The Fabric of the Cosmos: Space, Time, and the Texture of Reality (61 page)

Black holes have the universe's most inscrutable poker faces. From the outside, they appear just about as simple as you can get. The three distinguishing features of a black hole are its mass (which determines how big it is—the distance from its center to its event horizon, the enshrouding surface of no return), its electric charge, and how fast it's spinning. That's it. There are no more details to be gleaned from scrutinizing the visage that a black hole presents to the cosmos. Physicists sum this up with the saying "Black holes have no hair," meaning that they lack the kinds of detailed features that allow for individuality. When you've seen one black hole with a given mass, charge, and spin (though you've learned these indirectly, through their effect on surrounding gas and stars, since black holes are black), you've definitely seen them all.

Nevertheless, behind their stony countenances, black holes harbor the greatest reservoirs of mayhem the universe has ever known. Among all physical systems of a given size with
any
possible composition, black holes contain the highest possible entropy. Recall from Chapter 6 that one rough way to think about this comes directly from entropy's definition as a measure of the number of rearrangements of an object's internal constituents that have no effect on its appearance. When it comes to black holes, even though we can't say what their constituents actually are— since we don't know what happens when matter is crushed at the black hole's center—we can say confidently that rearranging these constituents will no more affect a black hole's mass, charge, or spin than rearranging the pages in
War and Peace
will affect the weight of the book. And since mass, charge, and spin fully determine the face that a black hole shows the external world,
all
such manipulations go unnoticed and we can say a black hole has maximal entropy.

Even so, you might suggest one-upping the entropy of a black hole in the following simple way. Build a hollow sphere of the same size as a given black hole and fill it with gas (hydrogen, helium, carbon dioxide, whatever) that you allow to spread through its interior. The more gas you pump in, the greater the entropy, since more constituents means more possible rearrangements. You might guess, then, that if you keep on pumping and pumping, the entropy of the gas will steadily rise and so will eventually exceed that of the given black hole. It's a clever strategy, but general relativity shows that it fails. The more gas you pump in, the more massive the sphere's contents become. And before you reach the entropy of an equal-sized black hole, the increasingly large mass within the sphere will reach a critical value that causes the sphere and its contents to
become
a black hole.
There's just no way around it. Black holes have a monopoly on maximal disorder.

What if you try to further increase the entropy in the space inside the black hole itself by continuing to pump in yet more gas? Entropy will indeed continue to rise, but you'll have changed the rules of the game. As matter takes the plunge across a black hole's ravenous event horizon, not only does the black hole's entropy increase, but
its size increases as well.
The size of a black hole is proportional to its mass, so as you dump more matter into the hole, it gets heavier and bigger. Thus, once you max out the entropy in a region of space by creating a black hole, any attempt to further increase the entropy in that region will fail. The region just can't support more disorder. It's entropy-sated. Whatever you do, whether you pump in gas or toss in a Hummer, you will necessarily cause the black hole to grow and hence surround a larger spatial region. Thus, the amount of entropy contained within a black hole not only tells us a fundamental feature of the black hole, it also tells us something fundamental about space itself:
the maximum entropy that can be crammed into a
region of space—any region of space, anywhere, anytime—is equal to the
entropy contained within a black hole whose size equals that of the region
in question.

So, how much entropy does a black hole of a given size contain? Here is where things get interesting. Reasoning intuitively, start with something more easily visualized, like air in a Tupperware container. If you were to join together two such containers, doubling the total volume and number of air molecules, you might guess that you'd double the entropy. Detailed calculations confirm
¹
this conclusion and show that, all else being equal (unchanging temperature, density, and so on), the entropies of familiar physical systems are proportional to their volumes. A natural next guess is that the same conclusion would also apply to less familiar things, like black holes, leading us to expect that a black hole's entropy is also proportional to its volume.

But in the 1970s, Jacob Bekenstein and Stephen Hawking discovered that this isn't right. Their mathematical analyses showed that the entropy of a black hole is not proportional to its volume, but instead is proportional to the
area
of its event horizon—roughly speaking, to its surface area. This is a very different answer. Were you to double the radius of a black hole, its volume would increase by a factor of 8 (2
³
) while its surface area would increase by only a factor of 4 (2
²
); were you to increase its radius by a factor of a hundred, its volume would increase by a factor of a million (100
³
), while its surface area would increase only by a factor of 10,000 (100
²
). Big black holes have much more volume than they do surface area.
²
Thus, even though black holes contain the greatest entropy among all things of a given size, Bekenstein and Hawking showed that the amount of entropy they contain is less than what we'd naïvely guess.

That entropy is proportional to surface area is not merely a curious distinction between black holes and Tupperware, about which we can take note and swiftly move on. We've seen that black holes set a limit to the amount of entropy that, even in principle, can be crammed into a region of space: take a black hole whose size precisely equals that of the region in question, figure out how much entropy the black hole has, and that
is
the absolute limit on the amount of entropy the region of space can contain. Since this entropy, as the works of Bekenstein and Hawking showed, is proportional to the black hole's surface area—which equals the surface area of the region, since we chose them to have the same size—we conclude that the maximal entropy any given region of space can contain is proportional to the region's surface area.
³

The discrepancy between this conclusion and that found from thinking about air trapped in Tupperware (where we found the amount of entropy to be proportional to the Tupperware's
volume,
not its surface area) is easy to pinpoint: Since we assumed the air was uniformly spread, the Tupperware reasoning ignored gravity; remember, when gravity matters, things clump. To ignore gravity is fine when densities are low, but when you are considering large entropy, densities are high, gravity matters, and the Tupperware reasoning is no longer valid. Instead, such extreme conditions require the gravity-based calculations of Bekenstein and Hawking, with the conclusion that the maximum entropy potential for a region of space is proportional to its surface area, not its volume.

All right, but why should we care? There are two reasons.

First, the entropy bound gives yet another clue that ultramicroscopic space has an atomized structure. In detail, Bekenstein and Hawking found that if you imagine drawing a checkerboard pattern on the event horizon of a black hole, with each square being one Planck length by one Planck length (so each such "Planck square" has an area of about 10
^-66
square centimeters), then the black hole's entropy equals the number of such squares that can fit on its surface.
⁴
It's hard to miss the conclusion to which this result strongly hints: each Planck square is a minimal, fundamental unit of space, and each carries a minimal, single unit of entropy. This suggests that there is nothing, even in principle, that can take place
within
a Planck square, because any such activity could support disorder and hence the Planck square could contain more than the single unit of entropy found by Bekenstein and Hawking. Once again, then, from a completely different perspective we are led to the notion of an elemental spatial entity.
⁵

Second, for a physicist, the upper limit to the entropy that can exist in a region of space is a critical, almost sacred quantity. To understand why, imagine you're working for a behavioral psychiatrist, and your job is to keep a detailed, moment-to-moment record of the interactions between groups of intensely hyperactive young children. Every morning you pray that the day's group will be well behaved, because the more bedlam the children create, the more difficult your job. The reason is intuitively obvious, but it's worth saying explicitly: the more disorderly the children are, the more things you have to keep track of. The universe presents a physicist with much the same challenge. A fundamental physical theory is meant to describe everything that goes on—or could go on, even in principle—in a given region of space. And, as with the children, the more disorder the region can contain—even in principle—the more things the theory must be capable of keeping track of. Thus, the maximum entropy a region can contain provides a simple but incisive litmus test: physicists expect that a truly fundamental theory is one that is perfectly matched to the maximum entropy in any given spatial region. The theory should be so tightly in tune with nature that its maximum capacity to keep track of disorder
exactly
equals the maximum disorder a region can possibly contain, not more and not less.

The thing is, if the Tupperware conclusion had had unlimited validity, a fundamental theory would have needed the capacity to account for a volume's worth of disorder in any region. But since that reasoning fails when gravity is included—and since a fundamental theory must include gravity—we learn that a fundamental theory need only be able to account for a surface area's worth of disorder in any region. And as we showed with a couple of numerical examples a few paragraphs ago, for large regions the latter is much smaller than the former.

Thus, the Bekenstein and Hawking result tells us that a theory that includes gravity is, in some sense, simpler than a theory that doesn't. There are fewer "degrees of freedom"—fewer things that can change and hence contribute to disorder—that the theory must describe. This is an interesting realization in its own right, but if we follow this line of reasoning one step further, it seems to tell us something exceedingly bizarre. If the maximum entropy in any given region of space is proportional to the region's surface area and not its volume, then perhaps the true, fundamental degrees of freedom—the attributes that have the potential to give rise to that disorder
—actually
reside on the region's surface and not within
its volume.
Maybe, that is, the universe's real physical processes take place on a thin, distant surface that surrounds us, and all we see and experience is merely a projection of those processes. Maybe, that is, the universe is rather like a hologram.

This is an odd idea, but as we'll now discuss, it has recently received substantial support.

A hologram is a two-dimensional piece of etched plastic, which, when illuminated with appropriate laser light, projects a three-dimensional image.
⁶
In the early 1990s, the Dutch Nobel laureate Gerard 't Hooft and Leonard Susskind, the same physicist who coinvented string theory, suggested that the universe itself might operate in a manner analogous to a hologram. They put forward the startling idea that the comings and goings we observe in the three dimensions of day-to-day life might themselves be holographic projections of physical processes taking place on a distant, two-dimensional surface. In their new and peculiar-sounding vision, we and everything we do or see would be akin to holographic images. Whereas Plato envisioned common perceptions as revealing a mere shadow of reality, the holographic principle concurs, but turns the metaphor on its head. The shadows—the things that are flattened out and hence live on a lower-dimensional surface—are real, while what seem to be the more richly structured, higher-dimensional entities (us; the world around us) are evanescent projections of the shadows.
⁴⁵

Again, while it is a fantastically strange idea, and one whose role in the final understanding of spacetime is far from clear, 't Hooft and Susskind's so-called
holographic principle
is well motivated. For, as we discussed in the last section, the maximum entropy that a region of space can contain scales with the area of its surface, not with the volume of its interior. It's natural to guess, then, that the universe's most fundamental ingredients, its most basic degrees of freedom—the entities that can carry the universe's entropy much as the pages of
War and Peace
carry its entropy— would reside on a bounding surface and not in the universe's interior. What we experience in the "volume" of the universe—in the
bulk,
as physicists often call it—would be determined by what takes place on the bounding surface, much as what we see in a holographic projection is determined by information encoded on a bounding piece of plastic. The laws of physics would act as the universe's laser, illuminating the real processes of the cosmos—processes taking place on a thin, distant surface—and generating the holographic illusions of daily life.

We have not yet figured out how this holographic principle might be realized in the real world. One challenge is that in conventional descriptions the universe is imagined either to go on forever, or if not, to wrap back on itself like a sphere or a video game screen (as in Chapter 8), and hence it wouldn't have any edges or boundaries. So, where would the supposed "bounding holographic surface" be located? Moreover, physical processes certainly seem to be under our control, right here, deep in the universe's interior. It doesn't seem that something on a hard-to-locate boundary is somehow calling the shots regarding what happens here in the bulk. Does the holographic principle imply that
that
sense of control and autonomy is illusory? Or is it better to think of holography as articulating a kind of duality in which, on the basis of taste—not of physics— one can choose a familiar description in which the fundamental laws operate here in the bulk (which aligns with intuition and perception) or an unfamiliar description in which fundamental physics takes place on some kind of boundary of the universe, with each viewpoint being equally valid? These are essential questions that remain controversial.

But in 1997, building on earlier insights of a number of string theorists, the Argentinian physicist Juan Maldacena had a breakthrough that dramatically advanced thinking on these matters. His discovery is not directly relevant to the question of holography's role in our real universe, but in the time-honored fashion of physics, he found a hypothetical context—a hypothetical universe—in which abstract musings on holography could be made both concrete and precise using mathematics. For technical reasons, Maldacena studied a hypothetical universe with four large space dimensions and one time dimension that have uniform negative curvature—a higher dimensional version of the Pringle's potato chip, Figure 8.6c. Standard mathematical analysis reveals that this fivedimensional spacetime has a boundary
⁷
that, like all boundaries, has one dimension less than the shape it bounds: three space dimensions and one time dimension. (As always, higher-dimensional spaces are hard to envision, so if you want a mental picture, think of a can of tomato soup—the three-dimensional liquid soup is analogous to the five-dimensional spacetime, while the two-dimensional surface of the can is analogous to the four-dimensional spacetime boundary.) After including additional curled-up dimensions as required by string theory, Maldacena convincingly argued that the physics witnessed by an observer living within this universe (an observer in the "soup") could be completely described in terms of physics taking place on the universe's boundary (physics on the surface of the can).

Although it is not realistic, this work provided the first concrete and mathematically tractable example in which the holographic principle was explicitly realized.
⁸
In doing so, it shed much light on the notion of holography as applied to an entire universe. For instance, in Maldacena's work, the bulk description and the boundary description are on an absolutely equal footing. One is not primary and the other secondary. In much the same spirit as the relation between the five string theories, the bulk and boundary theories are translations of each other. The unusual feature of this particular translation, though, is that the bulk theory has more dimensions than the equivalent theory formulated on the boundary. Moreover, whereas the bulk theory includes gravity (since Maldacena formulated it using string theory), calculations show that the boundary theory doesn't. Nevertheless, any question asked or calculation done in one of the theories can be translated into an equivalent question or calculation in the other. While someone unfamiliar with the dictionary would think that the corresponding questions and calculations have absolutely nothing to do with each other (for example, since the boundary theory does not include gravity, questions involving gravity in the bulk theory are translated into very-different-sounding, gravity-less questions in the boundary theory), someone well versed in both languages—an expert on both theories—would recognize their relationship and realize that the answers to corresponding questions and the results of corresponding calculations must agree. Indeed, every calculation done to date, and there have been many, supports this assertion.

The details of all this are challenging to grasp fully, but don't let that obscure the main point. Maldacena's result is amazing. He found a concrete, albeit hypothetical, realization of holography within string theory. He showed that a particular quantum theory without gravity is a translation of—is indistinguishable from—another quantum theory that includes gravity but is formulated with one more space dimension. Vigorous research programs are under way to determine how these insights might apply to a more realistic universe, our universe, but progress is slow as the analysis is fraught with technical hurdles. (Maldacena chose the particular hypothetical example he did because it proved relatively easy to analyze mathematically; more realistic examples are much harder to deal with.) Nevertheless, we now know that string theory, at least in certain contexts, has the capacity to support the concept of holography. And, as with the case of geometric translations described earlier, this provides yet another hint that spacetime is not fundamental. Not only can the size and shape of spacetime change in translation from one formulation of a theory to another, equivalent form, but the
number
of space dimensions can change, too.

More and more, these clues point toward the conclusion that the form of spacetime is an adorning detail that varies from one formulation of a physical theory to the next, rather than being a fundamental element of reality. Much as the number of letters, syllables, and vowels in the word
cat
differ from those in
gato,
its Spanish translation, the form of spacetime—its shape, its size, and even the number of its dimensions—also changes in translation. To any given observer who is using one theory to think about the universe, spacetime may seem real and indispensable. But should that observer change the formulation of the theory he or she uses to an equivalent, translated version, what once seemed real and indispensable necessarily changes, too. Thus, if these ideas are right—and I should emphasize that they have yet to be rigorously proven even though theorists have amassed a great deal of supporting evidence—they strongly challenge the primacy of space and time.

Of all the clues discussed here, I'd pick the holographic principle as the one most likely to play a dominant role in future research. It emerges from a basic feature of black holes—their entropy—the understanding of which, many physicists agree, rests on firm theoretical foundations. Even if the details of our theories should change, we expect that any sensible description of gravity will allow for black holes, and hence the entropy bounds driving this discussion will persist and holography will apply. That string theory naturally incorporates the holographic principle—at least in examples amenable to mathematical analysis—is another strong piece of evidence suggesting the principle's validity. I expect that regardless of where the search for the foundations of space and time may take us, regardless of modifications to string/M-theory that may be waiting for us around the bend, holography will continue to be a guiding concept.