# Things I Learned While Eating Cookies: Black-Holes-as-the-Key-to-What-We-Can’t-Know Edition

February 19, 2011 1 Comment

So it’s taken me quite some time to decide what aspect of last week’s colloquium I could turn into a blog post. Although from a string theorist’s perspective I the material was “dumbed down” to be accessible to a general physics audience, it was a fairly technical talk and hard to excerpt nicely. Seeing as it is already time for the *next* cookie-filled colloquium, however, I’ve decided just have a go at explaining the broad overview of the talk rather than trying to pull out a particular detail.

The speaker last week (Princeton’s Steve Gubser, a real up-and-coming theorist and the quintessential American nerd) certainly knew how to play to an audience. His talk, titled “On the sometimes-happy relationship between string theory and heavy-ion physics” could not have been better tailored to the Stony Brook department, which is excellent in many areas but outstanding at string theory and nearly unparalleled in nuclear physics (shut up, people at U Michigan). The fact that our school excels in these two areas, however, is largely a coincidence, since up until a few years ago they were considered widely different disciplines of physics. But that was before the discovery of the aDS/CFT correspondence, which formed a bridge between the two and is the concept I want to explain in the rest of the post.

To understand the power of aDS/CFT, however, you have to understand something that many people don’t realize: just because physicists say they’ve developed a successful theory of something doesn’t mean they necessarily know how to *use* it. The fact of the matter is, there are many problems in physics which are often spoken of as though they are “solved” simply because we have developed equations that describe the problem. But unfortunately, just because you know the equation that governs something’s behavior doesn’t necessarily mean you can use that equation to get new information.

I think this tends to confuse people because most of us are used to encountering equations only when they can be explicitly solved. Understandably, when you’re doing homework assignments or working out the mathematics of everyday situations, you tend to encounter things like “x^2+ x = 0.” If you know the right tricks and can figure out the right series of steps, you can quickly figure out that, whatever “x” stands for, it should be equal to zero or negative one. But not all equations can be unpacked this easily and made to reveal the secret information they contain. Consider even the simple example “x = Sin(x).” Easy as this equation is to write down, there’s just about nothing you can do with just paper and pencil to tell me the solution. So if you encounter some situation in life and you determine that it’s described by this equation, you’ve only fought half the battle. You could argue that on some level you “understand” the situation now, because you know the mathematics that describes it. But you’re still not prepared to make a prediction about what “x” might be.

This is the predicament that physicists find themselves in with respect to a surprising number of aspects of their discipline. A lot of fundamental things in physics are governed by equations that are known very well, but which can only be solved in certain simple circumstances. By comparing the predictions of the equations to what happens in real life in these simple cases, we can become quite certain that the overall equations are correct.* But if we then want to use them to predict the behavior of things in more complicated scenarios, we often find ourselves out of luck.

When this happens, physicists resort to approximate solutions instead of exact ones, and this kind of approach works brilliantly for a large category of physical concepts including gravity, electricity and magnetism, various aspects of atomic physics, most of fluid dynamics, and so on. What happens is, the physicist makes an initial “guess” at the solution, then calculates a correction that brings his guess closer to the right answer. Then he calculates another, smaller correction to his new guess, and repeats until he has achieved the desired level of accuracy.

Unfortunately, every now and then this approximation scheme breaks down, because instead of getting smaller and smaller (and therefore more and more negligible), the corrections the physicist calculates will get larger at every step. That means it’s never safe to stop the approximation process, because you can’t just write off the next correction as “so small it’s no longer important.” And even though they have relatively meager social lives, few physicists have infinite time to sit around and calculate corrections. After all, they have conventions to attend!

Annoyingly for nuclear physicists, one of the subjects which falls into this category of “impossible to solve directly and also impossible to approximate” is the study of what holds quarks together into protons and neutrons, and what holds these protons and neutrons together in the hearts of the atoms. We have a marvelous theory to describe this kind of physics, called “quantum chromodynamics,” or often just “QCD” for short. But the equations prescribed by QCD simply intractable in all but the most boring of cases.

There are a number of “alternative” approximation tricks which have arisen which allow QCD calcultions to be done under certain situations, of course. But by far the newest and sexiest is the aDS/CFT correspondence. Remember that? It’s the thing you thought I was going to talk about when you started reading this post, before I took a long digression to tell you about all the ways that physicists’ knowledge of the universe is incomplete. But I had to do that to make sure you fully appreciate how exciting aDS/CFT is as a possible addition to the physicist’s toolkit. After all, I won’t really be able to explain the technical details of aDS/CFT because I don’t yet understand some of them myself, so a shared sense of excitement will be the best you can get.

Anyway, the key idea behind aDS/CFT is this: Imagine a strange kind of black hole, embedded in a different kind of spacetime than our own (this is the anti-De Sitter space that makes up the first half of the acronym. It’s a strange sort of space where everything gets further apart the farther you move from a central point.) The black hole, of course, has an event horizon, a sort of “outermost edge” that forms the boundary between the inside and outside of the hole. Now, throw some string theory objects into the center of the black hole. What’s been discovered is that, if you choose all the parameters correctly, the bits of the strings that reach out to the event horizon interact with each other in a manner that looks very much like particles interacting in QCD. If you didn’t know anything else about the black hole other than what was going on at it’s surface, you would think you were just looking at a bunch of particles obeying a particular Conformal Field Theory (the second half of the acronym). And it’s a conformal field theory that looks *almost *like QCD. Not quite, unfortunately, but remarkably close. And besides, it’s not out of the question that a slightly different choice of black hole might be found that reproduces QCD exactly on it’s surface.

So why is this cool? Well as I mentioned before, while QCD is in the category of theories whose equations are impossible to solve directly and also immune to most of our approximation schemes, gravity, as you’ll recall, was in the other category. And guess which force governs the behavior of things inside a black hole? You guessed it. Thus, even though we would have no way of predicting the behavior of particles on the surface of this imaginary black hole, we actually have *tons* of techniques for figuring out what’s going on inside, and once we know that we can easily figure out how this interior behavior affects what’s going on at the surface. It’s a brilliant back-door way of “sneaking up” on a problem that had proven intractable in the face of scores of more “direct” methods.

By the way, I tried for a while to think of a clever analogy for the way the aDS/CFT trick works, only to discover eventually that the only real analogy is the concept of an analogy itself. When I tell my students to think of the atoms in a hot gas like little people in a crowded room, it’s not because there’s any physical reality at all to the idea of microscopic human beings. It’s because it’s hard to know how the temperature of diatomic nitrogen will change when it’s compressed, but easy to imagine that if you have a bunch of little people in a tiny space, and you make the space even smaller, they’ll start bumping into each other, getting in each others faces, and generally starting to become flustered and angry. So if you also understand that, in the analogy, heated tempers stand for heated gas molecules, you can use the imaginary “little people” construction to figure out things you might otherwise have no intuition about. And that’s exactly how aDS/CFT works: It draws an analogy between the nuclear physics equations we don’t know what to do with and the equations of black hole behavior which are much more familiar to physicists. They can do all their thinking and calculating in “black hole language” and then translate the results back into “QCD language” only at the very end. Even though there are certainly no anti-De Sitter black holes in our universe any more than there are tiny people in my nitrogen balloon.

In addition to it’s powers to let us boldly calculate what no man has calculated before, there are several other things about aDS/CFT that have make it an important part of physics today. One such point is simply the fact that aDS/CFT is that it’s potential is still being unlocked. As time goes on, hopefully more unusual combinations of black holes and spacetimes can be found whose surfaces correspond to other problems we have trouble solving. It’s also been argued that the success of the aDS/CFT correspondence so far should actually count as experimental evidence in favor of string theory (which has been infamously “unprovable” for an uncomfortably long time) because the aDS/CFT calculations depend on treating the particles in nuclear physics as string theory objects instead of point particles. Of course, a lot of people object to this on philosophical grounds, including me. To me, the string theory elements of the aDS/CFT correspondence seem to be just mathematical tools which don’t necessarily have any more basis in reality than the imaginary black hole they’re living in. But then again, I don’t really have a deep enough understanding of the details to adjudicate this particular debate.

But to me perhaps the single most exciting thing about aDS/CFT is that it is *definitely* proof of the value of abstract, fundamental research, which can otherwise seem hard to justify. I hope to write a blog post addressing this point in more detail in the near future, but for now, suffice it to say that *no one* could have expected when they began q studying these types of black holes and strangely curved spacetimes that they would have any real-world application. There are a lot of people who would have used this fact to say that there was no need to fund people doing this kind of research, either with public or with private funds. But unfortunately, it’s simply the nature of deep, fundamental scientific and mathematical questions that they are so complex and mysterious we have no way of knowing which questions will have “useful” answers and which will “merely” enrich mankind’s understanding of his place in the cosmos. Granted, a lot of research projects produce results that still seem to have no bearing on “practical” science. But now that aDS/CFT can help us understand the equations that govern nuclear interactions, I hope it’s obvious that it has some very real “practical” applications indeed.

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*You may be wondering how we can know a theory is correct if we haven’t been able to test it in all cases. Of course, on principle, we can’t; but then again, it’s impossible to test something in *every *case. What we do is make a few assumptions about the way the universe works: basically, we assume that it plays fair, and that if something works for two particles, it works for four particles, which is just two groups of two particles, etc. So while we may be only able to test a theory by predicting what will happen in the two particle case, if that’s successful then there’s really no reason to believe the same thing won’t hold for four or more, even if the predictions are much much harder to caluclate in that case.

Does this mean that (0/1)==(0/23)? I always mix up my leptons and leprechauns. Existential import is both always and never unimportant.

“. . . of course, a lot of people object to this on philosophical grounds, including me.”

And me! To the string-ent ontology and any particular epistemology.