April 20, 1996

This Week's Finds in Mathematical Physics (Week 80)

John Baez

There are a number of interesting books I want to mention.

Huw Price's book on the arrow of time is finally out! It's good to see a philosopher of science who not only understands what modern physicists are up to, but can occaisionally beat them at their own game.

Why is the future different from the past? This has been vexing people for a long time, and the stakes went up considerably when Boltzmann proved his "H-theorem", which seems at first to show that the entropy of a gas always increases, despite the time-reversibility of the laws of classical mechanics. However, to prove the H-theorem he needed an assumption, the "assumption of molecular chaos". It says roughly that the positions and velocities of the molecules in a gas are uncorrelated before they collide. This seems so plausible that one can easily overlook that it has a time-asymmetry built into it --- visible in the word "before". In fact, we aren't getting something for nothing in the H-theorem; we are making a time-asymmetric assumption in order to conclude that entropy increases with time!

The "independence of incoming causes" is very intuitive: if we do an experiment on an electron, we almost always assume our choice of how to set the dials is not correlated to the state of the electron. If we drop this time-asymmetric assumption, the world looks rather different... but I'll let Price explain that to you.

Anyway, Price is an expert at spotting covertly time-asymmetric assumptions. You may remember from "week26" that he even got into a nice argument with Stephen Hawking about the arrow of time, thanks to this habit of his. You can read more about it in:

1) Huw Price, Time's Arrow and Archimedes' Point: New Directions for a Physics of Time, Oxford University Press, 1996.

Also, there is a new book out by Hawking and Roger Penrose on quantum gravity. First they each present their own ideas, and then they duke it out in a debate in the final chapter. This book is an excellent place to get an overview of some of the main ideas in quantum gravity. It helps if you have a little familiarity with general relativity, or differential geometry, or are willing to fake it.

There is even some stuff here about the arrow of time! Hawking has a theory of how it arose, starting from his marvelous "no-boundary boundary conditions", which say that the wavefunction of the universe is full of quantum fluctuations corresponding to big bangs which erupt and then recollapse in big crunches. The wavefunction itself has no obvious "time-asymmetry", indeed, time as we know it only makes sense within any one of the quantum fluctuations, one of which is presumably the world we know! But Hawking thinks that each of these quantum fluctuations, or at least most of them, should have an arrow of time. This is what Price raised some objections to. Hawking seems to argue that each quantum fluctuation should "start out" rather smooth near its big bang and develop more inhomogeneities as time passes, "winding up" quite wrinkly near its big crunch. But it's not at all clear what this "starting out" and "winding up" means. Possibly he is simply speaking vaguely, and all or most of the quantum fluctuations can be shown to have one smooth end and wrinkly at the other. That would be an adequate resolution to the arrow of time problem. But it's not clear, at least not to me, that Hawking really showed this.

Penrose, on the other hand, has some closely related ideas. His "Weyl curvature hypothesis" says that the Weyl curvature of spacetime goes to zero at initial singularities (e.g. the big bang) and infinity at final ones (e.g. black holes). The Weyl curvature can be regarded as a measure of the presence of inhomogeneity --- the "wrinkliness" I alluded to above. The Weyl curvature hypothesis can be regarded as a time-asymmetric law built into physics from the very start.

To see them argue it out, read

2) Stephen Hawking and Roger Penrose, The Nature of Space and Time, Princeton University Press, 1996.

There are also a couple of more technical books on general relativity that I'd been meaning to get ahold of for a long time. They both feature authors of that famous book,

3) Charles Misner, Kip Thorne and John Wheeler, Gravitation, Freeman Press, 1973,

which was actually the book that made me decide to work on quantum gravity, back at the end of my undergraduate days. They are:

4) Ignazio Ciufolini and John Archibald Wheeler, Gravitation and Inertia, Princeton University Press, 1995.


5) Kip Thorne, Richard Price and Douglas Macdonald, eds., Black Holes: The Membrane Paradigm, 1986.

The book by Ciufolini and Wheeler is full of interesting stuff, but it concentrates on "gravitomagnetism": the tendency, predicted by general relativity, for a massive spinning body to apply a torque to nearby objects. This is related to Mach's old idea that just as spinning a bucket pulls the water in it up to the edges, thanks to the centrifugal force, the same thing should happen if instead we make lots of stars rotate around the bucket! Einstein's theory of general relativity was inspired by Mach, but there has been a long-running debate over whether general relativity is "truly Machian" --- in part because nobody knows what "truly Machian" means. In any event, Ciufolini and Wheeler argue that gravitomagnetism exhibits the Machian nature of general relativity, and they give a very nice tour of gravitomagnetic effects.

That is fine in theory. However, the gravitomagnetic effect has never yet been observed! It was supposed to be tested by Gravity Probe B, a satellite flying at an altitude of about 650 kilometers, containing a superconducting gyroscope that should precess at a rate of 42 milliarcseconds per year thanks to gravitomagnetism. I don't know what ever happened with this, though: the following web page says "Gravity Probe B is expected to fly in 1995", but now it's 1996, right? Maybe someone can clue me in to the latest news.... I seem to remember some arguments about funding the program.

6) Gravity Probe B, http://stugyro.stanford.edu/RELATIVITY/GPB/

(Note added in 2002: now this webpage is gone; see http://einstein.stanford.edu/ for the latest story.)

Kip Thorne's name comes up a lot in conjuction with black holes and the LIGO --- or Laser-Interferometer Gravitational-Wave Observatory --- project. As pairs of black holes or neutron stars spiral emit gravitational radiation, they should spiral in towards each other. In their final moments, as they merge, they should emit a "chirp" of gravitational radiation, increasing in frequency and amplitude until their ecstatic union is complete. The LIGO project aims to observe these chirps, and any other sufficiently strong gravitational radiation that happens to be passing by our way. LIGO aims to do this by using laser interferometry to measure the distance between two points about 4 kilometers apart to an accuracy of about 10^{-18} meters, thus detecting tiny ripples in the spaceteim metric. For more on LIGO, try

7) LIGO project home page, http://www.ligo.caltech.edu/

Thorne helped develop a nice way to think of black holes by envisioning their event horizon as a kind of "membrane" with well-defined mechanical, electrical and magnetic properties. This is called the membrane paradigm, and is useful for calculations and understanding what black holes are really like. The book "Black Holes: The Membrane Paradigm" is a good place to learn about this.

Now let me return to the tale of 2-categories. So far I've said only that a 2-category is some sort of structure with objects, morphisms between objects, and 2-morphisms between morphisms. But I have been attempting to develop your intuition for Cat, the primordial example of a 2-category. Remember, Cat is the 2-category of all categories! Its objects are categories, its morphisms are functors, and its 2-morphisms are natural transformations --- these being defined in "week73" and again in "week75".

How can you learn more about 2-categories? Well, a really good place is the following article by Ross Street, who is one of the great gurus of n-category theory. For example, he was the one who invented ω-categories!

8) Ross Street, Categorical structures, in Handbook of Algebra, vol. 1, ed. M. Hazewinkel, Elsevier, 1996.

Physicists should note his explanation of the Yang-Baxter and Zamolodchikov equations in terms of category theory. If you have trouble finding this, you might try

9) G. Maxwell Kelly and Ross Street, Review of the elements of 2-categories, Springer Lecture Notes in Mathematics 420, Berlin, 1974, pp. 75-103.

I can't really compete with these for thoroughness, but at least let me give the definition of a 2-category. I'll give a pretty nuts-and-bolts definition; later I'll give a more elegant and abstract one. Readers who are familiar with Cat should keep this example in mind at all times!

This definition is sort of long, so if you get tired of it, concentrate on the pictures! They convey the basic idea. Also, keep in mind is that this is going to be sort of like the definition of a category, but with an extra level on top, the 2-morphisms.

So: first of all, a 2-category consists of a collection of "objects" and a collection of "morphisms". Every morphism f has a "source" object and a "target" object. If the source of f is X and its target is Y, we write f: X → Y. In addition, we have:

1) Given a morphism f: X → Y and a morphism g: Y → Z, there is a morphism fg: X → Z, which we call the "composite" of f and g.

2) Composition is associative: (fg)h = f(gh).

3) For each object X there is a morphism 1X: X → X, called the "identity" of X. For any f: X → Y we have 1X f = f 1Y = f.

You should visualize the composite of f: X → Y and g: Y → Z as follows:

                     f           g
               X ---->---- Y ---->---- Z

So far this is exactly the definition of a category! But a 2-category ALSO consists of a collection of "2-morphisms". Every 2-morphism T has a "source" morphism f and a target morphism g. If the source of T is f and its target is g, we write T: f => g. If T: f => g, we require that f and g have the same source and the same target; for example, f: x → y and g: x → y. You should visualize T as follows:
                 /         \
                x     T     y
                 \         /

People usually draw a double arrow like => going down next to the T, but I can't do that here.

In addition, we have:

1') Given a 2-morphism S: f => g and a 2-morphism T: g => h, there is a 2-morphism ST: f => h, which we call the "vertical composite" of S and T.

2') Vertical composition is associative: (ST)U = S(TU).

3') For each morphism f there is a 2-morphism 1f: f => f, called the "identity" of f. For any T: f => g we have 1f T = T 1g = T.

Note that these are just like the previous 3 rules. We draw the vertical composite of S: f => g and T: g => h like this:

                     /  S      \
                    /     g     \
                   x ----->----- y
                    \   T       /
                     \         /

Now for a twist. We also require that we can "horizontally" compose 2-morphisms as follows:
                      f           f'
                  ---->---    ---->---
                 /         \ /         \
                x     S     y     T     z
                 \         / \         /
                  ---->----   ---->----
                      g           g'

So we also demand:

1'') Given morphisms f,g: x → y and f',g': y → z, and 2-morphisms S: f => g and T: f' => g', there is a 2-morphism S.T: ff' => gg', which we call the "horizontal composite" of S and T.

2'') Horizontal composition is associative: (S.T).U = S.(T.U).

3'') The identities for vertical composition are also the identities for horizontal composition. That is, given f,g: x → y and T: f => g we have 11x.T = T.11y = T.

Finally, we demand the "exchange law" relating horizontal and vertical composition:

(ST).(S'T') = (S.S')(T.T')

This makes the following 2-morphism unambiguous:

                          f             f'
                      ---->----     ---->----
                     /  S      \   /  S'     \
                    /     g     \ /     g'    \
                   x ----->----- y ----->----- z
                    \   T       / \   T'      /
                     \         /   \         /
                      ---->----     ---->----
                          h             h'

We can think of it either as the result of first doing two vertical composites, and then one horizontal composite, or as the result of first doing two horizontal composites, and then one vertical composite!

Here we can really see why higher-dimensional algebra deserves its name. Unlike category theory, where we can visualize morphisms as 1-dimensional arrows, here we have 2-morphisms which are intrinsically 2-dimensional, and can be composed both vertically and horizontally.

Now if you are familiar with Cat, you may be wondering how we vertically and horizontally compose natural transformations, which are the 2-morphisms in Cat. Let me leave this as an exercise for now... there's a nice way to do it that makes Cat into a 2-category. This exercise is a good one to build up your higher-dimensional algebra muscles.

In fact, we could have invented the above definition of 2-category simply by thinking a lot about Cat and what you can do with categories, functors, and natural transformations. I'm pretty sure that's more or less what happened, historically! Thinking hard enough about nCat leads us on to the definition of (n+1)-categories....

But that's enough for now. Typing those diagrams is hard work.

To continue reading the `Tale of n-Categories', click here.

I thank Keith Harbaugh for catching lots of typos and other mistakes in "week73" - "week80".

© 1996 John Baez