This conflict of principles created a very serious dilemma. General relativity said that the bits enter the horizon and continue on their way, deep into the interior of the black hole. But the rules of quantum mechanics forbid information to be lost to the outside world. One possibility might resolve the dilemma. Let’s go back to the prison analogy. Suppose that at the entrance to the prison, a guard were stationed at a Xerox machine, and that every incoming message was xeroxed—one copy going into the prison and one sent back out after nonrandom shuffling. That should satisfy everyone. Inside the prison they would see the message entering as if it were undisturbed on its way in. Outside the observers would find that the information was never lost. Everyone is right.
Here the problem gets interesting. A very basic principle of quantum mechanics says that a quantum Xerox machine is impossible. Quantum information cannot be faithfully copied. No matter how well the machine copies some kinds of information, it will always fail badly with other kinds. I called this the No Quantum Xerox Principle. The quantum-information experts call it the No Cloning Theorem. What it states is that no physical system can ever function completely faithfully to replicate information in a quantum world.
Here is a way to understand the No Quantum Xerox Principle. Start with a single electron. The Heisenberg Uncertainty Principle tells us that it is never possible to know both the position of the electron and its velocity. But now suppose we could quantum Xerox the electron in exactly its original state. Then we could measure the position of one copy and the velocity of the other, thereby learning the forbidden knowledge.
So, here is the new dilemma: general relativity tells us that information should fall straight through the horizon toward the deep interior of the black hole. On the other hand, the principles of quantum mechanics tell us that the same information must remain outside the black hole. And the No Cloning Theorem assures us that only one copy of each bit is possible. That’s the confusing situation that Hawking, ’t Hooft, and I found ourselves in. By the early nineties the situation had reached a crisis: who is right? The observer on the outside who expects the rules of quantum mechanics to be respected? For him the bits of information should be located just above the horizon, where they are scrambled and then sent back out in the Hawking radiation. Or is the observer who falls through the horizon correct in expecting the bits to accompany her into the heart of the black hole?
The solution to the paradox was eventually provided by two new principles of physics that ’t Hooft and I introduced in the early 1990s. Both of them are very strange, far stranger than Hawking’s idea that information can be lost; so strange that, in fact, no one else besides ’t Hooft and me believed them at first. But as Sherlock Holmes once told Watson, “When you have eliminated all that is impossible, whatever remains must be the truth, no matter how improbable.”
Black Hole Complementarity
With the possible exception of Einstein, Niels Bohr was the most philosophical of the fathers of modern physics. To Bohr the philosophical revolution that accompanied the discovery of quantum mechanics was all about complementarity. The complementarity of quantum mechanics was manifest in many ways, but Bohr’s favorite example was the particle-wave duality that had been forced on physics by Einstein’s photon. Is light a particle? Or is it a wave? The two are so different that they seem totally irreconcilable.
Nevertheless, light is both a wave and a particle. Or more accurately, for certain kinds of experiments light behaves like particles. A very dilute beam of light falling on a photographic plate leaves tiny black dots: discrete evidence of the indivisible particle nature of the photon. On the other hand, those dots will eventually add up to a wavy interference pattern, a phenomenon that makes sense only for waves. It all depends on how you observe the light and what experiment you do. The two descriptions are complementary, not contradictory.
Another example of complementarity has to do with Heisenberg’s Uncertainty Principle. In classical physics the state of motion of a particle involves both its position and its momentum. But in quantum mechanics you either describe a particle by its position or its momentum—never both. The sentence, “A particle has a position AND a momentum,” must be replaced by, “A particle has a position OR a momentum.” Likewise, light is particles, OR light is waves. Whether you use one description or the other depends on the experiment.
Black hole complementarity is the new kind of complementarity that results from combining quantum mechanics with the theory of gravity. There is no single answer to the question, “Who is right? The observer who remains outside the black hole and sees all information radiated from just above the horizon? Or the observer who falls through with the bits that are heading toward the center of the black hole?” Each is right in its own context: they are complementary descriptions of two different experiments. In the first the experimenter stays outside the black hole. He may throw things in, collect photons as they come out, lower probes down to just above the horizon, observe the effects on the trajectories of particles passing near the black hole, and so on.
But in the second kind of experiment, the physicist prepares an experiment in her lab. Then, lab and all, she jumps into the black hole, crossing the horizon, while performing the experiment.
The complementary descriptions of the two experiments are so radically different that it hardly seems credible that they could both be right. The external observer sees matter fall toward the horizon, slow down, and hover just above it.3 The temperature just above the horizon is intense and reduces all matter to particles, which are finally radiated back out. In fact the external observer, monitoring the in-falling observer, sees her vaporized and reemitted as Hawking radiation.
But this is nothing like what the freely falling observer experiences. Instead she passes safely through the horizon without even noticing it. No bump or jolt, no high temperature, no warning of any kind signals the fact that she has passed the point of no return. If the black hole is big enough, let’s say, with a few million light-year radius, she would sail on for another million years with no discomfort. No discomfort, that is, until she reaches the heart of the black hole, where tidal forces—the distorting forces of gravity—eventually become so strong that… never mind, it’s too gruesome.
Two such different descriptions sound contradictory. But what we have learned from Bohr, Heisenberg, and others after them is that apparent paradoxes of this type signal genuine contradictions only when they lead to conflicting expectations for a single experiment. There is no danger of incompatible experimental results because the freely falling observer can never communicate her safe passage to the outside. Once she has safely passed the horizon, she is permanently out of contact with all observers who remain outside the black hole. Complementarity is strange but true.
The other major revolution of the early twentieth century was Einstein’s Theory of Relativity. Certain things are relative to the state of motion of the observer. Two different observers moving rapidly past each other will disagree about whether two events occurred at the same time. One observer might see two flashbulbs flash at exactly the same time. The other would see one flash take place before the other.
The Principle of Black Hole Complementarity is also a new and stronger relativity principle. Once again the description of events depends on the state of motion of the observer. Remaining at rest outside the black hole, you see one thing. Falling freely toward the interior of the black hole, you see the same events entirely differently.
Complementarity and relativity—the products of the great minds of the early twentieth century—are now being united in a radically new vision of space, time, and information.
The Holographic Principle
Perhaps the error that Hawking made is to think that a bit of information has a definite location in space. A simple example of a quantum bit is the polarization of a photon. Every photon has a screw sense to it. Imagine the electric field of a photon as it moves. The tip of the electric field moves with a helical
motion—a corkscrewlike motion. Think of yourself as following behind the light ray. The corkscrewing motion can be either clockwise or counterclockwise. In the first case the photons making up the beam are called right-handed photons; in the second case, they are left-handed. It’s like the direction that you would have to twist a screwdriver in order to drive a screw into the wall in front of you. Ordinary screws are right-handed, but no law of nature forbids left-handed screws. Photons come in both types. The distinction is called the circular polarization of the photon.
The polarization of a single photon is composed of a single quantum bit of information. Morse code messages could be sent in the form of a sequence of photons, the messages being coded in the sequence of polarizations instead of a sequence of dots and dashes.
What about the location of that bit of information? In quantum mechanics the location of a photon may not be definite. After all, you can’t specify both the location and momentum of the photon. Doesn’t that mean that the bit of information is not at a definite place?
You may not know exactly where the photon is, but you can measure its location if you choose. You just can’t measure both its position and its momentum. And once you measure the photon’s location, you know exactly where that bit of information is. Furthermore, in conventional quantum mechanics and relativity, every other observer will agree with you. In that sense the quantum bit of information has a definite location. At least that’s what was always thought to be the case.
But the Principle of Black Hole Complementarity says that the location of information is not definite, even in that sense. One observer finds the bits making up her own body are somewhere far behind the horizon. The other sees those same bits radiated back out from a region just outside the horizon. So it seems that the idea that information has a definite location in space is wrong.
There is an alternative way to think about it. In this view the bits have a location, but they’re not at all where you think they are. This is the holographic view of nature that grew out of thinking about black holes. How do holograms apply?
A picture, a photograph, or a painting is not the real world that it depicts. It’s flat, not full with three-dimensional depth like the real thing. Look at it from the side—almost edge on. It doesn’t look anything like the real scene viewed from an angle. In short it’s two-dimensional, while the world is three-dimensional. The artist, using perceptual sleight of hand, has conned you into producing a three-dimensional image in your brain, but in fact the information just isn’t there to form a three-dimensional model of the scene. There is no way to tell if that figure is a distant giant or a close midget. There is no way to tell if the figure is made of plaster or if it’s filled with blood and guts. The brain is providing information that is not really present in the painted strokes on the canvas or the darkened grains of silver on the photographic surface.
The screen of a computer is a two-dimensional surface filled with pixels. The actual data that are stored in a single image are in the form of some digital information about color and intensity—a collection of bits for each pixel. Like the painting or the photo, it is actually a very poor representation of the three-dimensional scene.
What would we have to do to faithfully store the full three-dimensional data, including depth information, as well as the “blood-and-guts” data about the interior of objects? The answer is obvious: instead of a collection of pixels filling two dimensions, we would need a space-filling collection of “voxels,” tiny elements that fill the volume of space.
Filling space with voxels is far more costly than filling a surface with pixels. For example, if your computer screen is a thousand pixels on a side, the total number of pixels is one thousand squared, or one million. But if we want to fill a volume of the same size with voxels, the required number would be one thousand cubed, or one billion.
That’s what makes holograms so surprising. A hologram is a two-dimensional image—an image on a piece of film—that allows you to unambiguously reconstruct full-blown three-dimensional images. You can walk around the reconstructed holographic image and see it from all sides. Your powers of depth perception allow you to determine which object in a hologram is closer or farther away. Indeed, if you move, the farther object can become the closer object. A hologram is a two-dimensional image, but one that has the full information of a three-dimensional scene. However, if you actually look closely at the two-dimensional film that contains the information, you see absolutely nothing recognizable. The image is thoroughly scrambled.
The information on a hologram, although scrambled, could be located on pixels. Of course nothing is for free. To describe the volume of space one thousand pixels on a side, the hologram would have to be composed of one billion pixels, not one million.
One of the strangest discoveries of modern physics is that the world is a kind of holographic image. But even more surprising, the number of pixels the hologram comprises is proportional only to the area of the region being described, not the volume. It is as though the full three-dimensional content of a region, one billion voxels in volume, can be described on a computer screen containing only a million pixels! Picture yourself in an enormous room bounded by walls, a ceiling, and a floor. Better yet, think of yourself as being in a large spherical space. According to the Holographic Principle, that fly just in front of your nose is really a kind of holographic image of data stored on the two-dimensional boundary of the room. In fact you and everything else in the room are images of data stored on a quantum hologram located on the boundary. The hologram is a two-dimensional array of tiny pixels—not voxels—each as big as the Planck length! Of course the nature of the quantum hologram and the way it codes three-dimensional data is very different from the way ordinary holograms work. But they do have in common that the three-dimensional world is completely scrambled.
What does this have to do with black holes? Let’s place a black hole in our large spherical room. Everything—black hole, space traveler, mother ship—is stored as information on the holographic walls of the space. The two different pictures that black hole complementarity tries to reconcile are simply two different reconstructions of the same hologram by two different reconstruction algorithms!
The Holographic Principle was not widely accepted when ’t Hooft and I put it forward in the early 1990s. My own view was that it was correct but that it would take many decades until we knew enough about quantum mechanics and gravity to confirm it in a precise way. But just three years later, in 1997, all that changed when a young theoretical physicist—Juan Maldacena—electrified the physics world with a paper titled “The Large N Limit of Superconformal Field Theories and Supergravity.” Never mind what the words mean. Maldacena, by cleverly using String Theory and Polchinski’s D-branes, had discovered a completely explicit holographic description of, if not our world, a world similar enough to make a convincing case for the Holographic Principle. Slightly later Ed Witten put his stamp of approval on the Holographic Principle with a follow-up to Maldacena’s paper titled “Anti De Sitter Space and Holography.” Since then the Holographic Principle has matured into one of the cornerstones of modern theoretical physics. It has been used in many ways to illuminate problems that, on the face of them, have nothing to do with black holes.
What does the Holographic Principle have to do with black hole complementarity? The answer is everything. Holograms are incredible scrambles of data that have to be decoded. That can be done by a mathematical algorithm or by shining laser light on the hologram. The laser light implements the mathematical algorithm.
Imagine a scene containing a large black hole and other things that might fall into the black hole as well as the radiation coming out. The entire scene can be described by a quantum hologram localized far away on some distant boundary of space. But now there are two possible ways—two algorithms—for decoding the hologram. The first reconstructs the scene as seen from outside the black hole, with the Hawking radiation carrying away all the bits that fell in. But the
second reconstruction shows the scene as it would be seen by someone falling into the black hole—one hologram, but two ways to reconstruct its content.
Bubbles All around Us
It’s probably too much to say that the three-dimensional world is a complete illusion. But the idea that the location of a bit of information is not necessarily where you might expect is now a widely accepted fact. What are its implications for the bubble bath universe of chapter 11? Let me remind you where we left off at the end of that chapter.
In the last chapter I explained the two views of history, one series and one parallel. According to the series view, every observer sees at most a small portion of the entire megaverse. The rest will never be seen because it is moving away so fast that light cannot bridge the gap. The boundary between what can and cannot be seen is the horizon. Unfortunately, the rest of the megaverse of pocket universes is all in this never-never land beyond the horizon. According to the classical principles of general relativity, we can wonder all we want about the existence and reality of these other worlds, but we can never know. They are irrelevant. They are meaningless in the scientific sense. They are metaphysics, not physics.
But exactly the same conclusion was incorrectly drawn about black hole horizons. Indeed, the cosmic event horizon of an eternally inflating universe is mathematically very similar to the horizon of a black hole. Let’s return to the infinite lake filled with boats and observers. The black hole was very much like the dangerous drain, the horizon being the point of no return. Let’s compare that situation with the eternally inflating lake, i.e., the lake fed by feeder tubes so that the floating observers all separate according to Hubble’s Law. If the lake is fed at a constant rate, it provides a precise analog for Eternal Inflation.
The Cosmic Landscape Page 36
