Posted on October 1, 2004

Soapbox Seminar #10

Can’t Win, Can’t Break Even, Can’t Get Out of the Game

Classical thermodynamics ... is the only physical theory
of a universal content concerning which I am convinced that,
within the framework of applicability of its basic concepts,
it will never be overthrown.

— Albert Einstein, Autobiographical Notes

Up till now, all the black holes we’ve looked at — from the tiny ones born in the Big Bang to the monsters gobbling up whole galaxies — have been what physicists call classical. Meaning that pretty much all their properties and behaviors can be understood within the framework of classical physics, more particularly Einstein’s relativistic brand of classical physics.

Remember John Wheeler’s “no hair” theorem from our Black Hole Primer? — the one that says that mass, spin, and charge are a black hole’s only distinguishing characteristics? Well, the classical-physics understanding of black holes is where that comes from.

And why it’s wrong.

Because, as we’ve seen, classical physics, even Uncle Albert’s far-out version of it, isn’t the ultimate truth about reality — quantum mechanics is.

Still, if leaving quantum mechanics out of the equation turned out to be an oversight, it was an understandable one. The quantum, after all, is a power to be reckoned with only at the very smallest scales, whereas the only black holes physicists knew about back in the sixties were these huge objects — twice the mass of the sun on up. General relativity might be just an approximation to some deeper, still undiscovered theory of “quantum gravity,” but on such astronomical scales it looked to be close enough.

It was John Wheeler himself who first realized there might be some “holes” in his own no-hair picture of black holes. But it wasn’t quantum effects John was worried about. No, it was one of the cornerstones of the classical physics worldview — entropy.

Entropy was a hot topic back in the nineteenth century, way back when steam power was the height of high tech. What with steam engines being the driving force behind the Industrial Revolution, you could make a boatload of money just by cranking their efficiency up another notch or two. Assuming you could crank it up, that is — nobody knew enough about the relationship between heat (or energy in general) and mechanical work to be sure. But it did get some folks to thinking, thinking hard. Folks like Joule and Carnot and Clausius and Kelvin.

What they came up with was a brand new science of energy. And, since they were still thinking in terms of the heat-energy needed to make steam, they went and called it thermo-dynamics, after the Greek word for heat.

And what thermodynamics said was that, in order to do useful work, you need to start off with an energy differential — a concentration, say, of high temperature against a cooler background.[1] And that, as part and parcel of doing the work, that concentration of energy gets spread out into its surroundings, making it that much less concentrated, and consequently less able to do work.

Take those steam engines that started the whole thing, for instance: Back in the nineteenth century, it was coal they burned to heat the water in their boilers and make the steam that drove looms and locomotives and such. But the reason that all works is that coal started out life as plants a long way back, plants that captured and distilled the energy in sunlight, and used it to build up complex hydrocarbon molecules through a process we call photosynthesis. And it’s the energy tied up in those molecular bonds that gets released when we burn a lump of coal.

But once that coal has burned, what we’re left with is waste heat, smoke, and ashes, none of which contains anywhere near the energy concentration of the original lump. It’s not like the energy has gone away. Heck, the First Law of Thermodynamics says we can’t destroy energy (or create it either, for that matter). All that’s gone is the differential (the boiler cools down to the same temperature as the rest of the boiler-room, even if the room itself is now a degree or so hotter). But that lack of a difference is enough to keep us from getting any more work out of the system.

It gets worse: Turns out there’s no way to put things back the way they were — to turn heat, smoke, and ashes back into usable fuel. Not without using up as much or more energy than you got from the fire in the first place. What that means is, you can never run your steam engine, or any other machine, at 100% efficiency; there’ll always be some heat loss or other form of energy dissipation that’ll leave things worse off than before, at least by a little. That’s the Second Law of Thermodynamics. And where the First Law said you can’t win (can’t create energy), the Second Law says you can’t break even either.[2]

Worse still, you don’t even need to be doing anything useful with the energy for it to dissipate like that. Leave Ma Nature to her own devices and, uh, stuff just sort of happens: iron rusts, ice melts, unrefrigerated food rots, piping hot coffee cools down to room temperature. The more our nineteenth-century physicists studied this sort of thing, the more they came to realize they were dealing with a universal tendency, inherent in all physical processes. So universal, in fact, that they went and gave it a fancy Greek name all its own: entropy. Which means, curiously enough, “inherent tendency.”[3]

Worst of all, if this entropy thing is really universal, that means the universe as a whole must be sliding downhill along with everything else — that the total entropy of the universe can only increase, never decrease, over time. Think about it: all of the energy in the universe diffusing into less and less usable forms, until it ends in a state of total thermal equilibrium where everything’s cooled to the same (very low) temperature and nothing can happen ever again because there’s no concentrations of energy left around to make it happen with.

Maximum entropy, they call it. Or, if you want it a little more dramatic: heat-death of the universe.

Fast-forward a century and then some, to the early 1970s.

Thermodynamics’ grim picture of a universe doomed to run down like a windup toy has long since become gospel, far as physical scientists are concerned. There’s no escaping it. Or is there?

Think about a black hole the way John Wheeler would have thought about it back then, as a place where spacetime comes to an end, an edge to the universe. Anything you throw into a black hole falls over that edge and is gone, forever.

So far, so good, but here’s the problem: What if the stuff you’re throwing in is in a state of high entropy to begin with — like, say, the burn products from that lump of coal we were talking about just a moment ago? Does that mean the entropy disappears over the edge along with the smoke and ashes and all? If it does, then we’ve found a way to commit what is, thermodynamically speaking, the perfect crime: a way to get rid of the evidence of an entropy increase, and never mind what the Second Law says.

It all comes down to that “no hair” theorem. According to John Wheeler, the only thing a black hole can ever tell you about itself is its mass, spin, and charge. But that’s no help at all here: A high-entropy system can have every bit as much mass or spin or charge as a low-entropy one. And once that system’s been sucked into a black hole, those three properties are all you’re ever going to get out of it. A black hole could’ve gobbled up all kinds of entropy and who’d be the wiser?

Now, most physicists didn’t see a problem here. Not much of a one, anyway. Sure, it was weird, but black holes are weird in general, so this newfound ability to swallow entropy whole was just the latest in a long line of weird things they did.

Jacob Bekenstein was not most physicists. He was just a lowly Princeton grad student back in late 1970, when Wheeler posed the black-hole entropy puzzler to him. As Wheeler himself recollects it:

In a joking mood one day in my office, I remarked to Jacob Bekenstein that I always felt like a criminal when I put a cup of hot tea next to a glass of iced tea and then let the two come to a common temperature, conserving the world’s energy but increasing the world’s entropy. My crime, I said to Jacob, 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?

It was just an offhand remark maybe, but it stuck in Jacob’s craw. This was serious stuff: the Second Law is such a bedrock principle of physics that saying black holes didn’t obey it might mean there really was no such thing as a black hole. Hell, it could mean that general relativity was in direct conflict with thermodynamics, that we didn’t understand the universe nearly as well as we thought we did.

John Wheeler’s little tea party threatened to become as disorienting and disastrous as the one in Alice in Wonderland.

Jacob pondered all this for months. Finally, he thought he had the answer. He showed up in Wheeler’s office one day and announced that you couldn’t remove entropy from the universe by throwing it into a black hole. Instead, he claimed, “the black hole already has entropy, and you only increase it” when you dump more in.

Jacob took it one giant step further — he said he knew what the black hole’s entropy was. It was the surface area of the hole’s event horizon.

When he said that, he was building on some 1971 work by Stephen Hawking, proving that the surface area of a black hole’s event horizon could only increase, never decrease, over time. See, the area of the event horizon is proportional to the black hole’s mass. If mass can only enter, never leave, that area can only increase. In fact, in you drop one black hole into another one, the combined hole has more surface area than the total sum of the original holes’ surface areas.[4]

Notice that I didn’t say what would happen if you split a large black hole into two smaller ones. Because you can’t. Once holes join, they’re joined for good. It’s irreversible.

”Irreversible” is a word that gets thrown around in thermodynamics a lot, with particular reference to entropy. So, you can’t blame Jacob for thinking he saw a connection. Heck, even Hawking himself had noted a similarity between his “area theorem” and the Second Law. Only he’d dismissed it as a coincidence.

There was a really good reason for not taking the resemblance too seriously: Remember how the whole notion of entropy grew out of studying the dynamics of heat? Well, that was no accident. You can’t measure entropy directly, you see — it’s a computed quantity. And, turns out the computation always involves temperature.

So, claiming black holes have entropy is the same as claiming they have a temperature. And, since temperature’s just a measure of radiation, claiming black holes have entropy is the same as claiming they radiate.

And there’s your problem in a nutshell. Radiation means giving off particles. That’s just how it’s done. Whereas black holes — classical ones, anyway — don’t give off anything, period.

No wonder Steve Hawking dropped this one like a “hot” potato!

But Jacob Bekenstein didn’t.

In fact, he devoted two whole chapters of his Ph.D. thesis to black hole thermodynamics. Then in the spring of 1972 he boiled it all down into a brief article, and — thinking it was maybe too controversial for the prestigious Physical Review Letters — submitted it to the Italian journal Lettere al Nuovo Cimento. In that article Jacob came right out and fingered the event horizon area as the only place where a black hole could be keeping its entropy, hiding it in plain sight, so to speak. And then he went and cited Stephen Hawking’s area theorem in support.

Well, if Jacob was trying to avoid controversy, that was not the way to do it. Then as now, the world of black hole physics was like one of those small West Texas towns where everybody knows everybody else’s business. News of the thesis, and preprints of the Nuovo Cimento article, spread like wildfire. Before long it’d reached the ears of Stephen Hawking himself.

Hawking was not pleased. In fact, he recalls experiencing “irritation with Bekenstein, because I felt he had misused my discovery of the increase of the area of the event horizon.” That’s maybe not so hard to understand: After all, Hawking had already rejected the area-to-entropy analogy on what seemed to him like pretty solid grounds — black holes couldn’t have a temperature, could they? — and now this novice physicist, this postdoc still wet behind the ears, comes along and says different.

So, Hawking got together with James Bardeen and Brandon Carter to try to do some damage control. Over the summer of 1972, the three of them worked out a response to Bekenstein’s upstart entropy proposal — what they called the “four laws of black hole mechanics.” Nowadays some folks trace the beginnings of black hole thermodynamics to this paper. Folks that haven’t read it, that is. Because right up front it makes it clear that the new Bardeen-Carter-Hawking laws are “similar to, but distinct from” those of thermodynamics.

The showdown came in August 1972, at a month-long summer school on black hole theory held at the Les Houches resort in the French Alps. There, with Mount Blanc towering in the background, Hawking, Bardeen, and Carter got up in front of an audience that just happened to include Jacob Bekenstein and presented their new laws of black hole “mechanics.”

All very gentlemanly, of course. This was theoretical physics, after all, not some barroom brawl.

But outside the lecture hall, the gloves came off. While the formal talks barely touched on the entropy question, in between the sessions a sort of behind-the-scenes campaign was starting up, aimed at ridiculing Jacob’s ideas. In that campaign, Bardeen and Hawking mostly stood back and let Brandon Carter take point. Not that he needed much encouragement. According to Jacob, “Carter was very vehemently against black hole thermodynamics.”

On one of their last evenings at Les Houches, Carter cornered Jacob in one of the side rooms and treated him to a long harangue about “How stupid the thermodynamic point of view is.” If there really was a one-to-one correlation between black-hole mechanics and thermodynamics, Carter said, then black holes would have to have a non-zero temperature, wouldn’t they? They’d have to emit particles, wouldn’t they? And everybody knows that’s just plain impossible.

Well, Jacob wasn’t ready to go up against “everybody” over this one. Especially since he had no more of a clue than anybody else as to how black holes could have a temperature. In the end, it was a fight where he couldn’t win, couldn’t even break even...

...But he also couldn’t get out of the game.

Feeling totally isolated — “lonely” and “intimidated” and “too bashful to take on such masters in public” is the way he puts it — he conceded in his next paper that, no, the notion of black holes having a temperature shouldn’t be taken literally. That doing so could, in fact, “easily lead to all sorts of paradoxes.”

But then — kind of like Galileo muttering “And yet it moves” after being forced to deny that the Earth goes round the sun — Jacob added: but somehow they still have entropy.

But to see why it’s wrong, we’ve got one more detour to take.

[Notes and Further Reading]

[1] Note that how hot and how cool don’t matter; only the difference does. You could make a perfectly good engine where the “hot” part of the cycle is at room temperature, long as the “cold” end is below that — say, down at the temperature of liquid nitrogen.[Return to text]

[2] And, yes, there’s a Third Law that says you can’t get out of the game. Not by cooling a system down to absolute zero temperature anyway — because it’s unattainable. [Return to text]

[3] There’s another way of talking about entropy; that’s entropy as a tendency toward increasing disorder. It all comes downto the same thing in the end, long as you keep in mind that terms like “orderly” and “messy” don’t represent value judgements here; they just refer to states of concentration and dissipation, respectively. But, with that proviso, thinking of entropy in terms of disorder can help explain what’s really driving the process.

Picture an egg inside its unbroken shell: yellow stuff in the middle, white stuff on the outside. If you could use differences in color to do real work, you’d have yourself a little engine right there. Now, drop that egg on the floor, and you’ll see that the state where every particle of yolk is concentrated in one spot is kind of unstable, and all too prone to slip into one of the other configurations where the yolk particles are mixed in with the albumen, a little or, more likely, a lot. It’s just that there are a lot more possible mixed states than unmixed ones, so any change is likely to be for the “worse.”

And, sure, it’s possible that your egg might survive its trip to the kitchen floor intact and unmixed. And cold coffee might reheat itself spontaneously, if you leave it alone for — oh, I don’t know — maybe a gazillion years. In that sense, entropy’s only a statistical certainty, not an absolute one. But, with odds like that, it’s definitely the way to bet. [Return to text]

[4] This is surprising! Take a lump of clay and chop it in two. Both faces of the “cut” are brand-new area. Pinch bits off a baseball-sized hunk of clay and roll them into a hundred clay marbles. The marbles have much more total surface area than the baseball did. Fuse them back into a baseball and surface disappears. But merge black-hole marbles together and the resultant black-hole baseball has more surface! [Return to text]

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James Bardeen, Brandon Carter, and Stephen W. Hawking, “The Four Laws of Black Hole Mechanics,” Communications in Mathematical Physics, vol. 31, no. 2 (1973), pp. 161-170.

Jacob D. Bekenstein, “Black Holes and the Second Law,” Lettere al Nuovo Cimento, 4, pp. 737-750 (1972).

Jacob D. Bekenstein, “Black holes and entropy,” Physics Review D (Particles and Fields) , vol. 7, no. 8 (1973), pp. 2333-2346.

Jacob D. Bekenstein, “Black-hole thermodynamics,” Physics Today, vol. 33, no. 1 (January 1980), pp. 24-31.

Jacob D. Bekenstein, Buchi neri, communicazione, energia, Rome: di Renzo, 2001.

Jacob D. Bekenstein, “Of black holes and entropy,” unpublished manuscript.

Stephen W. Hawking, “Fourth Lecture: Black Holes Ain’t So Black,” The Illustrated Theory of Everything: the Origin and Fate of the Universe, Beverly Hills CA: New Millennium, 2003.

Frank L. Lambert, “Entropy Is Simple — If We Avoid The Briar Patches!”, Version 3.0, May 2003, at: http://www.entropysimple.com/content.htm.

Kip S. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy, New York NY: Norton, 1994.

John Archibald Wheeler, A Journey into Gravity and Spacetime, New York NY: Scientific American Library, 1990.

John Archibald Wheeler (with Kenneth Ford), Geons, Black Holes & Quantum Foam: A Life in Physics, New York NY: Norton, 1998.

Special thanks to Professor Jacob D. Bekenstein for sharing his personal reminiscences of the discovery of black hole entropy, and its aftermath.

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