October 17, 1994

This Week's Finds in Mathematical Physics (Week 41)

John Baez

In the beginning of September I went to a conference at the Center for Gravitational Physics and Geometry at Penn State. This is the center run by Abhay Ashtekar, and it has Jorge Pullin and Lee Smolin as faculty, and Roger Penrose as a part-time visitor --- so it's a great place to visit if you're interested in quantum gravity. There are a lot of good postdocs and such there, too. I've been too busy to say much so far about what happened at this conference, but I'd like to now.

One talk I enjoyed a lot was Steve Carlip's, on the entropy of black holes. This has subsequently come out as a preprint, available electronically:

1) The Statistical Mechanics of the (2+1)-Dimensional Black Hole, by Steve Carlip, 12 pages available as gr-qc/9409052.

It's well-known by know that in certain situations it makes sense to speak of the "entropy" of a black hole, but the real meaning of this entropy is still mysterious. In particular, since the entropy of a black hole is (often, but not always) proportional to the area of its event horizon, it would be very satisfying if the entropy corresponded somehow to degrees of freedom that "lived at the event horizon". Steve Carlip has done a pretty credible calculation along these lines (though not without various subtle difficulties) in the case of a black hole in 3-dimensional spacetime.

I should say a little bit about gravity in 3 dimensions and why people are interested in it. 3-dimensional gravity is drastically simpler than 4-dimensional gravity, since in 3 dimensions the vacuum Einstein's equations say the spacetime metric is flat, at least if the cosmological constant vanishes. Thus there can be no gravitational radiation (and in quantum theory no "gravitons"), and the metric produced by a static point mass is not like the Schwarschild metric, instead, on space it is just like that of a cone. Things are a bit different if the cosmological constant is nonzero; in particular, there are black-hole type solutions. But there is still no gravitational radiation.

Basically, people are interested in 3-dimensional quantum gravity because it's simple enough that one can compute something and hope it sheds some light on the 4-dimensional world we live in. For some issues this appears to be the case: primarily, conceptual issues having to do with theories in which there is no "background metric". Unfortunately, there are SEVERAL DIFFERENT WAYS to set up 3-dimensional quantum gravity, corresponding to different approaches people have to 4-dimensional quantum gravity. For this, check out Carlip's paper "Six ways to quantize (2+1)-dimensional gravity," mentioned in "week16". However, I think the "best" way to quantize gravity in 3 dimensions is the way involving Chern-Simons theory, because this way is the most closely related to Ashtekar's approach to quantizing gravity in 4 dimensions, hence it sheds the most light on the things I'm interested in --- and I also think it's the most beautiful. In this approach, you can compute a lot of things, and basically what Carlip has done is to show that associated to the event horizon there are degrees of freedom which should give entropy proportional to its area.

I suppose I can't say how he does it much more clearly than he says it, so I'll quote the introduction, taking the liberty of turning some of his LaTeX into English. If you get scared by the "Virasoro operator L_0" below, never fear --- in this context, it just amounts to the angular momentum operator, which generates rotations about the origin. So:

The basic argument is quite simple. Begin by considering general relativity on a manifold M with boundary. We ordinarily split the metric into true physical excitations and ``pure gauge'' degrees of freedom that can be removed by diffeomorphisms of M. But the presence of a boundary alters the gauge invariance of general relativity: the infinitesimal transformations [...] must now be restricted to those generated by vector fields [...] with no component normal to the boundary, that is, true diffeomorphisms that preserve the boundary of M. As a consequence, some degrees of freedom that would naively be viewed as ``pure gauge'' become dynamical, introducing new degrees of freedom associated with the boundary.

Now, the event horizon of a black hole is not a true boundary, although the black hole complementarity approach of Susskind et al. suggests that it might be appropriately treated as such. Regardless of one's view of that program, however, it is clear that in order to ask quantum mechanical questions about the behavior of black holes, one must put in ``boundary conditions'' that ensure that a black hole is present. This means requiring the existence of a hypersurface with particular metric properties---say, those of an apparent horizon.

The simplest way to do quantum mechanics in the presence of such a surface is to quantize fields separately on each side, imposing the appropriate correlations as boundary conditions. In a path integral approach, for instance, one can integrate over fields on each side, equate the boundary values, and finally integrate over those boundary values compatible with the existence of a black hole. But this process again introduces boundary terms that restrict the gauge invariance of the theory, leading once more to the appearance of new degrees of freedom at the horizon that would otherwise be treated as unphysical.

My suggestion is that black hole entropy is determined by counting these would-be gauge degrees of freedom. The resulting picture is similar to Maggiore's membrane model of the black hole horizon, but with a particular derivation and interpretation of the ``membrane'' degrees of freedom.

The analysis of this phenomenon is fairly simple in 2+1 dimensions. It is well known that (2+1)-dimensional gravity can be written as a Chern-Simons theory, and it is also a standard result that a Chern-Simons theory on a manifold with boundary induces a dynamical Wess-Zumino-Witten (WZW) theory on the boundary. In the presence of a cosmological constant Lambda = -1/L^2 appropriate for the (2+1)-dimensional black hole, one obtains a slightly modified SO(2,1) x SO(2,1) WZW model, with coupling constant

k = L sqrt(2)/8G

This model is not completely understood, but in the large k --- i.e., small Λ --- limit, it may be approximated by a theory of six independent bosonic oscillators. I show below that the Virasoro operator L_0 for this theory takes the form

L_0 ~ N - (r/4G)^2,

where N is a number operator and r is the horizon radius. It is a standard result of string theory that the number of states of such a system behaves asymptotically as

n(N) ~ exp(π sqrt 4 N)

If we demand that L_0 vanish --- physically, requiring states to be independent of the choice of origin of the angular coordinate at the horizon --- we thus obtain

log n(r) ~ (2 π r)/4G ,

precisely the right expression for the entropy of the (2+1)-dimensional black hole.

Also, Carlo Rovelli spoke about describing the dynamics of quantum gravity coupled to a scalar field in terms of "spin network" states. I think this was based on work he did in collaboration with Lee Smolin, and I don't think it's out yet. I'm just about to finish up a little paper on spin network states myself, since they seem like very useful things in quantum gravity. The simplest sort of spin network is just a trivalent graph (i.e., 3 edges adjacent to each vertex) with edges labelled by "spins" 0,1/2,1,3/2,..., and satisfying the "triangle inequality" at each vertex:

         j1 + j2 <= j3,      j2 + j3 <= j1,      j3 + j1 <= j2,

where j1, j2, j3 are the spins labelling the edges adjacent to the given vertex. Really, the spins should be thought of as irreducible representations of SU(2), and the triangle inequalities is necessary for the representation j3 to appear as a summand in the tensor product of the representations j1 and j2. (If the last sentence was meaningless to you, reading "week5" will help a little, though probably not quite enough.)

Penrose introduced spin networks as part of a purely combinatorial approach to spacetime in the paper:

2) Angular momentum; an approach to combinatorial space time, by Roger Penrose, in "Quantum Theory and Beyond," ed. T. Bastin, Cambridge University Press, Cambridge, 1971.

It is somehow satisfying, therefore, to see that spin networks arise naturally as a convenient description of states in the loop representation of quantum gravity, which STARTS mainly with Einstein's equations and the principles of quantum mechanics. Certainly there is a lot more we need to learn about them.... One place worth reading about them is:

3) Conformal field theory, spin geometry, and quantum gravity, by Louis Crane, Phys. Lett. B259 (1991), 243-248.

I will be coming out with a paper on them next week if I get my act together, and I may say a bit more about them in future "weeks".

Rovelli also mentioned an interesting paper he wrote about the problem of time in quantum gravity with the operator-algebra/ noncommutative- geometry guru Alain Connes:

4) Von Neumann algebra automorphisms and time-thermodynamics relation in general covariant quantum theories, by A. Connes and C. Rovelli, 25 pages in LaTex format available as gr-qc/9406019.

The problem of time in quantum gravity is a bit tricky to describe, since it takes different guises in different approaches to quantum gravity, but I have attempted to give a rough introduction to it in "week11" and "week27". One way to get a feeling for it is to realize that anything you are used to doing with Hamiltonians in quantum mechanics or quantum field theory, you CAN'T do in quantum gravity, at least not in any simple way, because there is no Hamiltonian in general relativity, but only a ``Hamiltonian constraint'' --- which in quantum gravity becomes the Wheeler-DeWitt equation

                         H Ψ = 0.

Now, people know there is a mystical relationship between time and temperature that might be written

                         it = 1/kT

where t is time, T is temperature, and k is Boltzmann's constant. This equation is a bit of an exaggeration! But the point is that in quantum theory, when there is a Hamiltonian H around one evolves states using the operator


while the Gibbs state, that is, the equilibrium state at temperature T, is given by the density matrix


It is this fact that relates statistical mechanics and quantum field theory so closely.

Now, in quantum gravity things aren't so simple, since there isn't a Hamiltonian (just a Hamiltonian constraint). However, people do know that there are all sorts of funny relationships between statistical mechanics and quantum gravity. For example, an accelerating observer in Minkowski space will see the vacuum as a heat bath with temperature proportional to her acceleration, so in curved spacetime, where there are no truly inertial frames, there really is no well-defined notion of a vacuum; in some vague sense, all there are is "thermal" states. This fact is also somehow related to Hawking radiation, and to the notion of black hole entropy... but really, there is a lot that nobody understands about all these connections!

In any event, Rovelli was prompted to use thermodynamics to DEFINE time in quantum gravity as follows. Given a mixed state with density matrix D, find some operator H such that D is the Gibbs state exp(-H/kT). In lots of cases this isn't hard; it basically amounts to

                        H = -kT ln D

Of course, H will depend on T, but this really is just saying that fixing your units of temperature fixes your units of time!

Operator theorists have pondered this notion very carefully for a long time and generalized it into something called the Tomita-Takesaki theorem, which Connes and Rovelli explain. This gives a very general way to cook up a Hamiltonian (hence a notion of time evolution) from a state of a quantum system! For example, one can use this trick to start with a Robertson-Walker universe full of blackbody radiation, and recover a notion of "time". This is very intriguing, and it may represent some real progress in understanding the deep relations between time, thermodynamics, and gravity. There are, of course, lots of problems and puzzles to deal with.

Another intriguing talk at the conference was given by Viqar Husain, on the subject of the following paper:

5) The affine symmetry of self-dual gravity, by Viqar Husain, 17 pages in ReVTeX format available as hep-th/9410072.

Let me simply quote the abstract, since I don't feel I really understand the essence of this business well enough to say anything useful yet:

Self-dual gravity may be reformulated as the two dimensional chiral model with the group of area preserving diffeomorphisms as its gauge group. Using this formulation, it is shown that self-dual gravity contains an infinite dimensional hidden symmetry algebra, which is the Affine (Kac-Moody) algebra associated with the Lie algebra of area preserving diffeomorphisms. This result provides an observable algebra and a solution generating technique for self-dual gravity.

A couple more things before I wrap this up.... First, in case any mathematicians out there are wondering what this "knots and quantum gravity" business is all about, here's something I wrote to review the subject:

6) Knots and Quantum Gravity: Progress and Prospects, John Baez, 22 pages in LaTeX format, available as gr-qc/9410018.

My abstract:

Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and 4-dimensional generally covariant physics. We review how some of these relationships arise from a `ladder of field theories' including quantum gravity and BF theory in 4 dimensions, Chern-Simons theory in 3 dimensions, and the G/G gauged WZW model in 2 dimensions. We also describe the relation between link (or multiloop) invariants and generalized measures on the space of connections. In addition, we pose some research problems and describe some new results, including a proof (due to Sawin) that the Chern-Simons path integral is not given by a generalized measure.

Finally, let me draw people's attention to "Matters of Gravity", the newsletter Jorge Pullin puts together at considerable effort, to keep people informed about general relativity and the like, experimental and theoretical:

7) "Matters of Gravity", a newsletter for the gravity community, Number 4, edited by Jorge Pullin, 24 pages in Plain TeX, available as gr-qc/9409004, or from WWW by http://vishnu.nirvana.phys.psu.edu/

Here's the table of contents of this issue:

Gravity News:
 Report on the APS topical group in gravitation, Beverly Berger.
Research briefs:
 Gravitational microlensing and the search for dark matter, Bohdan Paczynski.
 Laboratory gravity: the G mystery, Riley Newman.
 LIGO project update, Stan Whitcomb.
Conference Reports
 PASCOS '94, Peter Saulson.
 The Vienna Meeting, P. Aichelburg, R. Beig.
 The Pitt binary black hole grand challenge meeting, Jeff Winicour.
 International symposium on experimental gravitation at Pakistan, 
Munawar Karim.
 10th Pacific coast gravity meeting, Jim Isenberg.

© 1994 John Baez