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I feel guilty for slacking off on This Week's Finds, so I should explain the reason. Lots of things have been building up that I'm dying to talk about, but new ones keep coming in at such a rapid rate that I never feel I have time!
I will have to be ruthless, and face up to the fact that a quick and imperfect exposition is better than none.
First of all, here at the Center for Gravitational Physics and Geometry there are a lot of interesting attempts going on to compute the entropy of black holes from first principles. Bekenstein, Hawking and many others have used a wide variety of semiclassical arguments to argue that black holes satisfy
S = A/4
where S is the entropy and A is the area of the event horizon, both measured in Planck's units, where G = c = ħ = 1.
For example, using purely classical reasoning (general relativity, but no quantum theory) one can prove the "2nd law of black hole thermodynamics", which says that A always increases. As Bekenstein noted, this suggests that the area of the event horizon is somehow analogous to entropy. However, by itself this does not determine the magic number 1/4, which can only be derived using quantum theory (as one can see by simple dimensional analysis).
By semiclassical reasoning - studying quantum electrodynamics in the Schwarzschild metric used to describe black holes - Hawking showed that black holes should radiate as if they had a temperature inversely proportional to their mass:
T = 1/(8 π M)
This made the analogy between entropy and event horizon area much more than an analogy, because it meant that one could assign a temperature to black holes and see if they satisfy the laws of thermodynamics. It turns out that if you consider A/4 to be the entropy of a black hole, you can eliminate seeming violations of the 2nd law that otherwise arise in thought experiments where you get rid of entropy by throwing it into a black hole. In other words, if you throw something with entropy S into a black hole, calculations seem to show that the area of the event horizon always increases by at least 4S!
So far nothing I've said is related to full-fledged quantum gravity, because in the semiclassical arguments one is still working in the approximation where the gravitational field is treated classically. An interesting test of any theory of quantum gravity is whether can use it to derive S = A/4. In a subject with no real experimental evidence, this is the closest we have to an "experimental result" that our theory should predict.
Recently the string theorists have done some calculations claiming to show that string theory predicts S = A/4. Personally I feel that while these calculations are interesting they are far from definitive. For example, they all involve taking calculations done using perturbative string theory on flat spacetime and extrapolating them drastically to the regime in which string theory approximates general relativity. One typically uses ideas from supersymmetry to justify such extrapolations; however, these ideas only seem to apply to "extremal black holes", having the maximum possible charge for a black hole of a given mass and angular momentum. Realistic black holes are far from extremal. In short, while exciting, these calculations still need to be taken with a grain of salt.
Of course, I am biased because I am interested in another approach to quantum gravity, the loop representation of quantum gravity, which folks are working on here at the CGPG, among other places. This is in many ways a more conservative approach. The idea is to simply take Einstein's equation for general relativity and quantize it, rather than trying to develop a theory of all particles and forces as in string theory. Of course, for various reasons it is not so easy to quantize Einstein's equation. String theorists think it's impossible without dragging in all sorts of other forces and particles, but folks working on the loop representation are more optimistic. This is an ongoing argument, but certainly a good test of the loop representation would be to try to use it to derive Hawking's formula S = A/4. If the loop representation is really any good, this should be possible, because many different lines of reasoning using only general relativity and quantum theory lead to this formula.
I've already mentioned a few attempts to do this in "week56","week57", and "week87". These were promising, but they didn't get the magical number 1/4. Also, they are rather rough, in that they do computations on some region with boundary, but don't use anything that ensures the boundary is an event horizon.
Recently Kirill Krasnov has made some progress:
1) Kirill Krasnov, On statistical mechanics of Schwarzschild black hole, preprint available as gr-qc/9605047.
This paper still doesn't get the magic number 1/4, and Krasnov later realized it has a few mistakes in it, but it does something very cool. It notes that the boundary conditions holding on the event horizon of a Schwarzschild black hole are closely related to Chern-Simons theory. Now is not the time for me to go into Chern-Simons theory, but basically, it lets you apply a lot of neat mathematics to calculate everything to your heart's content, very much as Carlip did on his work on the toy model of a 2+1-dimensional black hole (see "week41"). Also, it sheds new light on the relationship between topological quantum field theory and quantum gravity, something I am always trying to understand better.
While I'm at it, I should note the existence of a paper that reworks Carlip's calculation from a slightly different angle:
2) Maximo Banados and Andres Gomberoff, Black hole entropy in the Chern-Simons formulation of 2+1 gravity, preprint available as gr-qc/9611044.
2+1-dimensional quantum gravity is very simple compared to the 3+1-dimensional quantum gravity we'd really like to understand: in a sense it's "exactly solvable". But there are still some puzzling things about Carlip's computation of the entropy of a black hole in 2+1 dimensions which need figuring out, so every paper on the subject is worth looking at, if you're interested in black hole entropy.
Speaking of topological quantum field theory and quantum gravity, I just finished a paper on these topics:
3) John Baez, Degenerate solutions of general relativity from topological field theory, preprint available as gr-qc/9702051 or in Postscript form at http://math.ucr.edu/home/baez/deg.ps.
Let me just summarize the basic idea, resisting the temptation to become insanely technical.
A while ago Rovelli and Smolin introduced Penrose's notion of "spin network" into the loop representation of quantum gravity. I described spin networks pretty carefully in "week43", but here let me just say that they are graphs embedded in space with edges labelled by spins j = 0, 1/2, 1, 3/2, and so on, just as in the quantum mechanics of angular momentum, and with vertices labelled by "intertwining operators", which are other gadgets that come up in the study of angular momentum. In the loop representation these spin networks form a basis of states. Geometrical observables like the area of surfaces and the volumes of regions have been quantized and their matrix elements computed in the spin network basis, giving us a nice picture of "quantum 3-geometries", that is, the possible geometries of space in the context of quantum gravity. In this picture, the edges of spin networks play the role of quantized flux tubes of area, much as the magnetic field comes in quantized flux tubes in a type II superconductor. To work out the area of a surface in some spin network state, you just total up contributions from each edge of the spin network that pokes through the surface. An edge labelled with spin j carries an area equal to sqrt(j(j+1)) times the Planck length squared. What's cool is that this is not merely postulated, it's derived from fairly standard ideas about how you turn observables into operators in quantum mechanics.
However, the dynamics of quantum gravity is more obscure. Technical issues aside, the main problem is that while we have a nice picture of quantum 3-geometries, we don't have a similar picture of the 4-dimensional, or spacetime, aspects of the theory. To represent a physical state of quantum gravity, a spin network state (or linear combination thereof) has to satisfy something called the Wheeler-DeWitt equation. This is sort of the quantum gravity analog of the Schrodinger equation. There is a lot of controversy over the Wheeler-DeWitt equation and what's the right way to write it down in the loop representation. The really annoying thing, however, is that even if you feel you know how to write it down - for example, Thomas Thiemann has worked out one way (see "week85") - and can find solutions, you still don't necessarily have a good intuition as to what the solutions mean. For example, almost everyone seems to agree that spin networks with no vertices should satisfy the Wheeler-DeWitt equation. These are just knots or links with edges are labelled by spins. We know these states are supposed to represent "quantum 4-geometries" satisfying the quantized Einstein equations. But how should we visualize these states in 4-dimensional terms?
In search of some insight into the 4-dimensional interpretation of these states, I turn to classical general relativity. In my paper, I construct solutions of the equations of general relativity which at a typical fixed time look like "flux tubes of area" reminiscent of the loop states of quantum gravity. These are "degenerate solutions", meaning that the "3-metric", the tensor you use to measure distances in 3-dimensional space, is zero in lots of regions of space. Here I should warn you that ordinary general relativity doesn't allow degenerate metrics like this. The loop representation works with an extension of general relativity that covers the case of degenerate metrics; for more on this, see "week88".
More precisely, if you look at these "flux tube" solutions at a typical time, the 3-metric vanishes outside a collection of solid tori embedded in space, while inside any of these solid tori the metric is degenerate in the longitudinal direction, but nondegenerate in the two transverse directions.
Now since these are classical solutions - no quantum theory in sight! - there is no problem with understanding what they do as time passes. We can solve Einstein's equation and get a 4-metric, a metric on spacetime. The 4-dimensional picture is as follows: given any surface Σ embedded in spacetime, I get solutions for which the 4-metric vanishes outside a neighborhood of Σ. Inside this neighborhood, the 4-metric is zero in the two directions tangent to Σ but nondegenerate in the two transverse directions. In the 4-geometry defined by one of these solutions, the area of a typical surface Σ' intersecting Σ in some isolated points is a sum of contributions from the points where Σ and Σ' intersect.
The solutions I study are inspired by the work of Mike Reisenberger, who studied a solution for which the metric vanishes outside a neighborhood of a sphere embedded in R^4. I consider more general surfaces embedded in more general 4-manifolds, so I need to worry a lot more about topological issues. Also, I allow the possibility of a nonzero cosmological constant (this being a parameter in Einstein's equation that determines the energy density of the vacuum). A lot of the most interesting stuff happens for nonzero cosmological constant, and this case actually helps one understand the case of vanishing cosmological constant as a kind of limiting case.
It turns out that the interesting degrees of freedom of the metric living on the surface Σ in spacetime are described by fields living on this surface. In fact, these fields are solutions of a 2-dimensional topological field theory called BF theory. To prove this, I take advantage of the relation between general relativity and BF theory in 4 dimensions, together with the fact that BF theory behaves in a simple manner under dimensional reduction.
Another neat thing is that to get a solution of general relativity this way, we need to pick a "framing" of Σ. Roughly speaking, this means we need to pick a way of thickening up the surface Σ to a neighborhood that looks like Σ x D^2, where D^2 is the 2-dimensional disc. This is precisely the 4-dimensional analog of a framing of a knot or link in 3-dimensions. People who know about topological quantum field theory know that framings are very important. In fact, I can show that my solutions of general relativity are closely related to Chern-Simons theory, a 3-dimensional topological field theory famous for giving invariants of framed knots and links. What's beginning to emerge is a picture that makes the spacetime aspects of framings easier to understand.
Now before I plunge into some even more esoteric stuff, let me briefly return to reality and answer the question you've all been secretly dying to ask: how does general relativity impact the world of big business?
In plain terms: is all this fancy physics just an excuse to have fun visualizing evolving spin networks in terms of quantum field theories on surfaces embedded in 4-dimensional spacetime, etcetera etcetera... or does it actually contribute to the well-being of the corporations upon which we depend?
Well, you may be surprised to know that general relativity plays an significant role in a $200-million business. Surprised? Read on! What follows is taken from the latest issue of "Matters of Gravity", the newsletter put out by Jorge Pullin. More precisely, it's from:
4) Neil Ashby, General relativity in the global positioning system, in Matters of Gravity, ed. Jorge Pullin, no. 9, available at http://www.phys.lsu.edu//mog/mog9/node9.html.
I will simply quote some excerpts from this fascinating article:
"The Global Position System (GPS) consists of 24 earth-orbiting satellites, each carrying accurate, stable atomic clocks. Four satellites are in each of six different orbital planes, of inclination 55 degrees with respect to earth's equator. Orbital periods are 12 hours (sidereal), so that the apparent position of a satellite against the background of stars repeats in 12 hours. Clock-driven transmitters send out synchronous time signals, tagged with the position and time of the transmission event, so that a receiver near the earth can determine its position and time by decoding navigation messages from four satellites to find the transmission event coordinates, and then solving four simultaneous one-way signal propagation equations. Conversely, γ-ray detectors on the satellites could determine the space-time coordinates of a nuclear event by measuring signal arrival times and solving four one-way propagation delay equations. Apart possibly from high-energy accelerators, there are no other engineering systems in existence today in which both special and general relativity have so many applications. The system is based on the principle of the constancy of c in a local inertial frame: the Earth-Centered Inertial or ECI frame. Time dilation of moving clocks is significant for clocks in the satellites as well as clocks at rest on earth. The weak principle of equivalence finds expression in the presence of several sources of large gravitational frequency shifts. Also, because the earth and its satellites are in free fall, gravitational frequency shifts arising from the tidal potentials of the moon and sun are only a few parts in 10^16 and can be neglected. [...] At the time of launch of the first NTS-2 satellite (June 1977), which contained the first Cesium clock to be placed in orbit, there were some who doubted that relativistic effects were real. A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by GR, then the synthesizer could be turned on bringing the clock to the coordinate rate necessary for operation. The atomic clock was first operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that interval was +442.5 parts in 10^12 faster than clocks on the ground; if left uncorrected this would have resulted in timing errors of about 38,000 nanoseconds per day. The difference between predicted and measured values of the frequency shift was only 3.97 parts in 10^12, well within the accuracy capabilities of the orbiting clock. This then gave about a 1% validation of the combined motional and gravitational shifts for a clock at 4.2 earth radii. [...] This system was intended primarily for navigation by military users having access to encrypted satellite transmissions which are not available to civilian users. Uncertainty of position determination in real time by using the Precise Positioning code is now about 2.4 meters. Averaging over time and over many satellites reduces this uncertainty to the point where some users are currently interested in modelling many effects down to the millimeter level. Even without this impetus, the GPS provides a rich source of examples for the applications of the concepts of relativity. New and surprising applications of position determination and time transfer based on GPS are continually being invented. Civilian applications include for example, tracking elephants in Africa, studies of crustal plate movements, surveying, mapping, exploration, salvage in the open ocean, vehicle fleet tracking, search and rescue, power line fault location, and synchronization of telecommunications nodes. About 60 manufacturers now produce over 350 different commercial GPS products. Millions of receivers are being made each year; prices of receivers at local hardware stores start in the neighborhood of $200."Pretty cool, eh?
Okay, now for something completely different - homotopy theory! Well, everything I write about is actually secretly part of my grand plan to see how everything interesting is related to everything else, but let me not delve into how homotopy theory is related to topological quantum field theory and thus quantum gravity. Let me simply mention the existence of this great book:
5) "Handbook of Algebraic Topology", ed. I. M. James, North-Holland, the Netherlands, 1995, 1324 pages.
Occaisionally you come across a book that you wish you just download into your brain; for me this is one of those books. It is probably not a good idea to read it if you are just wanting to get started on algebraic topology; it assumes you are pretty familiar with the basic ideas already, and it goes into a lot of depth, mainly in hardcore homotopy theory. A lot of it is too technical for me to appreciate, but let me list a few chapters that I can understand and like.
Chapter 1, "Homotopy types" by Hans-Joachim Baues, is a great survey of different models of homotopy types. Remember, we say two topological spaces X and Y are homotopy equivalent if there are continuous functions f: X → Y and g: Y → X that are inverses "up to homotopy". In other words, we don't require that fg and gf are equal to identity functions, but merely that they can both be continuously deformed to identity functions. So for example the circle and an annulus are homotopy equivalent, and we say therefore that they represent the same "homotopy type".
The cool thing is that there turn out to be very elegant algebraic and combinatorial ways of describing homotopy types that don't mention topology at all. Perhaps the most beautiful way of all is a way that in a sense hasn't been fully worked out yet: namely, thinking of homotopy types as "ω-groupoids". The idea is this. An "ω-category" is something that has
objects like x
morphisms between objects like f: x → y
2-morphisms between morphisms like F: f → g
3-morphisms between 2-morphisms like T: F → G
...
and so on ad infinitum. There should be some ways of composing these, and these should satisfy some axioms, and that of course is the tricky part. But the basic idea is that if you hand me a topological space X, I can cook up an ω-category whose
objects are points in X
morphisms are paths between points in X
2-morphisms are continuous 1-parameter families of paths in X, i.e. "paths of paths" in X
3-morphisms are "paths of paths of paths" in X
...
and so on. This is better than your garden-variety ω-category because all the morphisms and 2-morphisms and 3-morphisms and so on have inverses, at least "up to homotopy". We call it an "ω-groupoid". This ω-groupoid keeps track of the homotopy type of X in a very nice way. (If this "ω" stuff is too mind-boggling, you may want to start by reading a bit about plain old categories and groupoids in "week74".)
Conversely, given any ω-groupoid there should be a nice way to cook up a homotopy corresponding to it. This is just the infinite- dimensional generalization of something I described in "week75". There, I showed how you could get a groupoid from a "homotopy 1-type" and vice versa. Here there 1-morphisms but no interesting 2-morphisms, 3-morphisms, and so on, because the topology of a "homotopy 1-type" is boring in dimensions greater than 1. (In case any experts are reading this, what I mean is that its higher homotopy groups are trivial; its higher homology and cohomology groups can be very interesting.)
So we can - and should - think of homotopy theory as, among other things, the study of ω-groupoids, and thus a very useful warmup to the study of ω-categories. In my occaisional series on This Week's Finds called "the tale of n-categories", I have tried to explain why n-categories, and ultimately ω-categories, should serve as a powerful unifying approach to lots of mathematics and physics. In trying to understand this subject, I find time and time again that homotopy theorists are the ones to listen to.
Chapter 2, "Homotopy theories and model categories", by W. G. Dwyer and J. Spalinski, is a nice introduction to the formal idea of using different "models" for homotopy types. For example, above I was sketching how one might do homotopy theory using the "model category" of ω-groupoids. Other model categories include gadgets like Kan complexes, CW complexes, simplicial complexes, and so on.
Chapter 6, "Modern foundations for stable homotopy theory", by A. D. Elmendorf, I. Kriz, M. Mandell and J. P. May describes a very nice approach to spectra. Loosely speaking, we can think of a spectrum as a Z-groupoid, where Z denotes the integers. In other words, in addition to j-morphisms for all natural numbers j, we also have j-morphisms for negative j! This may seem bizarre, but it's a lot like how in homology theory one is interested in chain complexes that extend in both the positive and negative directions. In fact, we can think of a chain complex as a very special sort of Z-groupoid or spectrum. The study of spectra is called stable homotopy theory.
Chapter 13, "Stable homotopy and iterated loop spaces", by G. Carlsson and R. J. Milgram, is packed with handy information about stable homotopy theory.
Chapter 21, "Classifying spaces of compact Lie groups and finite loop spaces", by D. Notbohm, is a good source of heavy-duty information on classifying spaces of your favorite Lie groups. To study vector bundles and the like one really needs to become comfortable with classifying spaces, and I'm finally doing this, and I hope eventually I'll be comfortable enough with them to really understand all these results.
There is a lot more, but I will stop here.
© 1997 John Baez
baez@math.removethis.ucr.andthis.edu
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