The Fabric of the Cosmos: Space, Time, and the Texture of Reality (30 page)

Using a slight variation on the balloon model, we can now understand more precisely how symmetry in space, even though space is expanding, yields a notion of time that applies uniformly across the cosmos. Imagine replacing each penny by an identical clock, as in Figure 8.3. We know from relativity that identical clocks will tick off time at different rates if they are subject to different physical influences—different motions, or different gravitational fields. But the simple yet key observation is that the complete symmetry among all Lincolns on the inflating balloon translates to complete symmetry among all the clocks. All the clocks experience identical physical conditions, so all tick at exactly the same rate and record identical amounts of elapsed time. Similarly, in an expanding universe in which there is a high degree of symmetry among all the galaxies,
clocks that move along with one or another galaxy must also tick at the
same rate and hence record an identical amount of elapsed time.
How could it be otherwise? Each clock is on a par with every other, having experienced, on average, nearly identical physical conditions. This again shows the stunning power of symmetry. Without any calculation or detailed analysis, we realize that the uniformity of the physical environment, as evidenced by the uniformity of the microwave background radiation and the uniform distribution of galaxies throughout space,
⁸
allows us to infer uniformity of time.

Although the reasoning here is straightforward, the conclusion may nevertheless be confusing. Since the galaxies are all rushing apart as space expands, clocks that move along with one or another galaxy are also rushing apart. What's more, they're moving relative to each other at an enormous variety of speeds determined by the enormous variety of distances between them. Won't this motion cause the clocks to fall out of synchronization, as Einstein taught us with special relativity? For a number of reasons, the answer is no; here is one particularly useful way to think about it.

Recall from Chapter 3 that Einstein discovered that clocks that move
through
space in different ways tick off time at different rates (because they divert different amounts of their motion through time into motion through space; remember the analogy with Bart on his skateboard, first heading north and then diverting some of his motion to the east). But the clocks we are now discussing are
not
moving through space at all. Just as each penny is glued to one point on the balloon and only moves relative to other pennies because of the swelling of the balloon's surface, each galaxy occupies one region of space and, for the most part, only moves relative to other galaxies because of the expansion of space. And this means that, with respect to space itself, all the clocks are actually stationary, so they tick off time identically. It is precisely these clocks
—clocks
whose only motion comes from the expansion of space—
that provide the synchronized cosmic clocks used to measure the age of the universe.

Figure 8.3 Clocks that move along with galaxies—whose motion, on average, arises only from the expansion of space—provide universal cosmic timepieces. They stay synchronized even though they separate from one another, since they move
with
space but not
through
space.

Notice, of course, that you are free to take your clock, hop aboard a rocket, and zip this way and that across space at enormous speeds, undergoing motion significantly in excess of the cosmic flow from spatial expansion. If you do this, your clock
will
tick at a different rate and you
will
find a different length of elapsed time since the bang. This is a perfectly valid point of view, but it is completely individualistic: the elapsed time measured is tied to the history of your particular whereabouts and states of motion. When astronomers speak of the universe's age, though, they are seeking something universal—they are seeking a measure that has the same meaning everywhere. The uniformity of change throughout space provides a way of doing that.
⁹

In fact, the uniformity of the microwave background radiation provides a ready-made test of whether you actually are moving with the cosmic flow of space. You see, although the microwave radiation is homogeneous across space, if you undertake additional motion beyond that from the cosmic flow of spatial expansion, you will not observe the radiation to be homogeneous. Just as the horn on a speeding car has a higher pitch when approaching and a lower pitch when receding, if you are zipping around in a spaceship, the crests and troughs of the microwaves heading toward the front of your ship will hit at a higher frequency than those traveling toward the back of your ship. Higher-frequency microwaves translate into higher temperatures, so you'd find the radiation in the direction you are heading to be a bit warmer than the radiation reaching you from behind. As it turns out, here on "spaceship" earth, astronomers
do
find the microwave background to be a little warmer in one direction in space and a little colder in the opposite direction. The reason is that not only does the earth move around the sun, and the sun move around the galactic center, but the entire Milky Way galaxy has a small velocity, in excess of cosmic expansion, toward the constellation Hydra. Only when astronomers correct for the effect these relatively slight additional motions have on the microwaves we receive does the radiation exhibit the exquisite uniformity of temperature between one part of the sky and another. It is this uniformity, this overall symmetry between one location and another, that allows us to speak sensibly of time when describing the entire universe.

A few subtle points in our explanation of cosmic expansion are worthy of emphasis. First, remember that in the balloon metaphor, it is only the balloon's
surface
that plays any role—a surface that is only two-dimensional (each location can be specified by giving two numbers analogous to latitude and longitude on earth), whereas the space we see when we look around has three dimensions. We make use of this lower-dimensional model because it retains the concepts essential to the true, three-dimensional story but is far easier to visualize. It's important to bear this in mind, especially if you have been tempted to say that there
is
a special point in the balloon model: the center point in the interior of the balloon away from which the whole rubber surface is moving. While this observation is true, it is meaningless in the balloon analogy because any point not on the balloon's surface plays no role. The surface of the balloon represents
all
of space; points that do not lie on the surface of the balloon are merely irrelevant by-products of the analogy and do not correspond to any location in the universe.
¹⁹

Second, if the speed of recession is larger and larger for galaxies that are farther and farther away, doesn't that mean that galaxies that are sufficiently distant will rush away from us at a speed greater than the speed of light? The answer is a resounding, definite yes. Yet there is no conflict with special relativity. Why? Well, it's closely related to the reason clocks moving apart due to the cosmic flow of space stay synchronized. As we emphasized in Chapter 3, Einstein showed that nothing can move
through
space faster than light. But galaxies, on average, hardly move through space at all. Their motion is due almost completely to the
stretching of space itself.
And Einstein's theory does not prohibit space from expanding in a way that drives two points—two galaxies—away from each other at greater than light speed. His results only constrain speeds for which motion from spatial expansion has been subtracted out, motion in excess of that arising from spatial expansion. Observations confirm that for typical galaxies zipping along with the cosmic flow, such excess motion is minimal, fully in keeping with special relativity, even though their motion relative to each other, arising from the swelling of space itself, may exceed the speed of light.
²⁰

Third, if space is expanding, wouldn't that mean that in addition to galaxies being driven away from each other, the swelling space within each galaxy would drive all its stars to move farther apart, and the swelling space within each star, and within each planet, and within you and me and everything else, would drive all the constituent atoms to move farther apart, and the swelling of space within each atom would drive all the subatomic constituents to move farther apart? In short, wouldn't swelling space cause
everything
to grow in size, including our meter sticks, and in that way make it impossible to discern that any expansion had actually happened? The answer: no. Think again about the balloon-and-penny model. As the surface of the balloon swells, all the pennies are driven apart, but the pennies themselves surely do not expand. Of course, had we represented the galaxies by little circles drawn on the balloon with a black marker, then indeed, as the balloon grew in size the little circles would grow as well. But pennies, not blackened circles, capture what really happens. Each penny stays fixed in size because the forces holding its zinc and copper atoms together are far stronger than the outward pull of the expanding balloon to which it is glued. Similarly, the nuclear force holding individual atoms together, and the electromagnetic force holding your bones and skin together, and the gravitational force holding planets and stars intact and bound together in galaxies, are stronger than the outward swelling of space, and so none of these objects expands. Only on the largest of scales, on scales much larger than individual galaxies, does the swelling of space meet little or no resistance (the gravitational pull between widely separated galaxies is comparatively small, because of the large separations involved) and so only on such super galactic scales does the swelling of space drive objects apart.

If someone were to wake you in the middle of the night from a deep sleep and demand you tell them the shape of the universe—the overall shape of space—you might be hard pressed to answer. Even in your groggy state, you know that Einstein showed space to be kind of like Silly Putty and so, in principle, it can take on practically any shape. How, then, can you possibly answer your interrogator's question? We live on a small planet orbiting an average star on the outskirts of a galaxy that is but one of hundreds of billions dispersed throughout space, so how in the world can you be expected to know anything at all about the shape of the entire universe? Well, as the fog of sleep begins to lift, you gradually realize that the power of symmetry once again comes to the rescue.

If you take account of scientists' widely held belief that, over large-scale averages, all locations and all directions in the universe are symmetrically related to one another, then you're well on your way to answering the interrogator's question. The reason is that almost all shapes
fail
to meet this symmetry criterion, because one part or region of the shape fundamentally differs from another. A pear bulges significantly at the bottom but less so at the top; an egg is flatter in the middle but pointier at its ends. These shapes, although exhibiting some degree of symmetry, do not possess complete symmetry. By ruling out such shapes, and limiting yourself only to those in which every region and direction is like every other, you are able to narrow down the possibilities fantastically.

We've already encountered one shape that fits the bill. The balloon's spherical shape was the key ingredient in establishing the symmetry between all the Lincolns on its swelling surface, and so the three-dimensional version of this shape, the so-called
three-sphere,
is one candidate for the shape of space. But this is not the only shape that yields complete symmetry. Continuing to reason with the more easily visualized two-dimensional models, imagine an
infinitely
wide and
infinitely
long rubber sheet—one that is completely uncurved—with evenly spaced pennies glued to its surface. As the entire sheet expands, there once again is complete spatial symmetry and complete consistency with Hubble's discovery: every Lincoln sees every other Lincoln rush away with a speed proportional to its distance, as in Figure 8.4. Hence, a three-dimensional version of this shape, like an infinite expanding cube of transparent rubber with galaxies evenly sprinkled throughout its interior, is another possible shape for space. (If you prefer culinary metaphors, think of an infinitely large version of the poppy seed muffin mentioned earlier, one that is shaped like a cube but goes on forever, with poppy seeds playing the role of galaxies. As the muffin bakes, the dough expands, causing each poppy seed to rush away from the others.) This shape is called
flat space
because, unlike the spherical example, it has no curvature (a meaning of "flat" that mathematicians and physicists use, but that differs from the colloquial meaning of "pancake-shaped.")
¹¹

One nice thing about both the spherical and the infinite flat shapes is that you can walk endlessly and never reach an edge or a boundary. This is appealing because it allows us to avoid thorny questions: What is beyond the edge of space? What happens if you walk into a boundary of space? If space has no edges or boundaries, the question has no meaning. But notice that the two shapes realize this attractive feature in different ways. If you walk straight ahead in a spherically shaped space, you'll find, like Magellan, that sooner or later you return to your starting point, never having encountered an edge. By contrast, if you walk straight ahead in infinite flat space, you'll find that, like the Energizer Bunny, you can keep going and going, again never encountering an edge, but also never returning to where your journey began. While this might seem like a fundamental difference between the geometry of a curved and a flat shape, there is a simple variation on flat space that strikingly resembles the sphere in this regard.

Figure 8.4
(
a
)
The view from any penny on an infinite flat plane is the same as the view from any other.
(
b
)
The farther apart two pennies are in Figure 8.4a, the greater the increase in their separation when the plane expands.

Figure 8.5
(
a
)
A video game screen is flat (in the sense of "uncurved") and has a finite size, but contains no edges or boundaries since it "wraps around." Mathematically, such a shape is called a
two-dimensional torus.
(
b
)
A three-dimensional version of the same shape, called a
three-dimensional torus,
is also flat (in the sense of uncurved) and has a finite volume, and also has no edges or boundaries, because it wraps around. If you pass through one face, you enter the opposite face.

To picture it, think of one of those video games in which the screen appears to have edges but in reality doesn't, since you can't actually fall off: if you move off the right edge, you reappear on the left; if you move off the top edge, you reappear on the bottom. The screen "wraps around," identifying top with bottom and left with right, and in that way the shape is flat (uncurved) and has
finite
size, but has no edges. Mathematically, this shape is called a
two-dimensional torus;
it is illustrated in Figure 8.5a.
¹²
The three-dimensional version of this shape—a three-dimensional torus—provides another possible shape for the fabric of space. You can think of this shape as an enormous cube that wraps around along all three axes: when you walk through the top you reappear at the bottom, when you walk through the back, you reappear at the front, when you walk through the left side, you reappear at the right, as in Figure 8.5b. Such a shape is flat—again, in the sense of being uncurved, not in the sense of being like a pancake—three-dimensional, finite in all directions, and yet has no edges or boundaries.

Beyond these possibilities, there is still another shape consistent with the symmetric expanding space explanation for Hubble's discovery. Although it's hard to picture in three dimensions, as with the spherical example there is a good two-dimensional stand-in: an infinite version of a Pringle's potato chip. This shape, often referred to as a
saddle,
is a kind of inverse of the sphere: Whereas a sphere is symmetrically bloated outward, the saddle is symmetrically shrunken inward, as illustrated in Figure 8.6. Using a bit of mathematical terminology, we say that the sphere is
positively curved
(bloats outward), the saddle is
negatively curved
(shrinks inward), and flat space—whether infinite or finite—has
no curvature
(no bloating or shrinking).
²¹

Researchers have proven that this list—uniformly positive, negative, or zero—exhausts the possible curvatures for space that are consistent with the requirement of symmetry between all locations and in all directions. And that is truly stunning. We are talking about the shape of the
entire universe,
something for which there are endless possibilities. Yet, by invoking the immense power of symmetry, researchers have been able to narrow the possibilities sharply. And so, if you allow symmetry to guide your answer, and your late-night interrogator grants you a mere handful of guesses, you'll be able to meet his challenge.
¹³

All the same, you might wonder why we've come upon a variety of possible shapes for the fabric of space. We inhabit a single universe, so why can't we specify a unique shape? Well, the shapes we've listed are the only ones consistent with our belief that every observer, regardless of where in the universe they're located, should see on the largest of scales an identical cosmos. But such considerations of symmetry, while highly selective, are not able to go all the way and pick out a unique answer. For that we need Einstein's equations from general relativity.

Figure 8.6 Using the two-dimensional analogy for space, there are three types of curvature that are completely symmetric—that is, curvatures in which the view from any location is the same as that from any other. They are
(
a
)
positive
curvature, which uniformly bloats outward, as on a sphere;
(
b
)
zero
curvature, which does not bloat at all, as on an infinite plane or finite video game screen;
(
c
)
negative
curvature, which uniformly shrinks inward, as on a saddle.

As input, Einstein's equations take the amount of matter and energy in the universe (assumed, again by consideration of symmetry, to be distributed uniformly) and as output, they give the curvature of space. The difficulty is that for many decades astronomers have been unable to agree on how much matter and energy there actually is. If all the matter and energy in the universe were to be smeared uniformly throughout space, and if, after this was done, there turned out to be more than the so-called critical density of about .00000000000000000000001 (10
^-23
) grams in every cubic meter
²²
—about five hydrogen atoms per cubic meter—Einstein's equations would yield a positive curvature for space; if there were less than the critical density, the equations would imply negative curvature; if there were exactly the critical density, the equations would tell us that space has no overall curvature. Although this observational issue is yet to be settled definitively, the most refined data are tipping the scales on the side of no curvature—the flat shape. But the question of whether the Energizer Bunny could move forever in one direction and vanish into the darkness, or would one day circle around and catch you from behind—whether space goes on forever or wraps back like a video screen—is still completely open.
¹⁴

Even so, even without a final answer to the shape of the cosmic fabric, what's abundantly clear is that symmetry is
the
essential consideration allowing us to comprehend space and time when applied to the universe as a whole. Without invoking the power of symmetry, we'd be stuck at square one.