard model: Quantum mechanical theory composed of quantum chromodynamics and the electroweak theory; describes all matter and forces, except for gravity. Based on conception of point particles.

strong nuclear force: Force of nature that influences quarks; holds quarks together inside protons and neutrons.

string theory: Theory based on one-dimensional vibrating filaments of energy (see superstring theory), but which does not necessarily incorporate supersymmetry. Sometimes used as shorthand for superstring theory.

superstring theory: Theory in which fundamental ingredients are one-dimensional loops (closed strings) or snippets (open strings) of vibrating energy, which unites general relativity and quantum mechanics; incorporates supersymmetry.

supersymmetry: A symmetry in which laws are unchanged when particles with a whole number amount of spin (force particles) are interchanged with particles that have half of a whole number amount of spin (matter particles).

symmetry: A transformation on a physical system that leaves the system's appearance unchanged (e.g., a rotation of a perfect sphere about its center leaves the sphere

unchanged); a transformation of a physical system that has no effect on the laws describing the system.

time-reversal symmetry: Property of the accepted laws of nature in which laws make no distinction between one direction in time and the other. From any given moment, the laws treat past and future in exactly the same way.

time slice: All of space at one moment of time; a single slice through the spacetime block or loaf.

translational invariance, translational symmetry: Property of accepted laws of nature in which the laws are applicable at any location in space.

uncertainty principle: Property of quantum mechanics in which there is a fundamental limit on how precisely certain complementary physical features can be measured or specified.

unified theory: A theory that describes all forces and all matter in a single theoretical structure.

vacuum: The emptiest that a region can be; the state of lowest energy.

vacuum field fluctuations: See
quantum fluctuations.
velocity: The speed and direction of an object's motion.

W and Z particles: The messenger particles of the weak nuclear force.

wavefunction: See
probability wave.

weak nuclear force: Force of nature, acting on subatomic scales, and responsible for phenomena such as radioactive decay.

which-path information: Quantum mechanical information delineating the path a particle took in going from source to detector.