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THE FABRIC of the COSMOS, Brian Greene, 2004
```(annotated and with added bold highlights by Epsilon=One)
```(annotated and with added bold highlights by Epsilon=One)
Chapter 16 - The Future of an Allusion
Wherefore the Entropy of Black Holes?
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 fora 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.
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 fora 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.