The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos

by Brian Greene

  Even so, a number of early researchers developed a partial union of quantum mechanics and general relativity by considering quantum fields (the quantum part) evolving in a fixed but curved spacetime environment (the general relativity part). As I pointed out in Chapter 4, a full union would, at the very least, consider not only the quantum jitters of fields within spacetime but the jitters of spacetime itself. To facilitate progress, the early work steadfastly avoided this complication. Hawking embraced the partial union and studied how quantum fields would behave in a very particular spacetime arena: that created by the presence of a black hole. What he found knocked physicists clear off their seats.

  A well-known feature of quantum fields in ordinary, empty, uncurved spacetime is that their jitters allow pairs of particles, for instance an electron and its antiparticle the positron, to momentarily erupt out of the nothingness, live briefly, and then smash into each other, with mutual annihilation the result. This process, quantum pair production, has been intensively studied both theoretically and experimentally, and is thoroughly understood.

  A novel characteristic of quantum pair production is that while one member of the pair has positive energy, the law of energy conservation dictates that the other must have an equal amount of negative energy—a concept that would be meaningless in a classical universe.* But the uncertainty principle provides a window of weirdness whereby negative-energy particles are allowed as long as they don’t overstay their welcome. If a particle exists only fleetingly, quantum uncertainty establishes that no experiment will have adequate time, even in principle, to determine the sign of its energy. This is the very reason why the particle pair is condemned by quantum laws to swift annihilation. So, over and over again, quantum jitters result in particle pairs being created and annihilated, created and annihilated, as the unavoidable rumbling of quantum uncertainty plays itself out in otherwise empty space.

  Hawking reconsidered such ubiquitous quantum jitters not in the setting of empty space but near the event horizon of a black hole. He found that sometimes events look much as they ordinarily do. Pairs of particles are randomly created; they quickly find each other; they are destroyed. But every so often something new happens. If the particles are formed sufficiently close to the black hole’s edge, one can get sucked in while the other careens into space. In the absence of a black hole this never happens, because if the particles failed to annihilate each other then the one with negative energy would outlive the protective haze of quantum uncertainty. Hawking realized that the black hole’s radical twisting of space and time can cause particles that have negative energy, as determined by anyone outside the hole, to appear to have positive energy to any unfortunate observer inside the hole. In this way, a black hole provides the negative energy particles a safe haven, and so eliminates the need for a quantum cloak. The erupting particles can forgo mutual annihilation and blaze their own separate trails.4

  The positive-energy particles shoot outward from just above the black hole’s event horizon, so to someone watching from afar they look like radiation, a form since named Hawking radiation. The negative-energy particles are not directly seen, because they fall into the black hole, but they nevertheless have a detectable impact. Much as a black hole’s mass increases when it absorbs anything that carries positive energy, so its mass decreases when it absorbs anything that carries negative energy. In tandem, these two processes make the black hole resemble a piece of burning coal: the black hole emits a steady outward stream of radiation as its mass gets ever smaller.5 When quantum considerations are included, black holes are thus not completely black. This was Hawking’s bolt from the blue.

  Which is not to say that your average black hole is red hot, either. As particles stream from just outside the black hole, they fight an uphill battle to escape the strong gravitational pull. In doing so, they expend energy and, because of this, cool down substantially. Hawking calculated that an observer far from the black hole would find that the temperature for the resulting “tired” radiation was inversely proportional to the black hole’s mass. A huge black hole, like the one at the center of our galaxy, has a temperature that’s less than a trillionth of a degree above absolute zero. A black hole with the mass of the sun would have a temperature less than a millionth of a degree, minuscule even compared with the 2.7-degree cosmic background radiation left to us by the big bang. For a black hole’s temperature to be high enough to barbecue the family dinner, its mass would need to be about a ten-thousandth of the earth’s, extraordinarily small by astrophysical standards.

  But the magnitude of a black hole’s temperature is secondary. Although the radiation coming from distant astrophysical black holes won’t light up the night sky, the fact that they do have a temperature, that they do emit radiation, suggests that the experts had too quickly rejected Bekenstein’s suggestion that black holes do have entropy. Hawking then nailed the case. His theoretical calculations determining a given black hole’s temperature and the radiation it emits gave him all the data he needed to determine the amount of entropy the black hole should contain, according to the standard laws of thermodynamics. And the answer he found is proportional to the surface area of the black hole, just as Bekenstein had proposed.

  So by the end of 1974, the Second Law was law once again. The insights of Bekenstein and Hawking established that in any situation, total entropy increases, as long as you account for not only the entropy of ordinary matter and radiation but also that contained within black holes, as measured by their total surface area. Rather than being entropy sinks that subvert the Second Law, black holes play an active part in upholding the law’s pronouncement of a universe with ever-increasing disorder.

  The conclusion provided a welcome relief. To many physicists, the Second Law, emerging from seemingly unassailable statistical considerations, came as close to sacred as just about anything in science. Its restoration meant that, once again, all was right with the world. But, in time, a vital little detail in the entropy accounting made it clear that the Second Law’s balance sheet was not the deepest issue in play. That honor went to identifying where entropy is stored, a matter whose importance becomes clear when we recognize the deep link between entropy and the central theme of this chapter: information.

  Entropy and Hidden Information

  So far, I’ve described entropy, loosely, as a measure of disorder and, more quantitatively, as the number of rearrangements of a system’s microscopic constituents that leave its overall macroscopic features unchanged. I’ve left implicit, but will now make explicit, that you can think of entropy as measuring the gap in information between the data you have (those overall macroscopic features) and the data you don’t (the system’s particular microscopic arrangement). Entropy measures the additional information hidden within the microscopic details of the system, which, should you have access to it, would distinguish the configuration at a micro level from all the macro look-alikes.

  To illustrate, imagine that Oscar has straightened up his room, except that the thousand silver dollars he won in last week’s poker game remain scattered across the floor. Even after he gathers them in a neat cluster, Oscar sees only a haphazard assortment of dollar coins, some heads and others tails. Were you to randomly change some heads to tails and other tails to heads, he’d never notice—evidence that the thousand-dropped-silver-dollar system has high entropy. Indeed, this example is so explicit that we can do the entropy counting. If there were only two coins, there’d be four possible configurations: (heads, heads), (heads, tails), (tails, heads), and (tails, tails)—two possibilities for the first dollar, times two for the second. With three coins, there’d be eight possible arrangements: (heads, heads, heads), (heads, heads, tails), (heads, tails, heads), (heads, tails, tails), (tails, heads, heads), (tails, heads, tails), (tails, tails, heads), (tails, tails, tails), arising from two possibilities for the first, times two for the second, times two for the third. With a thousand coins, the number of possibilities follows exactly the same pattern—a f
actor of 2 for each coin—yielding a total of 21000, which is . The vast majority of these heads-tails arrangements would have no distinguishing features, so they would not stand out in any way. Some would, for instance, if all 1,000 coins were heads or all were tails, or if 999 were heads, or 999 tails. But the number of such unusual configurations is so extraordinarily small, compared with the huge total number of possibilities, that removing them from the count would hardly make a difference.*

  From our earlier discussion, you’d deduce that the number 21000 is the entropy of the coins. And, for some purposes, that conclusion would be fine. But to draw the strongest link between entropy and information, I need to sharpen up the description I gave earlier. The entropy of a system is related to the number of indistinguishable rearrangements of its constituents, but properly speaking is not equal to the number itself. The relationship is expressed by a mathematical operation called a logarithm; don’t be put off if this brings back bad memories of high school math class. In our coin example, it simply means that you pick out the exponent in the number of rearrangements—that is, the entropy is defined as 1,000 rather than 21000.

  Using logarithms has the advantage of allowing us to work with more manageable numbers, but there’s a more important motivation. Imagine I ask you how much information you’d need to supply in order to describe one particular heads-tails arrangement of the 1,000 coins. The simplest response is that you’d need to provide the list—heads, heads, tails, heads, tails, tails …—that specifies the disposition of each of the 1,000 coins. Sure, I respond, that would tell me the details of the configuration, but that wasn’t my question. I asked how much information is contained in that list.

  So, you start to ponder. What actually is information, and what does it do? Your response is simple and direct. Information answers questions. Years of research by mathematicians, physicists, and computer scientists have made this precise. Their investigations have established that the most useful measure of information content is the number of distinct yes-no questions the information can answer. The coins’ information answers 1,000 such questions: Is the first dollar heads? Yes. Is the second dollar heads? Yes. Is the third dollar heads? No. Is the fourth dollar heads? No. And so on. A datum that can answer a single yes-no question is called a bit—a familiar computer-age term that is short for binary digit, meaning a 0 or a 1, which you can think of as a numerical representation of yes or no. The heads-tails arrangement of the 1,000 coins thus contains 1,000 bits’ worth of information. Equivalently, if you take Oscar’s macroscopic perspective and focus only on the coins’ overall haphazard appearance while eschewing the “microscopic” details of the heads-tails arrangement, the coins’ “hidden” information content is 1,000 bits.

  Notice that the value of the entropy and the amount of hidden information are equal. That’s no accident. The number of possible heads-tails rearrangements is the number of possible answers to the 1,000 questions—(yes, yes, no, no, yes, …) or (yes, no, yes, yes, no, …) or (no, yes, no, no, no, …), and so on—namely, 21000. With entropy defined as the logarithm of the number of such rearrangements—1,000 in this case—entropy is the number of yes-no questions any one such sequence answers.

  I’ve focused on the 1,000 coins so as to offer a specific example, but the link between entropy and information is general. The microscopic details of any system contain information that’s hidden when we take account of only macroscopic, overall features. For instance, you know the temperature, pressure, and volume of a vat of steam, but did an H2O molecule just hit the upper right-hand corner of the box? Did another just hit the midpoint of the lower left edge? As with the dropped dollars, a system’s entropy is the number of yes-no questions that its microscopic details have the capacity to answer, and so the entropy is a measure of the system’s hidden information content.6

  Entropy, Hidden Information, and Black Holes

  How does this notion of entropy, and its relation to hidden information, apply to black holes? When Hawking worked out the detailed quantum mechanical argument linking a black hole’s entropy to its surface area, he not only brought quantitative precision to Bekenstein’s original suggestion, he also provided an algorithm for calculating it. Take the event horizon of a black hole, Hawking instructed, and divide it into a gridlike pattern in which the sides of each cell are one Planck length (10–33 centimeters) long. Hawking proved mathematically that the black hole’s entropy is the number of such cells needed to cover its event horizon—the black hole’s surface area, that is, as measured in square Planck units (10–66 square centimeters per cell). In the language of hidden information, it’s as if each such cell secretly carries a single bit, a 0 or a 1, that provides the answer to a single yes-no question delineating some aspect of the black hole’s microscopic makeup.7 This is schematically illustrated in Figure 9.2.

  Figure 9.2 Stephen Hawking showed mathematically that the entropy of a black hole equals the number of Planck-sized cells that it takes to cover its event horizon. It’s as if each cell carries one bit, one basic unit of information.

  Einstein’s general relativity, as well as the black hole no-hair theorems, ignores quantum mechanics and so completely misses this information. Choose values for its mass, its charge, and its angular momentum, and you’ve uniquely specified a black hole, says general relativity. But the most straightforward reading of Bekenstein and Hawking tells us you haven’t. Their work established that there must be many different black holes with the same macroscopic features that, nevertheless, differ microscopically. And much as is the case in more commonplace settings—coins on the floor, steam in a vat—the black hole’s entropy reflects information hidden within the finer details.

  Exotic as black holes may be, these developments suggested that, when it comes to entropy, black holes behave much like everything else. But the results also raised puzzles. Although Bekenstein and Hawking tell us how much information is hidden within a black hole, they don’t tell us what that information is. They don’t tell us the specific yes-no questions the information answers, nor do they even specify the microscopic constituents that the information is meant to describe. The mathematical analyses pinned down the quantity of information a given black hole contains, without providing insight into the information itself.8

  These were—and remain—perplexing issues. But there’s yet another puzzle, one that seems even more basic: Why would the amount of information be dictated by the area of the black hole’s surface? I mean, if you asked me how much information was stored in the Library of Congress, I’d want to know about the available space inside the Library of Congress. I’d want to know the capacity, within the library’s cavernous interior, for shelving books, filing microfiche, and stacking maps, photographs, and documents. The same goes for the information in my head, which seems tied to the volume of my brain, the available space for neural interconnections. And it goes for the information in a vat of steam, which is stored in the properties of the particles that fill the container. But, surprisingly, Bekenstein and Hawking established that for a black hole, the information storage capacity is determined not by the volume of its interior but by the area of its surface.

  Prior to these results, physicists had reasoned that since the Planck length (10–33 centimeters) was apparently the shortest length for which the notion of “distance” continues to have meaning, the smallest meaningful volume would be a tiny cube whose edges were each one Planck length long (a volume of 10–99 cubic centimeters). A reasonable conjecture, widely believed, was that irrespective of future technological breakthroughs, the smallest possible volume could store no more than the smallest unit of information—one bit. And so the expectation was that a region of space would max out its information storage capacity when the number of bits it contained equaled the number of Planck cubes that could fit inside it. That Hawking’s result involved the Planck length was therefore not surprising. The surprise was that the black hole’s storehouse of hidden information was determined by the number o
f Planck-sized squares covering its surface and not by the number of Planck-sized cubes filling its volume.

  This was the first hint of holography—information storage capacity determined by the area of a bounding surface and not by the volume interior to that surface. Through twists and turns across three subsequent decades, this hint would evolve into a dramatic new way of thinking about the laws of physics.

  Locating a Black Hole’s Hidden Information

  The Planckian chessboard with 0s and 1s scattered across the event horizon, Figure 9.2, is a symbolic illustration of Hawking’s result for the amount of information harbored by a black hole. But how literally can we take the imagery? When the math says that a black hole’s store of information is measured by its surface area, does that merely reflect a numerical accounting, or does it mean that the black hole’s surface is where the information is actually stored?

  It’s a deep issue and has been pursued for decades by some of the most renowned physicists.* The answer depends sensitively on whether you view the black hole from the outside or from the inside—and from the outside, there’s good reason to believe that information is indeed stored at the horizon.

  To anyone familiar with the finer details of how general relativity depicts black holes, this is an astoundingly odd claim. General relativity makes clear that were you to fall through a black hole’s event horizon, you would encounter nothing—no material surface, no signposts, no flashing lights—that would in any way mark your crossing the boundary of no return. It’s a conclusion that derives from one of Einstein’s simplest but most pivotal insights. Einstein realized that when you (or any object) assume free-fall motion, you become weightless; jump from a high diving board, and a scale strapped to your feet falls with you and so its reading drops to zero. In effect, you cancel gravity by giving in to it fully. From this, Einstein leaped to an immediate consequence. Based on what you experience in your immediate environment, there’s no way for you to distinguish between freely falling toward a massive object and freely floating in the depths of empty space: in both situations you are perfectly weightless. Sure, if you look beyond your immediate environment and see, say, the earth’s surface rapidly getting closer, that’s a pretty good clue that it’s time to pull your parachute cord. But if you are confined to a small, windowless capsule, the experiences of free fall and free float are indistinguishable.9