by Brian Greene
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 space 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 massive as the sun to be a black hole, it would need to be squeezed into a ball about three kilometers across; a body as massive 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 mass is about three million times that of the sun. There’s even a chance, as discussed in Chapter 4, that the Large Hadron Collider may produce tiny black holes in the laboratory by packing the mass (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 hello.” 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 or, in its more modern incarnation, 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 illustration.
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 coarser 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. 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 constituents 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 constituents 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 constituents 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).
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 d
etailed state of either’s molecular constituents. 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 assessing 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 quantitatively. 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 pass through higher-entropy states since, simply put, there are more of them. 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 clustered 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. Glass 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 analysis—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 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 glass, 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 spacetime extending to the boundary of no return—the event horizon—as in Figure 9.1. With no atoms or molecules wafting this way and that, and thus no constituents to rearrange, a black hole would seem to be entropy-free.
Figure 9.1 A black hole comprises a region of spacetime surrounded by a surface of no return, the event horizon.
In the 1970s, this view was reinforced by the so-called 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 mass, charge, and angular momentum (rate of rotation) are 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 damning consideration that seemed to definitively undercut Bekenstein’s idea. According to basic thermodynamics, there’s a close association between entropy and temperature. Temperature is a measure of the average motion of an object’s constituents: hot objects have fast-moving constituents, cold objects have slow-moving constituents. Entropy is a measure of the possible rearrangements of these constituents that, from a macroscopic viewpoint, would go unnoticed. Both entropy and temperature thus depend on aggregate features of an object’s constituents; 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 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 masses 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. By “surface area,” Hawking meant the area of each black hole’s event horizon. Now, there are many results in physics that ensure quantities don’t change over time (conservation of energy, conservation of charge, conservation of momentum, and so on), but there are very few that require quantities 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 analogy, 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 classical physics. But proper treatment of matter and radiation—of particles like photons, neutrinos, and electrons that can carry mass, energy, and entropy from one location to another—requires quantum physics. To fully assess 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.