Third, physics is not just about making predictions. If one day we were to find a black box that always and accurately predicted the outcome of our particle physics experiments and our astronomical observations, the existence of the box would not bring inquiry in these fields to a close. There's a difference between making making predictions and predictions and understanding understanding them. The beauty of physics, its raison d'etre, is that it offers insights into them. The beauty of physics, its raison d'etre, is that it offers insights into why why things in the universe behave the way they do. The ability to predict behavior is a big part of physics' power, but the heart of physics would be lost if it didn't give us a deep understanding of the hidden reality underlying what we observe. And should the Many Worlds approach be right, what a spectacular reality our unwavering commitment to understanding predictions will have uncovered. things in the universe behave the way they do. The ability to predict behavior is a big part of physics' power, but the heart of physics would be lost if it didn't give us a deep understanding of the hidden reality underlying what we observe. And should the Many Worlds approach be right, what a spectacular reality our unwavering commitment to understanding predictions will have uncovered.

I don't expect theoretical or experimental consensus to come in my lifetime concerning which version of reality-a single universe, a multiverse, something else entirely-quantum mechanics embodies. But I have little doubt that future generations will look back upon our work in the twentieth and twenty-first centuries as having n.o.bly laid the basis for whatever picture finally emerges.

*For simplicity, we won't consider the electron's position in the vertical direction-we focus solely on its position on a map of Manhattan. Also, let me re-emphasize that while this section will make clear that Schrodinger's equation doesn't allow waves to undergo an instantaneous collapse as in Figure 8.6 Figure 8.6, waves can can be carefully prepared by the experimenter in a spiked shape (or, more precisely, very close to a spiked shape). be carefully prepared by the experimenter in a spiked shape (or, more precisely, very close to a spiked shape).

*For a mathematical depiction, see note 4 note 4.

*This non-chancy perspective would argue strongly for abandoning the colloquial terminology that I've used, "probability wave," in favor of the technical name, "wavefunction."

CHAPTER 9.

Black Holes and Holograms.

The Holographic Multiverse.

Plato likened our view of the world to that of an ancient forebear watching shadows meander across a dimly lit cave wall. He imagined our perceptions to be but a faint inkling of a far richer reality that flickers beyond reach. Two millennia later, it seems that Plato's cave may be more than a metaphor. To turn his suggestion on its head, reality-not its mere shadow-may take place on a distant boundary surface, while everything we witness in the three common spatial dimensions is a projection of that faraway unfolding. Reality, that is, may be akin to a hologram. Or, really, a holographic movie.

Arguably the strangest parallel world entrant, the holographic principle holographic principle envisions that all we experience may be fully and equivalently described as the comings and goings that take place at a thin and remote locus. It says that if we could understand the laws that govern physics on that distant surface, and the way phenomena there link to experience here, we would grasp all there is to know about reality. A version of Plato's shadow world-a parallel but thoroughly unfamiliar encapsulation of everyday phenomena-would envisions that all we experience may be fully and equivalently described as the comings and goings that take place at a thin and remote locus. It says that if we could understand the laws that govern physics on that distant surface, and the way phenomena there link to experience here, we would grasp all there is to know about reality. A version of Plato's shadow world-a parallel but thoroughly unfamiliar encapsulation of everyday phenomena-would be be reality. reality.

The journey to this peculiar possibility combines developments deep and far flung-insights from general relativity; from research on black holes; from thermodynamics; quantum mechanics; and, most recently, string theory. The thread linking these diverse areas is the nature of information in a quantum universe.

Information.

Beyond John Wheeler's knack for finding and mentoring the world's most gifted young scientists (besides Hugh Everett, Wheeler's students included Richard Feynman, Kip Thorne, and, as we will shortly see, Jacob Bekenstein), he had an uncanny ability to identify issues whose exploration could change our fundamental paradigm of nature's workings. During a lunch we had at Princeton in 1998, I asked him what he thought the dominant theme in physics would be in the decades going forward. As he had already done frequently that day, he put his head down, as if his aging frame had grown weary of supporting such a ma.s.sive intellect. But now the length of his silence left me wondering, briefly, whether he didn't want to answer or whether, perhaps, he had forgotten the question. He then slowly looked up and said a single word: "Information."

I wasn't surprised. For some time, Wheeler had been advocating a view of physical law quite unlike what a fledgling physicist learns in the standard academic curriculum. Traditionally, physics focuses on things things-planets, rocks, atoms, particles, fields-and investigates the forces that affect their behavior and govern their interactions. Wheeler was suggesting that things things-matter and radiation-should be viewed as secondary, as carriers of a more abstract and fundamental ent.i.ty: information. It's not that Wheeler was claiming that matter and radiation were somehow illusory; rather, he argued that they should be viewed as the material manifestations of something more basic. He believed that information-where a particle is, whether it is spinning one way or another, whether its charge is positive or negative, and so on-forms an irreducible kernel at the heart of reality. That such information is instantiated in real particles, occupying real positions, having definite spins and charges, is something like an architect's drawings being realized as a skysc.r.a.per. The fundamental information is in the blueprints. The skysc.r.a.per is but a physical realization of the information contained in the architect's design.

From this perspective, the universe can be thought of as an information processor. It takes information regarding how things are now and produces information delineating how things will be at the next now, and the now after that. Our senses become aware of such processing by detecting how the physical environment changes over time. But the physical environment itself is emergent; it arises from the fundamental ingredient, information, and evolves according to the fundamental rules, the laws of physics.

I don't know whether such an information-theoretic stance will reach the dominance in physics that Wheeler envisioned. But recently, driven largely by the work of physicists Gerard 't Hooft and Leonard Susskind, a major shift in thinking has resulted from puzzling questions regarding information in one particularly exotic context: black holes.

Black Holes.

Within a year of general relativity's publication, the German astronomer Karl Schwarzschild found the first exact solution to Einstein's equations, a result that determined the shape of s.p.a.ce and time in the vicinity of a ma.s.sive spherical object such as a star or a planet. Remarkably, not only had Schwarzschild found his solution while calculating artillery trajectories on the Russian front during World War I, but also he had beaten the master at his own game: to that point, Einstein had found only approximate solutions to the equations of general relativity. Impressed, Einstein publicized Schwarzschild's achievement, presenting the work before the Prussian Academy, but even so he failed to appreciate a point that would become Schwarzschild's most tantalizing legacy.

Schwarszchild's solution shows that familiar bodies like the sun and the earth produce a modest curvature, a gentle depression in the otherwise flat s.p.a.cetime trampoline. This matched well the approximate results Einstein had managed to work out earlier, but by dispensing with approximations, Schwarzschild could go further. His exact solution revealed something startling: if enough ma.s.s were crammed into a small enough ball, a gravitational abyss would form. The s.p.a.cetime curvature would become so extreme that anything venturing too close would be trapped. And because "anything" includes light, such regions would fade to black, a characteristic that inspired the early term "dark stars." The extreme warping would also bring time to a grinding halt at the star's edge; hence another early label, "frozen stars." Half a century later, Wheeler, who was nearly as adept at marketing as he was at physics, popularized such stars both within and beyond the scientific community with a new and more memorable name: black holes. It stuck.

When Einstein read Schwarzschild's paper, he agreed with the mathematics as applied to ordinary stars or planets. But as to what we now call black holes? Einstein scoffed. In those early days it was a challenge, even for Einstein, to fully understand the intricate mathematics of general relativity. While the modern understanding of black holes was still decades away, the intense folding of s.p.a.ce and time already apparent in the equations was, in Einstein's view, too radical to be real. Much as he would resist cosmic expansion a few years later, Einstein refused to believe that such extreme configurations of matter were anything more than mathematical manipulations-based on his own equations-run amok.1 When you see the numbers that are involved, it's easy to come to a similar conclusion. For a star as ma.s.sive as the sun to be a black hole, it would need to be squeezed into a ball about three kilometers across; a body as ma.s.sive as the earth would become a black hole only if squeezed to a centimetor across. The idea that there might be such extreme arrangements of matter seems nothing short of ludicrous. Yet, in the decades since, astronomers have gathered overwhelming observational evidence that black holes are both real and plentiful. There is wide agreement that a great many galaxies are powered by an enormous black hole at their center; our very own Milky Way galaxy is believed to revolve around a black hole whose ma.s.s is about three million times that of the sun. There's even a chance, as discussed in Chapter 4 Chapter 4, that the Large Hadron Collider may produce tiny black holes in the laboratory by packing the ma.s.s (and energy) of violently colliding protons into such a minuscule volume that Schwarzschild's result again applies, though on microscopic scales. Extraordinary emblems of math's ability to illuminate the dark corners of the cosmos, black holes have become the cynosures of modern physics.

Besides serving as a boon for observational astronomy, black holes have also been a fertile source of inspiration for theoretical research by providing a mathematical playground in which physicists can push ideas to their limits, conducting pen-and-paper explorations of one of nature's most extreme environments. As a weighty case in point, in the early 1970s Wheeler realized that when the venerable Second Law of Thermodynamics-a guiding light for over a century in understanding the interplay between energy, work, and heat-was considered in the vicinity of a black hole, it seemed to flounder. The fresh thinking of Wheeler's young graduate student Jacob Bekenstein came to the rescue, and in doing so planted the seeds of the holographic proposal.

The Second Law.

The aphorism "less is more" takes many forms. "Let's have the executive summary." "Just the facts." "TMI." "You had me at h.e.l.lo." These idioms are so common because every moment of every day we're bombarded with information. Thankfully, in most cases our senses pare down the details to those that really matter. If I'm out on the savanna and encounter a lion, I don't care about the motion of every photon reflecting off his body. Way TMI. I just want particular overall features of those photons, the very ones our eyes have evolved to sense and our brains to rapidly decode. Is the lion coming toward me? Is he crouched and stalking? Provide me with a moment-to-moment catalog of every reflected photon and, sure, I'll be in possession of all the details. What I won't have is any understanding. Less would indeed be very much more.

Similar considerations play a central role in theoretical physics. Sometimes we want to know every microscopic detail of a system we're studying. At the locations along the Large Hadron Collider's seventeen-mile-long tunnel where particles are steered into head-on collisions, physicists have placed mammoth detectors capable of tracking, with extreme precision, the motion of the particle fragments produced. Essential for gaining insight into the fundamental laws of particle physics, the data are so detailed that a year's worth would fill a stack of DVDs about fifty times as tall as the Empire State Building. But, as in that impromptu meeting with a lion, there are other situations in physics where that level of detail would obscure, not clarify. A nineteenth-century branch of physics called thermodynamics thermodynamics or, in its more modern incarnation, or, in its more modern incarnation, statistical mechanics statistical mechanics, focuses on such systems. The steam engine, the technological innovation that initially drove thermodynamics-as well as the Industrial Revolution-provides a good ill.u.s.tration.

The core of a steam engine is a vat of water vapor that expands when heated, driving the engine's piston forward, and contracts when cooled, returning the piston to its initial position, ready to drive forward once again. In the late nineteenth and early twentieth centuries, physicists worked out the molecular underpinnings of matter, which among other things provided a microscopic picture of the steam's action. As steam is heated, its H2O molecules pick up increasing speed and career into the underside of the piston. The hotter they are, the faster they go and the bigger the push. A simple insight, but one essential to thermodynamics, is that to understand the steam's force we don't need the details of which particular molecules happen to have this or that velocity or which happen to hit the piston precisely here or there. Provide me with a list of billions and billions of molecular trajectories, and I'll look at you just as blankly as I would if you listed the photons bouncing off the lion. To figure out the piston's push, I need only the average number of molecules that will hit it in a given time interval, and the average speed they'll have when they do. These are much coa.r.s.er data, but it's exactly such pared-down information that's useful.

In crafting mathematical methods for systematically sacrificing detail in favor of such higher-level aggregate understanding, physicists honed a wide range of techniques and developed a number of powerful concepts. One such concept, encountered briefly in earlier chapters, is entropy entropy. Initially introduced in the mid-nineteenth century to quantify energy dissipation in combustion engines, the modern view, emerging from Ludwig Boltzmann's work in the 1870s, is that entropy provides a characterization of how finely arranged-or not-the const.i.tuents of a given system need to be for it to have the overall appearance that it does.

To get a feel for this, imagine that Felix is frantic because he believes the apartment he shares with Oscar has been broken into. "They've ransacked us!" he tells Oscar. Oscar brushes him off-surely Felix is having one of his moments. To make his point, Oscar throws open the door to his bedroom, revealing clothing, empty pizza boxes, and crushed beer cans strewn everywhere. "It looks just like it always does," Oscar barks. Felix isn't swayed. "Of course it looks the same-ransack a pigsty and you get a pigsty. But look at my room." And he throws open his own door. "Ransacked," mocks Oscar; "it's neater than a straight whiskey." "Neat, yes. But the intruders have left their mark. My vitamin bottles? Not lined up in order of size. My collected works of Shakespeare? Out of alphabetical order. And my sock drawer? Look at this-some black pairs are in the blue bin! Ransacked, I tell you. Obviously ransacked."

Putting Felix's hysteria aside, the scenario makes plain a simple but essential point. When something is highly disordered, like Oscar's room, a great many possible rearrangements of its const.i.tuents leave its overall appearance intact. Grab the twenty-six crumpled shirts that were scattered across the bed, floor, and dresser, and toss them this way and that, fling the forty-two crushed beer cans randomly here and there, and the room will look the same. But when something is highly ordered, like Felix's room, even small rearrangements are easily detected.

This distinction underlies Boltzmann's mathematical definition of entropy. Take any system and count the number of ways its const.i.tuents can be rearranged without affecting its gross, overall, macroscopic appearance. That number is the system's entropy.* If there's a large number of such rearrangements, then entropy is high: the system is highly disordered. If the number of such rearrangements is small, entropy is low: the system is highly ordered (or, equivalently, has low disorder). If there's a large number of such rearrangements, then entropy is high: the system is highly disordered. If the number of such rearrangements is small, entropy is low: the system is highly ordered (or, equivalently, has low disorder).

For more conventional examples, consider a vat of steam and a cube of ice. Focus only on their overall macroscopic properties, those you can measure or observe without accessing the detailed state of either's molecular const.i.tuents. When you wave your hand through the steam, you rearrange the positions of billions upon billions of H2O molecules, and yet the vat's uniform haze looks undisturbed. But randomly change the positions and speeds of that many molecules in a piece of ice, and you'll immediately see the impact-the ice's crystalline structure will be disrupted. Fissures and fractures will appear. The steam, with H2O molecules randomly flitting through the container, is highly disordered; the ice, with H2O molecules arranged in a regular, crystalline pattern, is highly ordered. The entropy of the steam is high (many rearrangements will leave it looking the same); the entropy of the ice is low (few rearrangements will leave it looking the same).

By a.s.sessing the sensitivity of a system's macroscopic appearance to its microscopic details, entropy is a natural concept in a mathematical formalism that focuses on aggregate physical properties. The Second Law of Thermodynamics developed this line of insight quant.i.tatively. The law states that, over time, the total entropy of a system will increase.2 Understanding why requires only the most elementary grasp of chance and statistics. By definition, a higher-entropy configuration can be realized through many more microscopic arrangements than a lower-entropy configuration. As a system evolves, it's overwhelmingly likely to pa.s.s through higher-entropy states since, simply put, there are more of them. Understanding why requires only the most elementary grasp of chance and statistics. By definition, a higher-entropy configuration can be realized through many more microscopic arrangements than a lower-entropy configuration. As a system evolves, it's overwhelmingly likely to pa.s.s through higher-entropy states since, simply put, there are more of them. Many Many more. When bread is baking, you smell it throughout the house because there are trillions more arrangements of the molecules streaming from the bread that are spread out, yielding a uniform aroma, than there are arrangements in which the molecules are all tightly packed in a corner of the kitchen. The random motions of the hot molecules will, with near certainty, drive them toward one of the numerous spread-out arrangements, and not toward one of the few cl.u.s.tered configurations. The collection of molecules evolves, that is, from lower to higher entropy, and that's the Second Law in action. more. When bread is baking, you smell it throughout the house because there are trillions more arrangements of the molecules streaming from the bread that are spread out, yielding a uniform aroma, than there are arrangements in which the molecules are all tightly packed in a corner of the kitchen. The random motions of the hot molecules will, with near certainty, drive them toward one of the numerous spread-out arrangements, and not toward one of the few cl.u.s.tered configurations. The collection of molecules evolves, that is, from lower to higher entropy, and that's the Second Law in action.

The idea is general. Gla.s.s shattering, a candle burning, ink spilling, perfume pervading: these are different processes, but the statistical considerations are the same. In each, order degrades to disorder and does so because there are so many ways to be disordered. The beauty of this kind of a.n.a.lysis-the insight provided one of the most potent "Aha!" moments in my physics education-is that, without getting lost in the microscopic details, we have a guiding principle to explain why a great many phenomena unfold the way they do.

Notice, too, that, being statistical, the Second Law does not say that entropy can't can't decrease, only that it is extremely unlikely to do so. The milk molecules you just poured into your coffee might, as a result of their random motions, coalesce into a floating figurine of Santa Claus. But don't hold your breath. A floating milk Santa has very low entropy. If you move around a few billion of his molecules, you'll notice the result-Santa will lose his head or an arm, or he'll disperse into abstract white tendrils. By comparison, a configuration in which the milk molecules are uniformly spread around has enormously more entropy: a vast number of rearrangements continue to look like ordinary coffee with milk. With a huge likelihood, then, the milk poured into your dark coffee will turn it a uniform tan, with nary a Santa in sight. Similar considerations hold for the vast majority of high-to-low-entropy evolutions, making the Second Law appear inviolable. decrease, only that it is extremely unlikely to do so. The milk molecules you just poured into your coffee might, as a result of their random motions, coalesce into a floating figurine of Santa Claus. But don't hold your breath. A floating milk Santa has very low entropy. If you move around a few billion of his molecules, you'll notice the result-Santa will lose his head or an arm, or he'll disperse into abstract white tendrils. By comparison, a configuration in which the milk molecules are uniformly spread around has enormously more entropy: a vast number of rearrangements continue to look like ordinary coffee with milk. With a huge likelihood, then, the milk poured into your dark coffee will turn it a uniform tan, with nary a Santa in sight. Similar considerations hold for the vast majority of high-to-low-entropy evolutions, making the Second Law appear inviolable.

The Second Law and Black Holes.

Now to Wheeler's point about black holes. Back in the early 1970s, Wheeler noticed that when black holes amble onto the scene, the Second Law appears compromised. A nearby black hole seems to provide a ready-made and reliable means for reducing overall entropy. Throw whatever system you're studying-smashed gla.s.s, burned candles, spilled ink-into the hole. Since nothing escapes from a black hole, the system's disorder would appear permanently gone. Crude the approach may be, but it seems easy to lower total entropy if you have a black hole to work with. The Second Law, many thought, had met its match.

Wheeler's student Bekenstein was not convinced. Perhaps, Bekenstein suggested, entropy is not lost to the black hole but merely transferred to it. After all, no one claimed that, in gorging themselves on dust and stars, black holes provide a mechanism for violating the First Law of Thermodynamics, the conservation of energy. Instead, Einstein's equations show that when a black hole gorges, it gets bigger and heftier. The energy in a region can be redistributed, with some falling into the hole and some remaining outside, but the total is preserved. Maybe, Bekenstein suggested, the same idea applies to entropy. Some entropy stays outside a given black hole and some entropy falls in, but none gets lost.

This sounds reasonable, but experts shot Bekenstein down. Schwarzschild's solution, and much work that followed, seemed to establish that black holes are the epitome of order. Infalling matter and radiation, however messy and disordered, are crushed to infinitesimal size at a black hole's center: a black hole is the ultimate in orderly trash compaction. True, no one knows exactly what happens during such powerful compression, because the extremes of curvature and density disrupt Einstein's equations; but there just doesn't seem to be any capacity for a black hole's center to harbor disorder. And outside the center, a black hole is nothing but an empty region of s.p.a.cetime extending to the boundary of no return-the event horizon-as in Figure 9.1 Figure 9.1. With no atoms or molecules wafting this way and that, and thus no const.i.tuents to rearrange, a black hole would seem to be entropy-free.

Figure 9.1 A black hole comprises a region of s.p.a.cetime surrounded by a surface of no return, the event horizon A black hole comprises a region of s.p.a.cetime surrounded by a surface of no return, the event horizon.

In the 1970s, this view was reinforced by the so-called no hair theorems no hair theorems, which established mathematically that black holes, much like the bald performers of Blue Man Group, have a dearth of distinguishing characteristics. According to the theorems, any two black holes that have the same ma.s.s, charge, and angular momentum (rate of rotation) are identical identical. Lacking any other intrinsic traits-as the Blue Men lack bangs, mullets, or dreads-black holes seemed to lack the underlying differences that would harbor entropy.

By itself, this was a fairly convincing argument, but there was a yet more d.a.m.ning consideration that seemed to definitively undercut Bekenstein's idea. According to basic thermodynamics, there's a close a.s.sociation between entropy and temperature. Temperature is a measure of the average motion of an object's const.i.tuents: hot objects have fast-moving const.i.tuents, cold objects have slow-moving const.i.tuents. Entropy is a measure of the possible rearrangements of these const.i.tuents that, from a macroscopic viewpoint, would go unnoticed. Both entropy and temperature thus depend on aggregate features of an object's const.i.tuents; they go hand in hand. When worked out mathematically, it became clear that if Bekenstein was right and black holes carried entropy, they should also have a temperature.3 That That idea set off alarm bells. Any object with a nonzero temperature radiates. Hot coal radiates visible light; we humans, typically, radiate in the infrared. If a black hole has a nonzero temperature, the very laws of thermodynamics that Bekenstein was seeking to preserve state that it too should radiate. But that conflicts blatantly with the established understanding that nothing can escape a black hole's gravitational grip. Most everyone concluded that Bekenstein was wrong. Black holes do not have a temperature. Black holes do not harbor entropy. Black holes are entropy sinkholes. In their presence, the Second Law of Thermodynamics fails. idea set off alarm bells. Any object with a nonzero temperature radiates. Hot coal radiates visible light; we humans, typically, radiate in the infrared. If a black hole has a nonzero temperature, the very laws of thermodynamics that Bekenstein was seeking to preserve state that it too should radiate. But that conflicts blatantly with the established understanding that nothing can escape a black hole's gravitational grip. Most everyone concluded that Bekenstein was wrong. Black holes do not have a temperature. Black holes do not harbor entropy. Black holes are entropy sinkholes. In their presence, the Second Law of Thermodynamics fails.

Despite the evidence mounting against him, Bekenstein had one tantalizing result on his side. In 1971, Stephen Hawking realized that black holes obey a curious law. If you have a collection of black holes with various ma.s.ses and sizes, some engaged in stately orbital waltzes, others pulling in nearby matter and radiation, and still others crashing into each other, the total surface area of the black holes increases over time the total surface area of the black holes increases over time. By "surface area," Hawking meant the area of each black hole's event horizon. Now, there are many results in physics that ensure quant.i.ties don't change over time (conservation of energy, conservation of charge, conservation of momentum, and so on), but there are very few that require quant.i.ties to increase. It was natural, then, to consider a possible relation between Hawking's result and the Second Law. If we envision that, somehow, the surface area of a black hole is a measure of the entropy it contains, then the increase in total surface area could be read as an increase in total entropy.

It was an enticing a.n.a.logy, but no one bought it. The similarity between Hawking's area theorem and the Second Law was, in almost everyone's view, nothing more than a coincidence. Until, that is, a few years later, when Hawking completed one of the most influential calculations in modern theoretical physics.

Hawking Radiation.

Because quantum mechanics plays no role in Einstein's general relativity, Schwarzschild's black hole solution is based purely in cla.s.sical physics. But proper treatment of matter and radiation-of particles like photons, neutrinos, and electrons that can carry ma.s.s, energy, and entropy from one location to another-requires quantum physics. To fully a.s.sess the nature of black holes and understand how they interact with matter and radiation, we must update Schwarzschild's work to include quantum considerations. This isn't easy. Notwithstanding advances in string theory (as well as in other approaches we haven't discussed, such as loop quantum gravity, twistors, and topos theory), we are still at an early stage in our attempt to meld quantum physics and general relativity. Back in the 1970s, there was still less theoretical basis for understanding how quantum mechanics would affect gravity.

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 s.p.a.cetime environment (the general relativity part). As I pointed out in Chapter 4 Chapter 4, a full union would, at the very least, consider not only the quantum jitters of fields within s.p.a.cetime but the jitters of s.p.a.cetime 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 s.p.a.cetime 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 s.p.a.cetime 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 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 negative energy-a concept that would be meaningless in a cla.s.sical universe. energy-a concept that would be meaningless in a cla.s.sical 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 s.p.a.ce. 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 s.p.a.ce.

Hawking reconsidered such ubiquitous quantum jitters not in the setting of empty s.p.a.ce 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 s.p.a.ce. 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 s.p.a.ce and time can cause particles that have negative energy, as determined by anyone outside the hole, to appear to have positive 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. 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 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 ma.s.s increases when it absorbs anything that carries positive energy, so its ma.s.s 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 ma.s.s gets ever smaller.5 When quantum considerations are included, black holes are thus not completely black. This was Hawking's bolt from the blue. 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 ma.s.s. 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 ma.s.s 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 ma.s.s 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 do have a temperature, that they have a temperature, that they do do emit radiation, suggests that the experts had too quickly rejected Bekenstein's suggestion that black holes emit radiation, suggests that the experts had too quickly rejected Bekenstein's suggestion that black holes do 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. 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 p.r.o.nouncement of a universe with ever-increasing disorder.

The conclusion provided a welcome relief. To many physicists, the Second Law, emerging from seemingly una.s.sailable 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 That honor went to identifying honor went to identifying where entropy is stored 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 quant.i.tatively, as the number of rearrangements of a system's microscopic const.i.tuents 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 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. 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 ill.u.s.trate, 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 cl.u.s.ter, Oscar sees only a haphazard a.s.sortment 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 factor of 2 for each coin-yielding a total of 21000, which is[image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image][image] . The vast majority of these heads-tails arrangements would have no distinguishing features, so they would not stand out in any way. Some . The vast majority of these heads-tails arrangements would have no distinguishing features, so they would not stand out in any way. Some would 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 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 related to the number of indistinguishable rearrangements of its const.i.tuents, but properly speaking is not equal to the number itself. The relationship is expressed by a mathematical operation called a the number of indistinguishable rearrangements of its const.i.tuents, but properly speaking is not equal to the number itself. The relationship is expressed by a mathematical operation called a logarithm; logarithm; don't be put off if this brings back bad memories of high school math cla.s.s. 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 2 don't be put off if this brings back bad memories of high school math cla.s.s. 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 how much information is contained in that list. is contained in that list.

So, you start to ponder. What actually is 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 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 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 bit-a familiar computer-age term that is short for binary digit binary digit, meaning a 0 0 or a or a 1 1, which you can think of as a numerical representation of yes yes or or no 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 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, 2 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 is the number of yes-no questions any one such sequence answers. 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. 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 quant.i.tative 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 (1033 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 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 (1066 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. 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 ill.u.s.trated in This is schematically ill.u.s.trated in Figure 9.2 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 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 ma.s.s, 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 const.i.tuents that the information is meant to describe. The mathematical a.n.a.lyses pinned down the quant.i.ty quant.i.ty of information a given black hole contains, without providing insight into the information itself. 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 s.p.a.ce inside 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 doc.u.ments. The same goes for the information in my head, which seems tied to the volume of my brain, the available s.p.a.ce 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. 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 doc.u.ments. The same goes for the information in my head, which seems tied to the volume of my brain, the available s.p.a.ce 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 (1033 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 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 1099 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 s.p.a.ce 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 of Planck-sized squares covering its surface and not by the number of Planck-sized cubes filling its volume. 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 s.p.a.ce 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 of 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 Figure 9.2, is a symbolic ill.u.s.tration 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. 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) a.s.sume 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 ma.s.sive object and freely floating in the depths of empty s.p.a.ce: 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 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 s.p.a.ce. 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 s.p.a.ce.

This realization renders the black hole's entropy all the more puzzling. If as you pa.s.s through the horizon of a black hole you find nothing there, nothing at all to distinguish it from empty s.p.a.ce, 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 Figure 5.2 provides a good visual metaphor; mathematical examples come from the mirror shapes of string theory's extra dimensions ( provides a good visual metaphor; mathematical examples come from the mirror shapes of string theory's extra dimensions (Chapter 4) and the naively 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 pa.s.s 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 When the distant observer measures the Hawking radiation's temperature, she finds it to be tiny; let's say it's 1013 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. 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 s.p.a.ce. And one thing we know for sure is that when you float in empty s.p.a.ce, you don't suddenly burst into flames. So the conclusion is that from your perspective, you pa.s.s 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. your experience is indistinguishable from that of floating in empty s.p.a.ce. And one thing we know for sure is that when you float in empty s.p.a.ce, you don't suddenly burst into flames. So the conclusion is that from your perspective, you pa.s.s 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 or not alive. But this is no ordinary situa