November 3, 1997

This Week's Finds in Mathematical Physics (Week 112)

John Baez

This week I will talk about attempts to compute the entropy of a black hole by counting its quantum states, using the spin network approach to quantum gravity.

But first, before the going gets tough and readers start dropping like flies, I should mention the following science fiction novel:

1) Greg Egan, Distress, HarperCollins, 1995.

I haven't been keeping up with science fiction too carefully lately, so I'm not really the best judge. But as far as I can tell, Egan is one of the few practitioners these days who bites off serious chunks of reality - who really tries to face up to the universe and its possibilies in their full strangeness. Reality is outpacing our imagination so fast that most attempts to imagine the future come across as miserably unambitious. Many have a deliberately "retro" feel to them - space operas set in Galactic empires suspiciously similar to ancient Rome, cyberpunk stories set in dark urban environments borrowed straight from film noire, complete with cynical voiceovers... is science fiction doomed to be an essentially nostalgic form of literature?

Perhaps we are becoming too wise, having seen how our wildest imaginations of the future always fall short of the reality, blindly extrapolating the current trends while missing out on the really interesting twists. But still, science fiction writers have to try to imagine the unimaginable, right? If they don't, who will?

But how do we dare imagine what things will be like in, say, a century, or a millenium? Vernor Vinge gave apt expression to this problem in his novel featuring the marooned survivors of a "singularity" at which the rate of technological advance became, momentarily, infinite, and most of civilization inexplicably... disappeared. Those who failed to catch the bus were left wondering just where it went. Somewhere unimaginable, that's all they know.

"Distress" doesn't look too far ahead, just to 2053. Asexuality is catching on bigtime... as are the "ultramale" and "ultrafemale" options, for those who don't like this gender ambiguity business. Voluntary Autists are playing around with eliminating empathy. And some scary radical secessionists are redoing their genetic code entirely, replacing good old ATCG by base pairs of their own devising. Fundamental physics, thank god, has little new to offer in the way of new technology. For decades, it's drifted off introspectively into more and more abstract and mathematical theories, with few new experiments to guide it. But this is the year of the Einstein Centenary Conference! Nobel laureate Violet Masala will unveil her new work on a Theory of Everything. And rumors have it that she may have finally cracked the problem, and found - yes, that's right - the final, correct and true theory of physics.

As science reporter Andrew Worth tries to bone up for his interviews with Masala, he finds it's not so easy to follow the details of the various "All-Topology Models" that have been proposed to explain the 10-dimensionality of spacetime in the Standard Unified Field Theory. In one of the most realistic passages of imagined mathematical prose I've ever seen in science fiction, he reads "At least two conflicting generalized measures can be applied to T, the space of all topological spaces with countable basis. Perrini's measure [Perrini, 2012] and Saupe's measure [Saupe, 2017] are both defined for all bounded subsets of T, and are equivalent when restricted to M - the space of n-dimensional paracompact Hausdorff manifolds - but they yield contradictory results for sets of more exotic spaces. However, the physical significance (if any) of this discrepancy remains obscure...."

But, being a hardy soul and a good reporter, Worth is eventually able to explain to us readers what's at stake here, and why Masala's new work has everyone abuzz. But that's really just the beginning. For in addition to this respectable work on All-Topology Models, there is a lot of somewhat cranky stuff going on in "anthrocosmology", involving sophisticated and twisted offshoots of the anthropic principle. Some argue that when the correct Theory of Everything is found, a kind of cosmic self-referential feedback loop will be closed. And then there's no telling what will happen!

Well, I won't give away any more. It's fun: it made me want to run out and do a lot more mathematical physics. And it raises a lot of deep issues. At the end it gets a bit too "action-packed" for my taste, but then, my idea of excitement is lying in bed thinking about n-categories.

Now for the black holes.

In "week111", I left off with a puzzle. In a quantum theory of gravity, the entropy of a black hole should be the logarithm of the number of its microstates. This should be proportional to the area of the event horizon. But what are the microstates? String theory has one answer to this, but I'll focus on the loop representation of quantum gravity. This approach to quantum gravity is very geometrical, which suggests thinking of the black hole microstates as "quantum geometries" of the black hole event horizon. But how are these related to the description of the geometry of the surrounding space in terms of spin networks?

Starting in 1995, Smolin, Krasnov, and Rovelli proposed some answers to these puzzles, which I have already mentioned in "week56", "week57", and "week87". The ideas I'm going to talk about now are a further development of this earlier work, but instead of presenting everything historically, I'll just present the picture as I see it now. For more details, try the following paper:

2) Abhay Ashtekar, John Baez, Alejandro Corichi and Kirill Krasnov, Quantum geometry and black hole entropy, to appear in Phys. Rev. Lett., preprint available as gr-qc/9710007.

This is a summary of what will eventually be a longer paper with two parts, one on the "black hole sector" of classical general relativity, and one on the quantization of this sector. Let me first say a bit about the classical aspects, and then the quantum aspects.

One way to get a quantum theory of a black hole is to figure out what a black hole is classically, get some phase space of classical states, and then quantize that. For this, we need some way of saying which solutions of general relativity correspond to black holes. This is actually not so easy. The characteristic property of a black hole is the presence of an event horizon - a surface such that once you pass it you can never get back out without going faster than light. This makes it tempting to find "boundary conditions" which say "this surface is an event horizon", and use those to pick out solutions corresponding to black holes.

But the event horizon is not a local concept. That is, you can't tell just by looking at a small patch of spacetime if it has an event horizon in it, since your ability to "eventually get back out" after crossing a surface depends on what happens to the geometry of spacetime in the future. This is bad, technically speaking. It's a royal pain to deal with nonlocal boundary conditions, especially boundary conditions that depend on *solving the equations of motion to see what's going to happen in the future just to see if the boundary conditions hold*.

Luckily, there is a purely local concept which is a reasonable substitute for the concept of event horizon, namely the concept of "outer marginally trapped surface". This is a bit technical - and my speciality is not this classical general relativity stuff, just the quantum side of things, so I'm no expert on it! - but basically it works like this.

First consider an ordinary sphere in ordinary flat space. Imagine light being emitted outwards, the rays coming out normal to the surface of the sphere. Clearly the cross-section of each little imagined circular ray will expand as it emanates outwards. This is measured quantitatively in general relativity by a quantity called... the expansion parameter!

Now suppose your sphere surrounds a spherically symmetric black hole. If the sphere is huge compared to the size of the black hole, the above picture is still pretty accurate, since the light leaving the sphere is very far from the black hole, and gravitational effects are small. But now imagine shrinking the sphere, making its radius closer and closer to the Schwarzschild radius (the radius of the event horizon). When the sphere is just a little bigger than the Schwarzschild radius, the expansion of light rays going out from the sphere is very small. This might seem paradoxical - how can the outgoing light rays not expand? But remember, spacetime is seriously warped near the event horizon, so your usual flat spacetime intuitions no longer apply. As we approach the event horizon itself, the expansion parameter goes to zero!

That's roughly the definition of an "outer marginally trapped surface". A more mathematical but still rough definition is: "an outer marginally trapped surface is the boundary S of some region of space such that the expansion of the outgoing family of null geodesics normal to S is everywhere less than or equal to zero."

We require that our space have some sphere S in it which is an outer marginally trapped surface. We also require other boundary conditions to hold on this surface. I won't explain them in detail. Instead, I'll say two important extra features they have: they say the black hole is nonrotating, and they disallow gravitational waves falling into S. The first condition here is a simplifying assumption: we are only studying black holes of zero angular momentum in this paper! The second condition is only meant to hold for the time during which we are studying the black hole. It does not rule out gravitational waves far from the black hole, waves that might eventually hit the black hole. These should not affect the entropy calculation.

Now, in addition to their physical significance, the boundary conditions we use also have an interesting mathematical meaning. Like most other field theories, general relativity is defined by an action principle, meaning roughly that one integrates some quantity called the Lagrangian over spacetime to get an "action", and finds solutions of the field equations by looking for minima of this action. But when one studies field theories on a region with boundary, and imposes boundary conditions, one often needs to "add an extra boundary term to the action" - some sort of integral over the boundary - to get things to work out right. There is a whole yoga of finding the right boundary term to go along with the boundary conditions... an arcane little art... just one of those things theoretical physicists do, that for some reason never find their way into the popular press.

But in this case the boundary term is all-important, because it's...

                      THE CHERN-SIMONS ACTION!

So far we've been studying general relativity in honest 4-dimensional spacetime. Chern-Simons theory is a closely related field theory one dimension down, in 3-dimensional spacetime. As time passes, the surface of the black hole traces out a 3-dimensional submanifold of our 4-dimensional spacetime. When we quantize our classical theory of gravity with our chosen boundary conditions, the Chern-Simons term will give rise to a "Chern-Simons field theory" living on the surface of the black hole. This field theory will describe the geometry of the surface of the black hole, and how it changes as time passes.

Well, let's not just talk about it, let's do it! We quantize our theory using standard spin network techniques outside the black hole, and Chern-Simons theory on the event horizon, and here is what we get. States look like this. Outside the black hole, they are described by spin networks (see "week110"). The spin network edges are labelled by spins j = 0, 1/2, 1, and so on. Spin network edges can puncture the black hole surface, giving it area. Each spin-j edge contributes an area proportional to sqrt(j(j+1)). The total area is the sum of these contributions.

Any choice of punctures labelled by spins determines a Hilbert space of states for Chern-Simons theory. States in this space describe the intrinsic curvature of the black hole surface. The curvature is zero except at the punctures, so that classically, near any puncture, you can visualize the surface as a cone with its tip at the puncture. The curvature is concentrated at the tip. At the quantum level, where the puncture is labelled with a spin j, the curvature at the puncture is described by a number j_z ranging -j to j in integer steps.

Now we ask the following question: "given a black hole whose area is within ε of A, what is the logarithm of the number of microstates compatible with this area?" This should be the entropy of the black hole. To figure it out, first we work out all the ways to label punctures by spins j so that the total area comes within ε of A. For any way to do this, we then count the allowed ways to pick numbers j_z describing the intrinsic curvature of the black hole surface. Then we sum these up and take the logarithm.

That's roughly what we do, anyway, and for black holes much bigger than the Planck scale we find that the entropy is proportional to the area. How does this compare with the result of Bekenstein and Hawking, described in "week111"? Remember, they computed that

S = A/4

where S is the entropy and A is the area, measured in units where c = ħ = G = k = 1. What we get is

S = (ln 2 / 4 π γ sqrt(3)) A

To compare these results, you need to know what is that mysterious "γ" factor in the second equation! It's often called the Immirzi parameter, because many people learned of it from the following paper:

3) Giorgio Immirzi, Quantum gravity and Regge calculus, in Nucl. Phys. Proc. Suppl. 57 (1997) 65-72, preprint available as gr-qc/9701052.

However, it was first discovered by Barbero:

4) Fernando Barbero, Real Ashtekar variables for Lorentzian signature space-times, Phys. Rev. D51 (1995), 5507-5510, preprint available as gr-qc/9410014.

It's an annoying unavoidable arbitrary dimensionless parameter that appears in the loop representation, which nobody had noticed before Barbero. It's still rather mysterious. But it works a bit like this. In ordinary quantum mechanics we turn the position q into an operator, namely multiplication by x, and also turn the momentum p into an operator, namely -i d/dx. The important thing is the canonical commutation relations: pq - qp = -i. But we could also get the canonical commutation relations to hold by defining

p = -i γ d/dx

q = x/γ

since the gammas cancel out! In this case, putting in a γ factor doesn't affect the physics. One gets "equivalent representations of the canonical commutation relations". In the loop representation, however, the analogous trick does affect the physics - different choices of the Immirzi parameter give different physics! For more details try:

4) Carlo Rovelli and Thomas Thiemann, The Immirzi parameter in quantum general relativity, preprint available as gr-qc/9705059.

How does the Immirzi parameter affect the physics? It *determines the quantization of area*. You may notice how I keep saying "each spin-j edge of a spin network contributes an area proportional to sqrt(j(j+1)) to any surface it punctures"... without ever saying what the constant of proportionality is! Well, the constant is

8 π γ

Until recently, everyone went around saying the constant was 1. (As for the factor of 8π, I'm no good at these things, but apparently at least some people were getting that wrong, too!) Now Krasnov claims to have gotten these damned factors straightened out once and for all:

5) Kirill Krasnov, On the constant that fixes the area spectrum in canonical quantum gravity, preprint available as gr-qc/9709058.

So: it seems we can't determine the constant of proportionality in the entropy-area relation, because of this arbitrariness in the Immirzi parameter. But we can, of course, use the Bekenstein-Hawking formula together with our formula for black hole entropy to determine γ, obtaining

γ = ln(2) / sqrt(3) π

This may seem like cheating, but right now it's the best we can do. All we can say is this: we have a theory of the microstates of a black hole, which predicts that entropy is proportional to area for largish black holes, and which taken together with the Bekenstein-Hawking calculation allows us to determine the Immirzi parameter.

What do the funny constants in the formula

S = (ln 2 / 4 π γ sqrt(3)) A

mean? It's actually simple. The states that contribute most to the entropy of a black hole are those where nearly all spin network edges puncturing its surface are labelled by spin 1/2. Each spin-1/2 puncture can have either j_z = 1/2 or j_z = -1/2, so it contributes ln(2) to the entropy. On the other hand, each spin-1/2 edge contributes 4 π γ sqrt(3) to the area of the black hole. Just to be dramatic, we can call ln 2 the "quantum of entropy" since it's the entropy (or information) contained in a single bit. Similarly, we can call 4 π γ sqrt(3) the "quantum of area" since it's the area contributed by a spin-1/2 edge. These terms are a bit misleading since neither entropy nor area need come in integral multiples of this minimal amount. But anyway, we have

S = (quantum of entropy / quantum of area) A

What next? Well, one thing is to try to use these ideas to study Hawking radiation. That's hard, because we don't understand Hamiltonians very well in quantum gravity, but Krasnov has made some progress....

6) Kirill Krasnov, Quantum geometry and thermal radiation from black holes, preprint available as gr-qc/9710006.

Let me just quote the abstract:

"A quantum mechanical description of black hole states proposed recently within the approach known as loop quantum gravity is used to study the radiation spectrum of a Schwarzschild black hole. We assume the existence of a Hamiltonian operator causing transitions between different quantum states of the black hole and use Fermi's golden rule to find the emission line intensities. Under certain assumptions on the Hamiltonian we find that, although the emission spectrum consists of distinct lines, the curve enveloping the spectrum is close to the Planck thermal distribution with temperature given by the thermodynamical temperature of the black hole as defined by the derivative of the entropy with respect to the black hole mass. We discuss possible implications of this result for the issue of the Immirzi γ-ambiguity in loop quantum gravity."

This is interesting, because Bekenstein and Mukhanov have recently noted that if the area of a quantum black hole is quantized in evenly spaced steps, there will be large deviations from the Planck distribution of thermal radiation:

7) Jacob D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum black hole, preprint available as gr-qc/9505012.

However, in the loop representation the area is not quantized in evenly spaced steps: the area A can be any sum of quantities like 8 π γ sqrt(j(j+1)), and such sums become very densely packed for large A.

Let me conclude with a few technical comments about how Chern-Simons theory shows up here. For a long time I've been studying the "ladder of dimensions" relating field theories in dimensions 2, 3, and 4, in part because this gives some clues as to how n-categories are related to topological quantum field theory, and in part because it relates quantum gravity in spacetime dimension 4, which is mysterious, to Chern-Simons theory in spacetime dimension 3, which is well-understood. It's neat that one can now use this ladder to study black hole entropy. It's worth comparing Carlip's calculation of black hole entropy in spacetime dimension 3 using a 2-dimensional field theory (the Wess-Zumino-Witten model) on the surface traced out by the black hole event horizon - see "week41". Both the theories we use and those Carlip uses, are all part of the same big ladder of theories! Something interesting is going on here.

But there's a twist in our calculation which really took me by surprise. We do not use SU(2) Chern-Simons theory on the black hole surface, we use U(1) Chern-Simons theory! The reason is simple. The boundary conditions we use, which say the black hole surface is "marginally outer trapped", also say that its extrinsic curvature is zero. Thus the curvature tensor reduces, at the black hole surface, to the intrinsic curvature. Curvature on a 3-dimensional space is so(3)-valued, but the intrinsic curvature on the surface S is so(2)-valued. Since so(3) = su(2), general relativity has a lot to do with SU(2) gauge theory. But since so(2) = u(1), the field theory on the black hole surface can be thought of as a U(1) gauge theory.

(Experts will know that U(1) is a subgroup of SU(2) and this is why we look at all values of j_z going from -j to j: we are decomposing representations of SU(2) into representations of this U(1) subgroup.)

Now U(1) Chern-Simons theory is a lot less exciting than SU(2) Chern-Simons theory so mathematically this is a bit of a disappointment. But U(1) Chern-Simons theory is not utterly boring. When we are studying U(1) Chern-Simons theory on a punctured surface, we are studying flat U(1) connections modulo gauge transformations. The space of these is called a "Jacobian variety". When we quantize U(1) Chern-Simons theory using geometric quantization, we are looking for holomorphic sections of a certain line bundle on this Jacobian variety. These are called "theta functions". Theta functions have been intensively studied by string theorists and number theorists, who use them do all sorts of wonderful things beyond my ken. All I know about theta functions can be found in the beginning of the following two books:

8) Jun-ichi Igusa, Theta Functions, Springer-Verlag, Berlin, 1972.

9) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser, Boston, 1983-1991.

Theta functions are nice, so it's fun to see them describing states of a quantum black hole!