by Brian Greene
In the early years of the twentieth century, Einstein seized on this simple but profound interconnection between motion and gravity; after a decade of development, he leveraged it into his general theory of relativity. Our application here is more modest. Suppose you are in that capsule and are freely falling not toward the earth but toward a black hole. The very same reasoning ensures that there’s no way for your experience to be any different from floating in empty space. And that means that nothing special or unusual will happen as you freely fall through the black hole’s horizon. When you eventually hit the black hole’s center, you’ll no longer be in free fall, and that experience will certainly distinguish itself. And spectacularly so. But until then, you could just as well be aimlessly floating in the dark depths of outer space.
This realization renders the black hole’s entropy all the more puzzling. If as you pass through the horizon of a black hole you find nothing there, nothing at all to distinguish it from empty space, how can it store information?
An answer that has gained traction over the last decade resonates with the duality theme encountered in early chapters. Recall that duality refers to a situation in which there are complementary perspectives that seem completely different, and yet are intimately connected through a shared physical anchor. The Albert-Marilyn image of Figure 5.2 provides a good visual metaphor; mathematical examples come from the mirror shapes of string theory’s extra dimensions (Chapter 4) and the naïvely distinct yet dual string theories (Chapter 5). In recent years, researchers, led by Susskind, have realized that black holes present another context in which complementary yet widely divergent perspectives yield fundamental insight.
One essential perspective is yours, as you freely fall toward a black hole. Another is that of a distant observer, watching your journey through a powerful telescope. The remarkable thing is that as you pass uneventfully through a black hole’s horizon, the distant observer perceives a very different sequence of events. The discrepancy has to do with the black hole’s Hawking radiation.* When the distant observer measures the Hawking radiation’s temperature, she finds it to be tiny; let’s say it’s 10–13 K, indicating that the black hole is roughly the size of the one at the center of our galaxy. But the distant observer knows that the radiation is cold only because the photons, traveling to her from just outside the horizon, have expended their energy valiantly fighting against the black hole’s gravitational pull; in the description I gave earlier, the photons are tired. She deduces that as you get ever closer to the black hole’s horizon, you’ll encounter ever-fresher photons, ones that have only just begun their journey and so are ever more energetic and ever hotter. Indeed, as she watches you approach to within a hair’s breadth of the horizon, she sees your body bombarded by increasingly intense Hawking radiation, until finally all that’s left is your charred remains.
Happily, however, what you experience is much more pleasant. You don’t see or feel or otherwise obtain any evidence of this hot radiation. Again, because your free-fall motion cancels the effects of gravity,10 your experience is indistinguishable from that of floating in empty space. And one thing we know for sure is that when you float in empty space, you don’t suddenly burst into flames. So the conclusion is that from your perspective, you pass seamlessly through the horizon and (less happily) hurtle on toward the black hole’s singularity, while from the distant observer’s perspective, you are immolated by a scorching corona that surrounds the horizon.
Which perspective is right? The claim advanced by Susskind and others is that both are. Granted, this is hard to square with ordinary logic—the logic by which you are either alive or not alive. But this is no ordinary situation. Most saliently, the wildly different perspectives can never confront each other. You can’t climb out of the black hole and prove to the distant observer that you are alive. And, as it turns out, the distant observer can’t jump into the black hole and confront you with evidence that you’re not. When I said that the distant observer “sees” you immolated by the black hole’s Hawking radiation, that was a simplification. The distant observer, by closely examining the tired radiation that reaches her, can piece together the story of your fiery demise. But for the information to reach her takes time. And the math shows that by the time she can conclude you’ve burned, she won’t have enough time left to then hop into the black hole and catch up with you before you’re destroyed by the singularity. Perspectives can differ, but physics has a built-in fail-safe against paradoxes.
What about information? From your perspective, all your information, stored in your body and brain and in the laptop you’re holding, passes with you through the black hole’s horizon. From the perspective of the distant observer, all the information you carry is absorbed by the layer of radiation incessantly bubbling just above the horizon. The bits contained in your body, brain, and laptop would be preserved, but would become thoroughly scrambled as they joined, jostled, and intermingled with the sizzling hot horizon. Which means that to the distant observer, the event horizon is a real place, populated by real things that give physical expression to the information symbolically depicted in the chessboard, Figure 9.2.
The conclusion is that the distant observer—us—infers that a black hole’s entropy is determined by the area of its horizon because the horizon is where the entropy is stored. Said that way, it seems utterly sensible. But don’t lose sight of how unexpected it is that the storage capacity isn’t set by the black hole’s volume. And, as we will now see, this result doesn’t merely highlight a peculiar feature of black holes. Black holes don’t just tell us about how black holes store information. Black holes inform us about information storage in any context. This paves a direct path to the holographic perspective.
Beyond Black Holes
Consider any object or collection of objects—the collections of the Library of Congress, all of Google’s computers, the CIA’s archives—situated in some region of space. For ease, imagine that we highlight the region by surrounding it with an imaginary sphere, as in Figure 9.3a. Assume further that the total mass of the objects, compared with the volume they fill, is of such an ordinary run-of-the-mill magnitude that it’s nowhere near what it takes to create a black hole. That’s the setup. Now for the pivotal question: What is the maximum amount of information that can be stored within the region of space?
Figure 9.3 (a) A variety of objects that store information, situated within a well-marked region of space. (b) We augment the region’s capacity for storing information. (c) When the amount of matter crosses a threshold (whose value can be calculated from general relativity),11 the region becomes a black hole.
Those unlikely bedfellows, the Second Law and black holes, provide the answer. Imagine adding matter to the region, with the aim of augmenting its information storage capacity. You might insert high-capacity memory chips or voluminous hard drives into the bank of Google’s computers; you might provide books or jam-packed Kindles to augment the Library of Congress collection. Since even raw matter carries information—Are the steam’s molecules here or there? Are they moving at this speed or that?—you also cram every nook and cranny of the region with as much matter as you can get your hands on. Until you reach a critical juncture. At some point, the region will be so thoroughly stuffed that were you to add even a single grain of sand, the interior would go dark as the region turned into a black hole. When that happens, game over. A black hole’s size is determined by its mass, so if you try to increase the information storage capacity by adding yet more matter, the black hole will respond by growing larger. And since we want to focus on the information that can inhabit a given fixed volume of space, this result falls afoul of the basic setup. You can’t increase the black hole’s information capacity without forcing the black hole to enlarge.12
Two observations take us across the finish line. The Second Law ensures that entropy increases throughout the entire process, and so the information hidden within the hard drives, Kindles, old-fashioned paper books, and everything e
lse you packed into the region is less than that hidden in the black hole. From the results of Bekenstein and Hawking, we know that the black hole’s hidden information content is given by the area of its event horizon. Moreover, because you were careful not to overspill the original region of space, the black hole’s event horizon coincides with the region’s boundary, so the black hole’s entropy equals the area of this surrounding surface. We thus learn an important lesson. The amount of information contained within a region of space, stored in any objects of any design, is always less than the area of the surface that surrounds the region (measured in square Planck units).
This is the conclusion we’ve been chasing. Notice that although black holes are central to the reasoning, the analysis applies to any region of space, whether or not a black hole is actually present. If you max out a region’s storage capacity, you’ll create a black hole, but as long as you stay under the limit, no black hole will form.
I hasten to add that in any practical sense, the information storage limit is of no concern. Compared with today’s rudimentary storage devices, the potential storage capacity on the surface of a spatial region is humongous. A stack of five off-the-shelf terabyte hard drives fits comfortably within a sphere of radius 50 centimeters, whose surface is covered by about 1070 Planck cells. The surface’s storage capacity is thus about 1070 bits, which is about a billion, trillion, trillion, trillion, trillion terabytes, and so enormously exceeds anything you can buy. No one in Silicon Valley cares much about these theoretical constraints.
Yet, as a guide to how the universe works, the storage limitations are telling. Think of any region of space, such as the room in which I’m writing or the one in which you’re reading. Take a Wheelerian perspective and imagine that whatever happens in the region amounts to information processing—information regarding how things are right now is transformed by the laws of physics into information regarding how they will be in a second or a minute or an hour. Since the physical processes we witness, as well as those by which we’re governed, seemingly take place within the region, it’s natural to expect that the information those processes carry is also found within the region. But the results just derived suggest an alternative view. For black holes, we found that the link between information and surface area goes beyond mere numerical accounting; there’s a concrete sense in which information is stored on their surfaces. Susskind and ’t Hooft stressed that the lesson should be general: since the information required to describe physical phenomena within any given region of space can be fully encoded by data on a surface that surrounds the region, then there’s reason to think that the surface is where the fundamental physical processes actually happen. Our familiar three-dimensional reality, these bold thinkers suggested, would then be likened to a holographic projection of those distant two-dimensional physical processes.
If this line of reasoning is correct, then there are physical processes taking place on some distant surface that, much like a puppeteer pulls strings, are fully linked to the processes taking place in my fingers, arms, and brain as I type these words at my desk. Our experiences here, and that distant reality there, would form the most interlocked of parallel worlds. Phenomena in the two—I’ll call them Holographic Parallel Universes—would be so fully joined that their respective evolutions would be as connected as me and my shadow.
The Fine Print
That familiar reality may be mirrored, or perhaps even produced, by phenomena taking place on a faraway, lower-dimensional surface ranks among the most unexpected developments in all of theoretical physics. But how confident should we be that the holographic principle is right? We are navigating a realm deep in theoretical territory, and relying almost exclusively on developments that have not been experimentally tested, so there is surely grounds for skepticism. There are many places where the argument could be forced off course. Do black holes really have nonzero entropy and nonzero temperature, and, if so, do the values conform to theoretical predictions? Is the information capacity of a region of space really determined by the amount of information that can be stored on a surface that surrounds it? And on such a surface, is one bit per Planck area really the limit? We think the answer to each of these questions is yes because of the coherent, consistent, and carefully constructed theoretical edifice into which the conclusions perfectly fit. But since none of these ideas has been subject to the experimenter’s scalpel, it is certainly possible (though in my view highly unlikely) that future advances will convince us that one or more of these essential intermediate steps are wrong. That could lay to waste the holographic idea.
Another important point is that throughout the discussion, we’ve spoken of a region of space, of a surface that surrounds it, and of the information content of each. But since our focus has been on entropy and the Second Law—both of which concern themselves primarily with the quantity of information in a given context—we’ve not elaborated on the details of how that information is physically realized or stored. When we talk about information residing on a sphere surrounding a region of space, what does that really mean? How does the information manifest itself? What form does it take? To what extent can we develop an explicit dictionary that translates from phenomena taking place on the boundary to those taking place in the interior?
Physicists have yet to articulate a general framework for addressing these questions. Given that gravity and quantum mechanics are both central to the reasoning, you might expect that string theory would provide a potent context for theoretical explorations. But when ’t Hooft first formulated the holographic concept, he doubted that string theory would be able to advance the subject, noting, “Nature is much more crazy at the Planck scale than even string theorists could have imagined.”13 Less than a decade later, string theory proved ’t Hooft wrong by proving him right. In a landmark paper, a young theorist showed that string theory provides an explicit realization of the holographic principle.
String Theory and Holography
When I was called to the stage at the University of California, Santa Barbara, to give my talk at the annual international string theory conference in 1998, I did something I’d never done before and suspect will never do again. I faced the audience, threw my right hand to my left shoulder and my left to my right shoulder, and then with both hands in succession grabbed the seat of my pants, bunny-hopped, and made a quarter turn, followed, thankfully, by audience laughter, which covered the three remaining steps necessary to reach the podium, where I began my talk. The crowd got the joke. At the banquet the night before, the conference participants had performed a song-and-dance celebrating—as only physicists can—a spectacular result of the Argentinian string theorist Juan Maldacena. With lyrics like “Black holes used to be a great mystery; / Now we use D-branes to compute D-entropy,” the crowd had reveled in a string theory version of the 1990s momentary dance craze, the Macarena—a touch more animated than Al Gore’s version at the Democratic National Convention, a touch less mellifluous than Los del Rio’s original one-hit wonder, but second to none in passion. I was one of the few at the conference whose talk was not focused on Maldacena’s breakthrough, so when I took the stage the next morning I felt it only appropriate to preface my remarks with a personal gesture of appreciation.
Now, more than a decade later, many would agree that no work in string theory since is of comparable magnitude and influence. Of the numerous ramifications of Maldacena’s result, one is directly relevant to the line we’ve been following. In a particular hypothetical setting, Maldacena’s result realized explicitly the holographic principle, and in doing so provided the first mathematical example of Holographic Parallel Universes. Maldacena achieved this by considering string theory in a universe whose shape differs from ours but for the purpose at hand proves easier to analyze. In a precise mathematical sense, the shape has a boundary, an impenetrable surface that completely surrounds its interior. By zeroing in on this surface, Maldacena argued convincingly that everything taking place within the specified univ
erse is a reflection of laws and processes acting themselves out on the boundary.
Although Maldacena’s method may not seem directly applicable to a universe with the shape of ours, his results are decisive because they established a mathematical proving ground in which ideas regarding holographic universes could be made explicit and investigated quantitatively. The results of such studies won over a great many physicists who had previously eyed the holographic principle with much misgiving, and thus set off an avalanche of research that has yielded thousands of articles and considerably deeper understanding. Most exciting of all, there’s now evidence that a link between these theoretical insights and physics in our universe can be forged. In the next few years, that link may very well allow the holographic ideas to be experimentally tested.