Posted on October 10, 2004

Soapbox Seminar #11

Hot Enough for You?

When you get right down to it, that whole big controversy over black hole thermodynamics — the one that came to a head in the summer of ’72 — was all just a matter of degree.

Or degrees, rather. Degrees Kelvin. As in temperature.

Because, when push came to shove at Les Houches, the black-hole physics establishment was ready to flush the second law of thermodynamics down the toilet (or down the event horizon, same thing) rather than face up to the possibility that black holes might actually have a temperature.

Because that was the same thing as saying black holes might radiate.

... which was another way of saying they might emit particles.

...which kind of defeats the whole purpose of being a black hole, doesn’t it?

After all, black holes are supposed to be these permanent fixtures of the universe, right? Nothing, not even light, is supposed to be able to escape from them. And now we’re saying they can leak?

No wonder Stephen Hawking felt that flash of “irritation” at Jacob Bekenstein for “misusing” Hawking’s area theorem in pursuit of black hole entropy.

But that was nothing compared to the “annoyance” Hawking was about to experience over the course of the next year — an annoyance that nearly won him the Nobel Prize for Physics (and may still).

It happened like this.

In mid-1971, maybe six or eight months after — and halfway round the world from — where John Wheeler had first talked tea and entropy with Jacob Bekenstein, CalTech physicist Kip Thorne was rousted out of his Moscow hotel room bed by a 6 a.m. phone call.

The impatient early-morning caller was none other than Russia’s top black-hole researcher, and Kip’s good friend, Yakov Borisovich Zel’dovich.

Yakov, it seemed, thought he might’ve found a way to coax radiation out of a spinning black hole.

Later that same morning, a bleary-eyed Kip listened as Yakov explained how a rotating hole would amplify the short-lived gravitational fluctuations of the vacuum to the point where they’d become real gravity waves — an effect no one had ever predicted before. It didn’t help that Yakov backed up his claim with an analogy to how a rotating metal sphere would amplify electro- magnetic fluctuations and emit radiate real electromagnetic waves, since that was an effect no one had ever predicted before either.

So, okay, an unsubstantiated guess based on an equally shaky analogy, and all that before the morning’s second cup of coffee. Kip tried breaking it to his friend gently.

“That’s one of the craziest things I’ve ever heard,” he said.

And Kip was right — in general relativity terms, at least, where nothing gets out of a black hole alive.

The thing of it is, Yakov wasn’t playing on that old classical-physics side of the street anymore. No, he’d crossed over and gone off in a direction nobody’s thought to look in before: toward the application of quantum mechanics to black holes.

Because that’s what Zel’dovich’s vacuum fluctuations are — quintessentially quantum phenomena. In fact, they’re of a piece with the virtual particle-pairs we talked about way back when, remember? Those particles are perpetually popping into existence and then winking out again, even in the depths of supposably empty space (”created and annihilated, created and annihilated — what a waste of time,” as Dick Feynmann used to say). So the fact that gravity waves can do pretty much the same thing — fluctuate in and out of existence, that is — should come as no surprise, though it doesn’t make it any the less weird.

Now plunk down a spinning black hole in the path of those gravity fluctuations. That hole will grab hold of them and whip them around itself and fling them loose. Part of them. The fluctuations’ outer edge sucks up enough energy from the hole’s spin to boost it to the speed of light and away. The rest spirals down into the event horizon and is gone.

Seeing as how they’ve been packed off in two different directions like this, the two halves of the virtual fluctuation can’t come back together to do their mutual annihilation trick anymore. The half that speeds away from the rotating hole has become a real gravity wave. The spinning black hole is radiating.

Zel’dovich figured this kind of thing would go on as long as the hole had enough spin to pull the fluctuations apart. But the splitting operation itself would drain off rotational energy. Eventually, the hole would spin down to a stop and go quiet.

No more spin, no more radiation. Just a nice, normal, non-radiating Schwarzschild black hole, like in all the textbooks.

Still, the fact that the hole would eventually stop radiating was no comfort. It wasn’t supposed to be radiating at all, ever. Kip argued that point every way he knew how, but Yakov was no poor unknown grad student like Jacob Bekenstein. He was a world-class theorist at the top of his game. And he wouldn’t budge.

Finally, after going back and forth for hours, the two friends agreed to table the disagreement. And make a side-bet: Kip’s fifth of White Horse Scotch against Yakov’s decanter of fine Georgian cognac says spinning black holes can radiate.

And that’s how the matter rested for the next two years. Zel’dovich published a paper on the topic with his grad student Alexei Starobinskii in 1971. But it took Zel’dovich’s towering reputation to get it past the referees even in his native Russia, and outside of the Soviet Union nobody paid it any mind at all. It wasn’t mentioned once at the Les Houches summer school in 1972, where Jacob Bekenstein’s black hole thermodynamics was shot down in flames for even hinting that black holes might somehow radiate.

Seemed like nobody wanted to hear about spinning black holes giving off gravity waves or anything else.

What turned things around — and ultimately won Yakov his scotch — was when Stephen Hawking himself visited Moscow in the fall of 1973. There, in Hawking’s hotel room, with Kip Thorne serving as interpreter, Yakov and Alexei managed to convince Hawking that spinning black holes really could radiate. “I believed their arguments on physical grounds,” Hawking recalls, “but I did not like the mathematical way in which they calculated the emission.”

Now, it’s not something physicists like to talk about a whole lot, but we still haven’t figured out how to combine these two theories of quantum mechanics and general relativity — these two best understandings of how the universe works at the largest and smallest scales — into one big, totally consistent picture. But partial solutions are possible, and Zel’dovich and Starobinskii had patched one together so as to shore up their physical intuitions of emissions from rotating black holes.

It was this patchwork approach that Hawking didn’t like. When he got back to Cambridge, he decided he’d take his own crack at the problem, use some better approximations and maybe get it right this time.

What he did get gave him a sinking feeling — somewhere between “surprise and annoyance” and flat-out “horror,” depending on who’s telling the story. Because Hawking’s results not only confirmed that Zel’dovich’s spinning black holes would radiate. They said that all black holes would radiate, rotating or not. And they wouldn’t just give off gravity waves, they’d emit all kinds of particles in all kinds of configurations, maybe even sperm whales and TV sets and encyclopedias.

There had to be a mistake somewhere.

Hawking’s first thought was that there was a problem with one of the approximations he’d been using. And that he’d better find it and fix it but quick. Because, as Steve himself put it, “If Bekenstein found out about it, he would use it as a further argument to support his ideas about the entropy of black holes, which I still did not like.”

Well, those approximations were a necessary evil all right, seeing how little was known about how quantum mechanics and relativity really fit together. But, try as he might, Hawking couldn’t find any glaring errors in his math. He found something a lot stranger instead: the math was predicting that black holes would give off radiation in the same way (with the same spectrum, that is) that a normal hot object would.

In other words, the math was saying that black holes did have a temperature, after all. They gave off particles, which they’re not supposed to be able to do. And, as a direct result of that, they did something else they’re not supposed to be able to do.

They shrank.

To see why Steve Hawking was “surprised,” “annoyed,” or “horrified” — take your pick — by all this, recall that a black hole’s gravity is so strong that not even light can escape from it. Recall, too, that lightspeed is an upper limit: Nothing — no material body and no information — gets to travel faster.

And then try to figure how the thing could be emitting particles and losing mass.

Well, you can’t, of course. Not if you stick with classical relativity, you can’t. There’s just no way.

But, of course, Hawking wasn’t sticking with relativity pure and simple. Like Zel’dovich before him, he was trying to combine it with quantum mechanics. If nobody’d ever succeeded in doing that, it wasn’t for lack of trying — marrying (relativistic) gravity up with the other three (quantum) forces of nature in a “Unified Field Theory” had started out as Einstein’s fondest dream, after all

But then, nobody’d ever looked at black holes as a way to arrange the marriage either. Far as most folks were concerned, a black hole was about as close to a purely classical object as you could get, way too big and massive for quantum effects to put a dent in.

Well, Hawking went and showed how the right kind of quantum effects could not only ding a black hole, they could blow it all to hell and gone!

Fair warning, there are three or four different explanations — or physical interpretations, as us physicists like to call them — for what Hawking’s calculations were telling him. There’s virtual particle-pairs and quantum tunneling and travelling backwards in time or faster than light (just a skootch). They all add up to the same thing in the end, though. So, if it’s all the same to you, I’m going to stick with the one we’ve been working with up till now and explain “Hawking radiation” in terms of virtual particles.

We first met up with virtual particles back when we were looking at how the universe might have popped into existence as the biggest something-for- nothing scam of all time. And we ran into them again just now, in the form of Zel’dovich’s virtual gravity fluctuations, the ones that made it possible for a spinning black hole to give off gravity waves.

What all these virtual particles or waves have in common, you’ll recall, is that, left to their own devices, they’re gone almost before they get here. Because they’re created in matter/antimatter pairs (or the equivalent), they slam back together again and annihilate each other in a few dozen attoseconds.[1] It takes energy, and lots of it, to make them real.

That much energy’s hard to come by in empty space, but it’s easy to find in Mr. Black Hole’s neighborhood. It all comes down to — the tides.

The topic of tides came up a while back, when we were talking about what goes on inside a black hole. As we saw back then, tide happens. It happens whenever an object is sitting with one end closer to a concentration of mass than the other. What with gravity decreasing by the inverse square of the distance and all, that mass will pull harder on the near end of the object than on the far end. Which is to say, it’ll try to pull the object apart.

Now, most places in the universe, this tidal effect is so weak that it’s got to be working across really big distances —across the diameter of the whole earth, say, like the moon’s gravity does — before you’d notice anything at all.

Not in the vicinity of a black hole.

There, gravity can vary on microscopic scales, over distances short enough to split a virtual particle pair into two real particles.

Now, picture this happening right at the edge of the event horizon. If the hole’s gravity acted on the new particles as a unit, it’d suck them both in, of course. But they’re not a unit. One might be a trifle closer to the hole than the other. One might have a velocity vector towards the hole. The other might have a similar vector, or it might not. It’s all a crap shoot, is what I mean to say.

Sometimes, both particles will fall into the hole. Sometimes neither will (in which case they’ll only last long enough to re-combine and annihilate each other). But sometimes, one’ll fall in and the other’ll escape — with no partner to cancel it out. Random as a roulette wheel. But that kind of randomness is meat and potatoes for Statistical Thermodynamics. Gazillions of particle-pairs, each behaving at random, but their behavior overall is dead simple to predict.

Here in the outside world, we only see a particle if it doesn’t get either canceled out or sucked in. To us, that particle looks as if the black hole had just given it off, radiated it. Multiply the effect by the whole area of the event horizon, and pretty soon you’ve got a steady stream of particles (looking like they’re) coming off the thing.

Radiation, in other words. In other words, temperature.

Oh, not much of a temperature. Not for the big ones, anyhow. See, the chances of a virtual particle pair being pulled apart by gravity are — hmmm, how do I put this?

Okay. Suppose a really, really energetic pair of particles pops out of nothingness. They’ll vanish again (to balance the books) in nothing flat. So they can’t move very far apart in their brief lifespan. That means that gravity’ll affect them almost equally. Hardly any chance that one will fall in and the other will be “promoted” into Reality with a capital R.

But a low-energy pair will last longer. That gives gravity more time to yank them apart, maybe grab one or the other of them for good. So you’re always going to get more low-energy particles streaming off a hole than high-energy ones. And the lowest-energy particles there are are long-wavelength photons. You’d get a lot of them. That’s the kind of radiation a cold body puts out.

Okay, now take a hole the mass of the sun. Its event horizon’d be a couple of kilometers across. So, moving a centimeter closer to or further from the horizon isn’t going to make that much of a difference in the force of its gravity. It’s only changes in distance to the mass-center that change gravity’s strength, and a centimeter’s pretty small potatoes when we’re talking a kilometer or two.

Now, that’s not to say a stellar-size hole is pitch black, but you’d hardly notice the difference. A hole twice the mass of the sun is going to be pretty chilly — just a few billionths of a degree above absolute zero. Way, way below the cosmic background radiation. Big black holes are cold!

But we’re not talking big ones here, and the less a hole weighs, the easier it is for it to create particles. Skinny a black hole down to subatomic scales, and all kinds of stuff can pop out. Because then even a little difference in position between two particles could make one heck of a difference in the forces acting on them. On those scales the tide’ll amplify even the tiniest displacements. Making it much more likely that half of a pair’ll be sucked in so the other half can go free.

So a small hole produces much more radiation. I’m talking in absolute terms here, not just radiation per unit surface of the horizon. And the average particle will be more energetic, too. Hotter, that is. You’ll get short wavelength photons. Visible light, X-rays, gamma rays. You’ll get particles with rest-mass. Electrons and positrons. Protons and anti-protons. ’Course, the heavier they are, the fewer, but they’re there all the same. Skinny a black hole down enough and all kinds of stuff can pop out.

Remember those atom-size primordial black holes we were talking about a couple seminars back? Well, they’re the ones that can really pump out the heat. Enough to raise their effective temperature into the billions of degrees on up.

And every particle that escapes from the hole carries off a teensy bit of its energy with it. Or, to say the same thing a different way, every particle that falls in contributes a teensy bit of negative energy. No matter how you look at it, the hole is losing energy. And that, according to special relativity, is the same as losing mass.

Our radiating black hole is evaporating. Give it enough time and it’ll evaporate down to nothing!

Again, how much time depends on the size. A black hole just two times the mass of the sun can keep on trickling out radiation for a lot longer than the life expectancy of the universe. A billion, trillion, trillion, trillion, trillion times longer (give or take a weekend). Don’t hold your breath.

But a “small” hole (under a billion tons, say) changes appreciably over a short time — couple hundred million years. It shrinks, gets hotter, which makes it shrink even faster. What we call “positive feedback,” like miswiring your thermostat to squirt more fuel into your furnace when the house is already too hot. Positive feedback always ends in a disaster. Your furnace burns out. A mini-hole expends its final few megatons of mass in a fraction-of-a-second orgy of hard radiation. If one of them is within a dozen light years of earth when its clock runs out — well, we’d know about it!

It wouldn’t be the end of the world, mind you — nor even close. Still, the spy satellites that watch for secret A-bomb tests, they’d spot it easy enough. And that’s at a couple of light years out. We haven’t seen any so far (or if we have, we’re not telling), but the satellites haven’t been up there for all that long. It just means small holes are rare enough that none have “timed out” within a relatively small bubble of space surrounding us, over a relatively short period. Hardly proof they’re not there. We’ll only see them when they die. One that had millennia, or years, or even just weeks to go on the meter could be sitting right here in our own solar system, unnoticed, just ticking away.

The primordial black holes formed in the Big Bang could’ve come in all sizes. The tiniest would’ve radiated in the trillions of degrees, but only for picoseconds. Then they’d cut loose with a final burst of gamma radiation. Pretty spectacular. Some folks think their superhot lives and deaths might’ve actually managed to reheat the early universe for whole microseconds.

’Course, the law of averages says that at least some of the primordials would’ve been big enough to last a while. Any of them that started out massing more than a billion tonnes or so could still be around today.

It’s one of these long-lived critters that’s going to play the starring role in our Vurdalak Conjecture, and we’ll be getting to that real soon now.

But first we’ve got one more cosmological side-trip to make. Because, if I’m right, Vurdalak wasn’t just any old garden-variety primordial black hole.

Nossir, it was something really, really special.

[Notes and Further Reading]

[1] One attosecond = 10E-18 seconds = one quintillionth of a second, or one millionth of a millionth of a millionth of a second. Virtual particles usually last about 80 attoseconds.[Return to text]

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Timothy Ferris, The Whole Shebang: A State-of-the-Universe(s) Report, New York NY: Simon & Schuster, 1997.

Stephen W. Hawking, “Black Hole Explosions?”, Nature, vol. 248, March 1 1974, pp. 30-31.

Stephen W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43 (1975), pp. 199-220. Reprinted in Stephen W. Hawking, Hawking on the Big Bang and Black Holes, Singapore: World Scientific, 1993, pp. 85-106.

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.

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

Yakov B. Zel’dovich and Alexei A. Starobinskii, “Particle Production and Vacuum Polarization in an Anisotropic Gravitational Field,” Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki [Journal of Experimental and Theoretical Physics], vol. 61 (1971), p.2161.

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