November 20, 2000

This Week's Finds in Mathematical Physics (Week 160)

John Baez

Anyone who grew up on science fiction in the 1960s probably read a bunch about adventures on strange planets, and dreamt of our future in space. At least I did. Asimov, Clarke, Heinlein... they helped get me interested in science, but they also painted a romantic vision of human destiny. Only later did it become clear that for now, the real adventures will come from the microscopic realm: from applications of integrated circuits, biotechnology, nanotechnology, and the like. When you're trying to have lots of fun in a hurry, the speed limit is the speed of light - and this makes interstellar travel a drag.

Nonetheless, when you imbibe a romantic dream in childhood, it can be hard to shake it it as an adult. So I still like to read about strange planets, even if know rationally that I can have more fun at home.

So - let me start by talking about a world where it might rain methane!

1) Ralph D. Lorenz, The weather on Titan, Science 290 (October 20, 2000), 467-468.

Caitlin A. Griffith, Joseph L. Hall and Thomas R. Geballe, Detection of daily clouds on Titan, Science 290 (October 20, 2000), 509-513.

Titan is the largest moon of Saturn, and it's the only moon in our solar system with a significant atmosphere. Its atmosphere is mostly nitrogen, with a surface pressure 1.5 times that of the air pressure here on Earths' surface. However, there is also a fair amount of methane, and even some ethane. At the surface of Titan, it's cold enough for these compounds to liquefy. People have even seen what look like pitch-black oceans of hydrocarbon compounds, hundreds of kilometers in size!

However, 14 kilometers or more from the surface, it gets cold enough for methane to freeze. And the news is that recently Caitlin Griffith et al have spotted things that look like methane clouds. Compared to Earth, which is usually 30 percent covered with clouds, the cloud cover on Titan seems spotty. There's not really enough methane for lots of clouds. But there may be rain! The drops would be larger than terrestrial raindrops, and fall slowly in the gravity of Titan, which is like that of our moon. Since the near-surface atmosphere usually has a relative humidity of at most 60%, the drops would tend to evaporate before hitting the ground. (I've seen a similar thing in New Mexico.) However, in a big rainstorm the evaporation of the first drops might elevate the humidity to the point where later drops could reach the surface. So there might even be erosion on the surface of Titan. With any luck, the Cassini spacecraft will arrive at Saturn in 2004 and make about 40 flybys of Titan in the following 4 years, getting a good look at this stuff.

Now for a crazy speculation of my own. Once upon a time James Lovelock argued that you could tell there was life on earth simply by noting that the atmosphere contains lots of oxygen, despite the fact that oxygen is highly reactive. This means the atmosphere is far from equilibrium. Yet the percentage of oxygen in the atmosphere has remained fairly constant for long periods of time! So presumably there must be some homeostatic mechanism at work to keep it constant. Only life - he argued - could be responsible! Conversely, Lovelock guessed there is not life on Mars, because its atmosphere is in equilibrium.

Now, the methane in Titan's atmosphere is dissociated by sunlight, and this process is irreversible, since the resulting hydrogen flies off into space. At the rate this happens, the entire methane content of the atmosphere would be destroyed in only 10 million years if it were not renewed somehow. In the first article cited above, the author writes: "For the methane we see today not to be a bizarre fluke, it must be continuously resupplied from a surface reservoir or by cryovolcanism (that is, volcanism where the molten `rock' is just ice)." And this made me wonder: where is Lovelock when we need him? Maybe life is responsible for this out-of-equilibrium condition.

Or maybe not. After all, it really could be something else.

Next: a world where it might rain diamonds!

2) Richard A. Kerr, Neptune may crush methane into diamonds, Science 286 (October 1, 1999), 25.

Laura Robin Benedetti, Jeffrey H. Nguyen, Wendell A. Caldwell, Hongjian Liu, Michael Kruger, and Raymond Jeanloz, Dissociation of CH4 at high pressures and temperatures: diamond formation in giant planet interiors?, Science 286 (October 1, 1999), 100-102.

The atmosphere of Neptune is believed to contain lots of methane when you go 4000 kilometers or more beneath the cloud tops. And Neptune ain't no measly moon: it's a gas giant, so the atmospheric pressure becomes enormous as you go further in. Recently, people have been compressing methane under ridiculously high pressures, using techniques too fiendish to describe here. At sufficiently high pressures, it releases hydrogen and turns into diamond crystals! - together with lots of other crud, like ethane and acetylene. This could happen in Neptune at a depth of about 7000 kilometers below the cloud tops, where the pressure reaches 500,000 times that of the Earth's atmosphere. So in fact, there could be a steady rain of diamond crystals on Neptune! By the way, all these Science articles are available for free online here:

3) Science Magazine,

I also want to say a bit about spin foams. Papers continue to come out on this subject:

4) Alejandro Perez and Carlo Rovelli, A spin foam model without bubble divergences, available as gr-qc/0006107.

A while ago, De Pietri, Freidel, Krasnov and Rovelli showed how to get the Barrett-Crane model for Riemannian quantum gravity from a quantum field theory on a product of 4 copies of SO(4) - see "week140". This was based on earlier work by Boulatov and Ooguri, who did a similar thing for BF theory. The basic idea is to cook up a quantum field theory on a product of copies of Lie group, with a nice Lagrangian that encodes how simplices can stick together to form a spacetime. If you do a Feynman diagram expansion of this quantum field theory, the Feynman diagrams can be identified with spin foams, and the sum over Feynman diagrams becomes a sum over spin foams.

The sum over spin foams may diverge; this paper attempts to control those divergences. It makes some precise mathematical conjectures about the convergence of certain sums - mathematicians who like analysis and representation theory should get to work on these!

5) Alejandro Perez and Carlo Rovelli, Spin foam model for Lorentzian general relativity, available as gr-qc/0009021.

This paper modifies the De Pietri-Freidel-Krasnov-Rovelli construction to get the Lorentzian Barrett-Crane model from quantum field theory on a product of 4 copies of SO(3,1).

6) Alejandro Perez and Carlo Rovelli, 3+1 spinfoam model of quantum gravity with spacelike and timelike components, available as gr-qc/0011037.

In the original Lorentzian Barrett-Crane model, spacetime is made of 4-simplices whose triangular faces are space/timelike - in other words, like little bits of the xt plane in Minkowski spacetime. This model also allows 4-simplices whose triangular faces are space/spacelike - in other words, like little bits of the xy plane. This amounts to using a different class of irreducible unitary representations of the Lorentz group to label the triangles.

7) Daniele Oriti and Ruth M. Williams, Gluing 4-simplices: a derivation of the Barrett-Crane spin foam model for Euclidean quantum gravity, available as gr-qc/0010031.

This gives an alternate derivation of the Riemannian Barrett-Crane spin foam model starting from the Lagrangian for Riemannian general relativity. This is good because it gives some more intuition for the relation between classical general relativity and the spin foam approach to quantum gravity.

Finally, if you're hopelessly confused about spin foams and other approaches to quantum gravity, you might enjoy the following little history of quantum gravity. It explains how many different approaches were tried, leading up to the research directions that people pursue now:

8) Carlo Rovelli, Notes for a brief history of quantum gravity, presented at the 9th Marcel Grossmann Meeting in Rome, July 2000. Available as gr-qc/0006061.

© 2000 John Baez