Trespassing on Einstein's Lawn (25 page)

She laughed again, and we made our way to our desks. I sat down, relieved to know that presumably I did look all right, but also a bit concerned about some of my previous responses.

The next morning, I opened the large glass door to the reception area and took a deep breath. I was ready to nail it this time.

The receptionist looked up and smiled. “Are you all right?”

I opened my mouth to repeat the phrase back to him, but I just couldn't do it. Answering a question with the same question was too advanced for my small-talk skills. Instead, I gave him a little upward tilt of the chin and replied, “Whassup?” He smiled, but it still didn't feel right.

Meanwhile, I needed to choose a topic for my thesis. I knew that this was my opportunity to delve deeply into a particular question. It was what I had come to London to do. I needed to choose carefully.

As I sifted through ideas, I kept coming back to the arrow of time. We had discussed the mystery of time's arrow extensively in my philosophy
of statistical mechanics class. Einstein's theories had supposedly put time and space on equal footing, sewing them together into one big block universe. Why, then, can we move backward in space but not in time? Relativity didn't have the answer, and particle physics wasn't any help, either. The laws of physics that govern particle interactions work the same way forward and backward. If particles don't see an arrow of time, why should we?

You need some kind of gross asymmetry on which to pin time's arrow. Luckily, there is one: entropy never decreases. And entropy, like the arrow of time, only exists here on the macro scale with us, not in the micro realm where particles live. I had always heard entropy described as a measure of disorder, but fundamentally, I learned, it's a measure of hidden information. If you want to describe a physical system, say a gas in a box, you have two options. You can track the constantly shuffling positions and momenta of every single molecule in the gas, or you can just take an average. Average the rate of shifting positions and call it temperature; average the changing momenta and call it pressure. Temperature and pressure are macroscopic shorthand—every last bit of information about the ever-changing micro states compressed into just two numbers.

There are tons of different possible microscopic arrangements that will all average out to the same macroscopic feature. The greater the number of microscopic possibilities, the harder it is for us to guess which arrangement is the real one, so the less precise our information about the micro state and the higher the entropy of the system. That's where the disorder comes in—there are way more microscopic configurations compatible with “disordered” states than with “ordered” ones. Countless arrangements of H
₂
O molecules correspond to a puddle of water; far fewer correspond to an intricately structured ice crystal. The puddle is messier—we have less information about its hidden inner workings, so it has more entropy. And more entropy means heat.

At first that didn't seem right. Why should our lack of information manifest itself as something as physical as heat?
We can literally be burned by our own ignorance?
I had written in my notebook. But it made sense once I thought about the scales involved. Temperature isn't part of bedrock reality—it's an emergent, collective macroscopic
feature. An individual molecule doesn't have a temperature. So if you choose to study a system at the level of individual molecules, temperature vanishes. Average out the microscopic information to look at swarm behavior and now you've got heat. It's a matter of choosing a scale—choose a larger one and you trade information for temperature.

When I pour milk into my coffee, it swirls around for a second, then dissipates, settling into a monochromatic mocha color. Why? Why doesn't it ever spontaneously settle into the shape of the word
hello
? Because that cup of coffee has something like 10
²⁴
molecules sloshing around, and the number of arrangements that correspond to a uniform mocha color vastly outweighs the number of configurations that correspond to
hello.
“Vastly” doesn't even begin to capture those odds. I could sit here and wait for billions of years as the configurations shift every split second and it still wouldn't be long enough for my coffee to offer a warm greeting. What were the odds of all the air molecules in my flat stumbling into the fleeting configuration of a rat? I wondered. How about just the tail?

So that was that, the second law of thermodynamics: entropy never decreases. It's merely statistical, but it's enough that physicists treat it as sacrosanct. It's enough that it treats us to an arrow of time. Entropy always increases because, statistically, high-entropy states are overwhelmingly more likely than low-entropy states. If entropy de-creased—if milk reswirled out of a mocha-colored coffee, if the air from my car's exhaust flew right back into the pipe whence it came, if cracked eggs healed—it would look like someone was rewinding the world. Like time was moving backward.

But saying that high entropy is more likely than low is not in itself enough to give you an arrow of time. After all, high entropy would be the most likely scenario in the past, too. Statistically, entropy should always be high—and once it's high enough it reaches a kind of mocha-colored equilibrium and there's nowhere else to go. In a universe in equilibrium, nothing would ever happen, save the rare statistical fluctuation every few billion years. But we don't live in equilibrium. We live in a universe where things happen all the time. Where entropy still has room to rise.

The only way to get an arrow of time is to assume that for some
unknown reason the universe began in an extraordinarily unlikely low-entropy state. Milk mixes into my morning coffee because 13.7 billion years ago the universe began in a rare configuration. Breakfast has cosmic significance. The mystery is no longer why there's an arrow of time; it's why did the universe start out so improbably?

When I first heard this issue about the mysterious low-entropy origin, it didn't seem to add up. What about the cosmic microwave background? It was a snapshot from nearly the beginning of time and it showed a universe that's perfectly smooth to one part in a hundred thousand. That sure looked like equilibrium to me. I searched some books for an explanation, and eventually I found one. The entropy that's low at the beginning of time isn't thermodynamic entropy—it's gravitational entropy. Once you get to large enough scales, thermodynamic entropy doesn't run the show. Gravity does. And gravitational entropy has its own arrow that points in the opposite direction. From gravity's point of view, a smooth universe is ripe for clumping. Gravity always attracts, so a world where nothing is clumped together is hugely unlikely. If gravity had its way, the whole universe would be a giant black hole—gravitational equilibrium.

The cosmic arrow of time depended on gravitational entropy, but when I tried to look into the matter further, I found that physicists didn't really know what gravitational entropy was. If entropy is a measure of missing information about the micro scale, what kind of microscopic information does gravity encode? Of course, if physicists knew the answer to that—if they knew the microscopic structure of gravity—they wouldn't be thinking about the arrow of time. They'd have found quantum gravity.

There is, however, one place where gravitational entropy
is
well-defined: on the event horizon of a black hole. Suddenly I realized that there was something amazing, something profound lurking there. I wasn't sure what it was, but I knew I had found the topic for my thesis.

Einstein had discovered that mass and energy wrinkle space, but he never anticipated that there would be places where space becomes so twisted that it turns on itself, like a snake swallowing its tail. When a
massive star burns through its fuel and buckles under its own weight, gravity ignites a runaway process of collapse. Growing denser and denser, the star can eventually collapse right through itself, burrowing into the very fabric of spacetime. In the aftermath of this implosive chain reaction, space and time are left warped beyond recognition, laying bare the bones of reality. It was Wheeler who gave this mangled bit of world a name: black hole.

Black holes bring together the three pillars of physics—general relativity, quantum theory, and thermodynamics—and force them to battle it out. If you're hunting ultimate reality, a black hole is the place to look. It is the breakdown of time and space, the beginning and end of the universe. It's where the shards are restored to symmetry. At the center lurks a singularity, a place where spacetime curvature approaches infinity and physics turns pathological. As the radius of spacetime curvature spirals down toward the Planck length, physics as we know it falls away, unveiling some terra incognita only quantum gravity can traverse.

Given the similarity of a black hole's singularity to the singularity at the origin of time, I had always assumed that singularities would be the most interesting aspects of black holes. I was wrong. I quickly learned that the real action was out on the edge: the event horizon. The horizon is the gravitational point of no return, a surface of spacetime where gravity's grip exactly counteracts the speed of light. It is a surface of light rays frozen in place by gravity. For an observer outside the black hole, the event horizon is a kind of cosmic wall. Since light can't escape it, an observer can never see anything on the other side.
For all intents and purposes, there is no other side.
It is fundamentally and eternally out of reach—nothing that takes place there can have any physical impact on the external world. It's what makes the black hole black. It carves the world into pieces.

The horizon is a one-way door: things go in, but they can never come out. For physicists, that was a huge problem. It meant entropy could go in and never come out, which would mean the entropy of the universe outside the black hole was decreasing. So much for the arrow of time.

The first step toward a solution came in 1970 when Stephen Hawking was getting ready for bed. Suddenly Hawking realized that in
order for horizons to be stable, the light rays that constitute them must never travel toward one another, only parallel or apart. That meant that the area of an event horizon could never shrink. If matter or radiation fell into a black hole, the area of its horizon would inevitably grow, and if two black holes merged, the horizon of the resulting black hole would be equal to or greater than the sum of the original two.

The area of an event horizon can never decrease. When Jacob Bekenstein, one of Wheeler's students at Princeton, heard about Hawking's area theorem, he couldn't help but notice that it bore a striking resemblance to the second law of thermodynamics. Could they be related? It was just a hunch, but he knew it was heretical, and he might have swept it under the rug if not for Wheeler.

“I always feel like a criminal when I put a hot cup of tea next to a glass of iced tea and then let the two come to a common room temperature, conserving the world's energy but increasing the world's entropy,” Wheeler told him. “My crime echoes down to the end of time, for there is no way to erase or undo it. But let a black hole swim by and let me drop the hot tea and the cold tea into it. Then is not all evidence of my crime erased forever?”

Jesus, I thought. Did Wheeler ever just speak in normal sentences?

Apparently Bekenstein understood what he meant. Months later he showed up in Wheeler's office with a bold claim: an event horizon's area, he said, is not only analogous to entropy, it
is
entropy. Wheeler replied, “Your idea is so crazy that it might just be right. Go ahead and publish it.”

When Hawking read Bekenstein's paper, he was pissed. He felt like Bekenstein had misused his area theorem to push an intrinsically flawed idea. The problem was obvious. Entropy means heat. Anything with entropy has a temperature—it radiates. Only black holes can't radiate. They're
black.

Annoyed, Hawking, along with physicists Brandon Carter and Jim Bardeen, wrote a paper explaining why Bekenstein couldn't possibly be right. But the idea kept nagging him, and after two years of calculations Hawking arrived at a shocking conclusion. In his now legendary 1975 paper “Particle Creation by Black Holes,” he showed that when quantum mechanics meets gravity at the horizon, particles appear.
That means that black holes do radiate heat, with a temperature inversely proportionate to their mass. If black holes can radiate, they must have entropy. The arrow of time had been saved, black holes were no longer quite so black, and Bekenstein had been vindicated. Hawking produced an equation showing that the entropy of a black hole is proportional to one-quarter the area of its event horizon. He requested that it be inscribed on his tombstone.

Soon other similarities between black hole physics and thermodynamics began to emerge. The so-called zeroth law of thermodynamics says that temperature is constant in a thermodynamic system in equilibrium. Likewise, gravity is constant across the surface of an event horizon. The first law of thermodynamics says that energy can change forms but is always conserved. Likewise in black hole physics. When an object is swallowed by a black hole, its mass and energy (which, via E = mc
²
, are interchangeable) are transferred to the black hole itself, so that the total energy of the system is always conserved. For each law of thermodynamics, there seemed to be an equivalent law of black hole physics. A deep connection between thermodynamics and gravity was beginning to emerge. For physicists it was an intriguing connection, to say the least. After all, thermodynamics is about matter and energy. Gravity is about space and time. Find the connections between the two and you're headed fast down the road to quantum gravity.

In hindsight, the notion that an event horizon would be marked by entropy isn't that surprising—after all, entropy is a measure of hidden information, and an event horizon's entire job is to hide information. But why would the entropy of the black hole, which represents all the hidden stuff inside the three-dimensional volume, scale with the two-dimensional area of the horizon? And where the hell were those particles coming from?