Read The Fabric of the Cosmos: Space, Time, and the Texture of Reality Online
Authors: Brian Greene
Tags: #Science, #Cosmology, #Popular works, #Astronomy, #Physics, #Universe
The Fabric of the Cosmos: Space, Time, and the Texture of Reality (64 page)
19
To go beyond the two-dimensional metaphor of a balloon's surface and have a spherical three-dimensional model is easy mathematically but difficult to picture, even for professional mathematicians and physicists. You might be tempted to think of a solid, three-dimensional ball, like a bowling ball without the finger holes. This, however, isn't an acceptable shape. We want all points in the model to be on a completely equal footing, since we believe that every place in the universe is (on average) just like any other. But the bowling ball has all sorts of different points: some are on the outside surface, others are embedded in the interior, one is right in the center. Instead, just as the two-dimensional surface of a balloon surrounds a
three-
dimensional spherical region (containing the balloon's air), an acceptable round three-dimensional shape would need to surround a
four-
dimensional spherical region. So the three-dimensional spherical surface of a balloon in a four-dimensional space is an acceptable shape. But if that still leaves you groping for an image, do what just about all professionals do: stick to the easy-to-visualize lower-dimensional analogies. They capture almost all of the essential features. A bit further on, we consider three-dimensional flat space, as opposed to the round shape of a sphere, and that flat space can be visualized.
20
Depending on whether the rate of the universe's expansion is speeding up or slowing down over time, the light emitted from such galaxies may fight a battle that would have made Zeno proud: the light may stream toward us at light speed while the expansion of space makes the distance the light has yet to cover ever larger, preventing the light from ever reaching us. See notes section for details.10
21
Just as the video game screen gives a finite-sized version of flat space that has no edges or boundaries, there are finite-sized versions of the saddle shape that also have no edges or boundaries. I won't discuss this further, save to note that it implies that all three possible curvatures (positive, zero, negative) can be realized by finite-sized shapes without edges or boundaries. (In principle, then, a space-faring Magellan could carry out a cosmic version of his voyage in a universe whose curvature is given by any of the three possibilities.)
22
Today, matter in the universe is more abundant than radiation, so it's convenient to express the critical density in units most relevant for mass—grams per cubic meter. Note too that while 10
-23
grams per cubic meter might not sound like a lot, there are many cubic meters of space out there in the cosmos. Moreover, the farther back in time you look, the smaller the space into which the mass/energy is squeezed, so the denser the universe becomes.
23
Even though a decrease in symmetry means that fewer manipulations go unnoticed, the heat released to the environment during these transformations ensures that overall entropy—including that of the environment—still increases.
24
The terminology isn't particularly important, but briefly, here's where it comes from. The valley in Figure 9.1c and 9.1d has a symmetric shape—it's circular—with every point being on a par with every other (each point denotes a Higgs field value of lowest possible energy). Yet, when the Higgs field's value slides down the bowl, it lands on
one
particular point on the circular valley, and in so doing "spontaneously" selects one location on the valley as special. In turn, the points on the valley are no longer all on an equal footing, since one has been picked out, and so the Higgs field disrupts or "breaks" the previous symmetry between them. Thus, putting the words together, the process in which the Higgs slides down to one particular nonzero value in the valley is called
spontaneous symmetrybreaking.
Later in the text, we will describe more tangible aspects of the reduction of symmetry associated with such a formation of a Higgs ocean.
7
25
You might think I've left out an "i" in the last syllable of "inflaton," but I haven't; physicists often give fields names, such as photon and gluon, which end with "on."
26
As the universe expands, the energy loss of photons can be directly observed because their wavelengths stretch—they undergo
redshift—
and the longer a photon's wavelength, the less energy it has. The microwave background photons have undergone such redshift for nearly 14 billion years, explaining their long—microwave—wavelengths, and their low temperature. Matter undergoes a similar loss of its kinetic energy (energy from particle motion), but the total energy bound up in the mass of particles (their
rest energy—
the energy equivalent of their mass, when at rest) remains constant.
27
While useful, the rubber-band analogy is not perfect. The inward, negative pressure exerted by the rubber bands impedes the expansion of the box, whereas the inflaton's negative pressure drives the expansion of space. This important difference illustrates the clarification emphasized on page 278: in cosmology, it is not that uniform negative pressure drives expansion (only pressure differences result in forces, so uniform pressure, whether positive or negative, exerts
no
force). Rather, pressure, like mass, gives rise to a gravitational force. And negative pressure gives rise to a repulsive gravitational force that drives expansion. This does not affect our conclusions.
28
Some researchers, including Alan Guth and Eddie Farhi, have investigated whether one might, hypothetically, create a new universe in the laboratory by synthesizing a nugget of inflaton field. Beyond the fact that we still don't have direct experimental verification that there is such a thing as an inflaton field, note that the twenty pounds of inflaton field would need to be crammed in a tiny space, roughly 10
-26
or so centimeters on a side, and hence the density would be enormous—some 10
67
times the density of an atomic nucleus—way beyond what we can produce, now or perhaps ever.
29
Don't get confused here: The inflationary stretching of quantum jitters discussed in the last section still produced a minuscule, unavoidable nonuniformity of about 1 part in 100,000. But that tiny nonuniformity overlaid an otherwise smooth universe. We are now describing how the latter—the underlying smooth uniformity—came to be.
30
For ease of writing, we'll consider only fields that reach their lowest energy when their values are zero. The discussion for other fields—Higgs fields—is identical, except the jitters fluctuate about the field's
nonzero,
lowest-energy value. If you are tempted to say that a region of space is empty only if there is no matter present and all fields are
absent,
not just that they have the value zero, see notes section.
2
31
The remainder of this chapter recounts the discovery of superstring theory and discusses the theory's essential ideas regarding unification and the structure of spacetime. Readers of
The Elegant Universe
(especially Chapters 6 through 8) will be familiar with much of this material, and should feel free to skim this chapter and move on to the next.
32
Remember, as noted in Chapter 9, even a puny magnet can overpower the pull of the entire earth's gravity and pick up a paper clip. Numerically, the gravitational force has about 10
-42
times the strength of the electromagnetic force.
33
I might note that the proponents of another approach for merging general relativity and quantum mechanics,
loop quantum gravity,
to be briefly discussed in Chapter 16, take a viewpoint that is closer to the former conjecture—that spacetime has a discrete structure on the smallest of scales.
34
The relationship to mass arising from a Higgs ocean will be discussed later in the chapter.
35
Were you to count left, right, clockwise, and counterclockwise all separately, you'd conclude that the worm can move in four directions. But when we speak of "independent" directions, we always group those that lie along the same geometrical axis—like left and right, and also clockwise and counterclockwise.
36
Let me prepare you for one relevant development we will encounter in the next chapter. String theorists have known for decades that the equations they generally use to mathematically analyze string theory are approximate (the exact equations have proven difficult to identify and understand). However, most thought that the approximate equations were sufficiently accurate to determine the required number of extra dimensions. More recently (and to the shock of most physicists in the field), some string theorists showed that the approximate equations
missed
one dimension; it is now accepted that the theory needs
seven
extra dimensions. As we will see, this does not compromise the material discussed in this chapter, but shows that it fits within a larger, in fact more unified, framework.
20
37
The more precise name for these sticky entities is
Dirichlet-p-branes,
or
D-p-branes
for short. We will stick with the shorter
p-brane.
38
There is even a proposal, from Lisa Randall, of Harvard, and Raman Sundrum, of Johns Hopkins, in which gravity too can be trapped, not by a sticky brane, but by extra dimensions that curve in just the right way, relaxing even further the constraints on their size.
39
One of these is the planned Laser Interferometer Space Antenna (LISA), a space-based version of LIGO comprising multiple spacecraft, separated by millions of kilometers, playing the role of LIGO's four-kilometer tubes. There are also other detectors that are playing a critical role in the search for gravitational waves, including the German-British detector GEO600, the French-Italian detector VIRGO, and the Japanese detector TAMA300.
40
Since teleportation starts with something here and seeks to make it appear at a distant location, in this section I will often speak as if particles have definite positions. To be more precise, I should always say, "starting with a particle that has a high likelihood of being located here" or "starting with a particle with a 99 percent chance of being located here," with similar language used where the particle is teleported, but for brevity's sake I will use the looser language.
41
For collections of particles—as opposed to individual particles—the quantum state also encodes the relationship of each particle in the collection to every other. So, by exactly reproducing the quantum state of the particles making up the DeLorean, we ensure that they all stand in the same relation to each other; the only change they experience is that their overall location would have been shifted from New York to London.
42
The fragility of the human body is another practical limitation: the acceleration required to reach such high speeds in a reasonable length of time is well beyond what the body can withstand. Note, too, that the slowing of time gives a strategy, in principle, for reaching distant locations in space. If a rocket were to leave earth and head for the Andromeda galaxy, traveling at 99.999999999999999999 percent of light speed, we'd have to wait nearly 6 million years for it to return. But at that speed, time on the rocket slows down relative to time on earth so dramatically that upon returning the astronaut would have aged only eight hours (setting aside the fact that he or she couldn't have survived the accelerations to get up to speed, turn back, and finally stop).
43
Of course, I really should say January 1, 1966, but let's not worry about that.
44
For details on geometrical duality involving both circles and Calabi-Yau shapes, see
The Elegant Universe,
Chapter 10.
45
If you're reluctant to rewrite Plato, the braneworld scenario gives a version of holography in which shadows are put back in their proper place. Imagine that we live on a three-brane that surrounds a region with four space dimensions (much as the two-dimensional skin of an apple surrounds the apple's three-dimensional interior). The holographic principle in this setting would say that our three-dimensional perceptions would be the shadows of four-dimensional physics taking place in the region surrounded by our brane.
Chapter 1
1. Lord Kelvin was quoted by the physicist Albert Michelson during his 1894 address at the dedication of the University of Chicago's Ryerson Laboratory (see D. Kleppner,
Physics Today,
November 1998).
2. Lord Kelvin, "Nineteenth Century Clouds over the Dynamical Theory of Heat and Light,"
Phil. Mag.
II—6th series, 1 (1901).
3. A. Einstein, N. Rosen, and B. Podolsky,
Phys. Rev.
47, 777 (1935).
4. Sir Arthur Eddington,
The Nature of the Physical World
(Cambridge, Eng.: Cambridge University Press, 1928).
5. As described more fully in note 2 of Chapter 6, this is an overstatement because there are examples, involving relatively esoteric particles (such as K-mesons and B-mesons), which show that the so-called weak nuclear force does not treat past and future fully symmetrically. However, in my view and that of many others who have thought about it, since these particles play essentially no role in determining the properties of everyday material objects, they are unlikely to be important in explaining the puzzle of time's arrow (although, I hasten to add, no one knows this for sure). Thus, while it is technically an overstatement, I will assume throughout that the error made in asserting that the laws treat past and future on equal footing is minimal—at least as far as explaining the puzzle of time's arrow is concerned.
6. Timothy Ferris,
Coming of Age in the Milky Way
(New York: Anchor, 1989).
Chapter 2
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, 1934), vol. 1, p. 10.
2. Ibid., p. 6.
3. Ibid.
4. Ibid., p. 12.
5. Albert Einstein, in Foreword to Max Jammer,
Concepts of Space: The Histories of
Theories of Space in Physics
(New York: Dover, 1993).
6. A. Rupert Hall,
Isaac Newton, Adventurer in Thought
(Cambridge, Eng.: Cambridge University Press, 1992), p. 27.
7. Ibid.
8. H. G. Alexander, ed.,
The Leibniz-Clarke Correspondence
(Manchester: Manchester University Press, 1956).
9. I am focusing on Leibniz as the representative of those who argued against assigning space an existence independent of the objects inhabiting it, but many others also strenuously defended this view, among them Christiaan Huygens and Bishop Berkeley.
10. See, for example, Max Jammer, p. 116.
11. V. I. Lenin, Materialism and Empiriocriticism: Critical Comments on a Reac
tionaryPhilosophy
(New York: International Publications, 1909). Second English ed. of
Materializm' i Empiriokrititsizm': Kriticheskia Zametki ob' Odnoi Reaktsionnoi Filosofii
(Moscow: Zveno Press, 1909).
Chapter 3
1. For the mathematically trained reader, these four equations are
denote the electric field, the magnetic field, the electric charge density, the electric current density, the permittivity of free space, and the permeability of free space, respectively. As you can see, Maxwell's equations relate the rate of change of the electromagnetic fields to the presence of electric charges and currents. It is not hard to show that these equations imply a speed for electromagnetic waves given by 1/sqrt 0 , which when evaluated is in fact the speed of light.
2. There is some controversy as to the role such experiments played in Einstein's development of special relativity. In his biography of Einstein,
Subtle Is the Lord: The Science and the Life of Albert Einstein
(Oxford: Oxford University Press, 1982), pp. 115-19, Abraham Pais has argued, using Einstein's own statements from his later years, that Einstein was aware of the Michelson-Morley results. Albrecht Fölsing in
Albert Einstein: A
Biography
(New York: Viking, 1997), pp. 217-20, also argues that Einstein was aware of the Michelson-Morley result, as well as earlier experimental null results in searching for evidence of the aether, such as the work of Armand Fizeau. But Fölsing and many other historians of science have also argued that such experiments played, at best, a secondary role in Einstein's thinking. Einstein was primarily guided by considerations of mathematical symmetry, simplicity, and an uncanny physical intuition.
3. For us to see anything, light has to travel to our eyes; similarly, for us to see light, the light itself would have to make the same journey. So, when I speak of Bart's seeing light that is speeding away, it is shorthand. I am imagining that Bart has a small army of helpers, all moving at Bart's speed, but situated at various distances along the path that he and the light beam follow. These helpers give Bart updates on how far ahead the light has sped and the time at which the light reached such distant locations. Then, on the basis of this information, Bart can calculate how fast the light is speeding away from him.
4. There are many elementary mathematical derivations of Einstein's insights on space and time arising from special relativity. If you are interested, you can, for example, take a look at Chapter 2 of
The Elegant Universe
(together with mathematical details given in the endnotes to that chapter). A more technical but extremely lucid account is Edwin Taylor and John Archibald Wheeler,
Spacetime Physics: Introduction to Special Relativity
(New York, W. H. Freeman & Co., 1992).
5. The stopping of time at light speed is an interesting notion, but it is important not to read too much into it. Special relativity shows that no material object can ever attain light speed: the faster a material object travels, the harder we'd have to push it to further increase its speed. Just shy of light speed, we'd have to give the object an essentially infinitely hard push for it to go any faster, and that's something we can't ever do. Thus, the "timeless" photon perspective is limited to
massless
objects (of which the photon is an example), and so "timelessness" is permanently beyond what all but a few types of particle species can ever attain. While it is an interesting and fruitful exercise to imagine how the universe would appear when moving at light speed, ultimately we need to focus on perspectives that material objects, such as ourselves, can reach, if we want to draw inferences about how special relativity affects our experiential conception of time.
6. See Abraham Pais,
Subtle Is the Lord,
pp. 113-14.
7. To be more precise, we
define
the water to be spinning if it takes on a concave shape, and not spinning if it doesn't. From a Machian perspective, in an empty universe there is no conception of spinning, so the water's surface would always be flat (or, to avoid issues of the lack of gravity pulling on the water, we can say that the tension on the rope tied between two rocks will always be slack). The statement here is that, by contrast, in special relativity there
is
a notion of spinning, even in an empty universe, so that the water's surface can be concave (and the tension on the rope tied between the rocks can be taut). In this sense, special relativity violates Mach's ideas.
8. Albrecht Fölsing,
Albert Einstein
(New York: Viking Press, 1997), pp. 208-10.
9. The mathematically inclined reader will note that if we choose units so that the speed of light takes the form of one space unit per one time unit (like one light-year per year or one light-second per second, where a light-year is about 6 trillion miles and a light-second is about 186,000 miles), then light moves through spacetime on 45-degree rays (because such diagonal lines are the ones which cover one space unit in one time unit, two space units in two time units, etc.). Since nothing can exceed the speed of light, any material object must cover less distance in space in a given interval of time than would a beam of light, and hence the path it follows through spacetime must make an angle with the centerline of the diagram (the line running through the center of the loaf from crust to crust) that is less than 45 degrees. Moreover, Einstein showed that the time slices for an observer moving with velocity
v
—all of space at one moment of such an observer's time—have an equation (assuming one space dimension for simplicity) given by t
moving
= (t
stationary
— (v/c
2
) x
stationary
), where = ( 1 — v
2
/c
2
)
-1/2
, and c is the velocity of light. In units where c = 1, we note that < 1 and hence a time slice for the moving observer—the locus where t
moving
takes on a fixed value—is of the form (t
stationary
— vx
stationary
) = constant. Such time slices are angled with respect to the stationary time slices (the loci of the form t
stationary
= constant), and because v < 1, the angle between them is less than 45 degrees.
10. For the mathematically inclined reader, the statement being made is that the geodesics of Minkowski's spacetime—the paths of extremal spacetime length between two given points—are geometrical entities that do not depend on any particular choice of coordinates or frame of reference. They are intrinsic, absolute, geometric spacetime features. Explicitly, using the standard Minkowski metric, the (timelike) geodesics are straight lines (whose angle with respect to the time axis is less than 45 degrees, since the speed involved is less than that of light).
11. There is something else of importance that all observers, regardless of their motion, also agree upon. It's implicit in what we've described, but it's worth stating directly. If one event is the cause of another (I shoot a pebble, causing a window to break), all observers agree that the cause happened
before
the effect (all observers agree that I shot the pebble
before
the window broke). For the mathematically inclined reader, it is actually not difficult to see this using our schematic depiction of spacetime. If event A is the cause of event B, then a line drawn from A to B intersects each of the time slices (time slices of an observer at rest with respect to A) at an angle that is
greater
than 45 degrees (the angle between the space axes—axes that lie on any given time slice—and the line between A and B is greater than 45 degrees). For instance, if A and B take place at the same location in space (the rubber band wrapped around my finger [A] causes my finger to turn white [B]) then the line connecting A and B makes a 90-degree angle relative to the time slices. If A and B take place at different locations in space, whatever traveled from A to B to exert the influence (my pebble traveling from slingshot to window) did so at less than light speed, which means the angle differs from 90 degrees (the angle when no speed is involved) by less than 45 degrees—i.e. the angle with respect to the time slices (the space axes) is greater than 45 degrees. (Remember from endnote 9 of this chapter that light speed sets the limit and such motion traces out 45-degree lines.) Now, as in endnote 9, the different time slicings associated with an observer in motion are angled relative to those of an observer at rest, but the angle is always
less
than 45 degrees (since the relative motion between two material observers is always less than the speed of light). And since the angle associated with causally related events is always
greater
than 45 degrees, the time slices of an observer, who necessarily travels at less than light speed, cannot first encounter the effect and then later encounter the cause. To all observers, cause will precede effect.
12. The notion that causes precede their effects (see the preceding note) would, among other things, be challenged if influences could travel faster than the speed of light.
13. Isaac Newton,
Sir Isaac Newton's Mathematical Principles of Natural Philosophy
and His System of the World,
trans. A. Motte and Florian Cajori (Berkeley: University of California Press, 1962), vol. 1, p. 634.
14. Because the gravitational pull of the earth differs from one location to another, a spatially extended, freely falling observer can still detect a residual gravitational influence. Namely, if the observer, while falling, releases two baseballs—one from his outstretched right arm and the other from his left—each will fall along a path toward the earth's center. So, from the observer's perspective, he will be falling straight down toward the earth's center, while the ball released from his right hand will travel downward and slightly toward the left, while the ball released from his left hand will travel downward and slightly toward the right. Through careful measurement, the observer will therefore see that the distance between the two baseballs slowly decreases; they move toward one another. Crucial to this effect, though, is that the baseballs were released in slightly different locations in space, so that their freely falling paths toward earth's center were slightly different as well. Thus, a more precise statement of Einstein's realization is that the smaller the spatial extent of an object, the more fully it can eliminate gravity by going into free fall. While an important point of principle, this complication can be safely ignored throughout the discussion.