Also available as http://math.ucr.edu/home/baez/week243.html
December 25, 2006
This Week's Finds in Mathematical Physics (Week 243)
John Baez
Today I'd like to talk a bit about the first stars in the Universe, and
some hotly contested possible observations of these stars. Then I want
to describe a new paper by my student Derek Wise. But first - if anyone
gave you a gift certificate for a bookstore this holiday season, here
are two suggestions.
The first one is really easy and fun:
1) William Poundstone, Fortune's Formula: The Untold Story of the
Scientific Betting System that Beat the Casinos and Wall Street,
Farrar, Strauss and Giroux, New York, 2005.
Packed with rollicking tales of gangsters, horse-racing, blackjack, and
insider trading, this is secretly the story of how Claude Shannon developed
information theory - and how he and his sidekick John Kelly Jr. used
it to make money in casinos and Wall Street. I'd known about Shannon's
work on information... but not that he beat 99.9% of mutual fund
managers, making an average compound return of 28% for many years -
as compared to 27% for Warren Buffett!
This book has just a few equations in it. I was delighted by one
discovered by Kelly, which I'd never seen before. Translating into my
own favorite notation, it goes like this:
S = log M
It's the fundamental equation relating gambling to information!
Let me explain it - in language far more complicated than you'll
see in Poundstone's book.
What's M? It's the best possible average growth of a gambler's money.
For example, if his best possible strategy lets him triple his money
on average, then M = 3.
What's S? This is the amount of "inside information" the gambler has:
information he has, that the people he's betting against don't.
Some technical stuff: First, the above "average" is a geometric mean,
not an arithmetic mean. Second, if we measure information in bits,
we need to use base 2 in the logarithm. Physicists would probably
prefer to use base e, which means measuring information in "nits".
It doesn't really matter, but let's use base 2 for now.
To get a feeling for why Kelly's theorem is true, it's best to start
with the simplest example. If S = 1, then M = 2. So, if a gambler
receives one bit of inside information, he can double his money!
This sounds amazing, but it's also obvious.
Suppose you have one bit of inside information: for example, whether a
flipped coin will land heads up or tails up. Then you can make a bet
with somebody where they give you $1,000,000 if you guess the coin
flip correctly, and you give them $1,000,000 if you guess wrong. This
is a fair bet, so they will accept. That is, they'll *think* it's
fair if they don't suspect you have inside information! But since you
do have this information, you'll win the bet, and double your money on
this coin flip.
Kelly's equation is usually phrased in terms of the *rate* at which
the gambler gets inside information, and the *rate* at which his money
grows. So, for example, to earn 12% interest annually, you only need
to receive
log(1.12) = 0.163
bits of inside information - and find some dupe willing to make bets
with you about this.
The last part is the hard part: the "inside information" really needs
to be information people don't believe you have. I must learn hundreds
of bits of information about math each year - stuff only I know - but
I haven't found anyone simultaneously smart enough to understand it
and dumb enough to make bets with me about it!
Still, I like this relation between information theory and gambling,
because one stream of Bayesian probability theory says probabilities
are subjectively defined in terms of the bets you would accept.
The argument for this is called the "Dutch book argument". It basically
shows how you can make money off someone who makes bets in ways that
correspond to stupid probabilities that don't add to 1, or fail to be
coherent in other ways:
2) Carlton M. Caves, Probabilities as betting odds and the Dutch book,
available at http://info.phys.unm.edu/~caves/reports/dutchbook.pdf
So, there's a deep relation between gambling and probability - no news
here, really.
But, there's also a deep relation between probability and information
theory, discovered by Shannon. Briefly, it goes like this: the
information you obtain by learning the value of a random variable is
S = - Sum_i p_i log(p_i)
where the sum is taken over all the possible values of this random
variable, and p_i is the probability that it takes its ith value.
So, for example, if you flip a fair coin, where p_1 = p_2 = 1/2,
the information you get by looking at the coin is
-[1/2 log(1/2) + 1/2 log(1/2)] = 1
One bit!
So: gambling is related to probability, and probability is related to
information. Kelly's result closes the circle by providing a direct
relation between gambling and information!
But, apparently some of Kelly's ideas are still controversial in the
world of economics and stock trading. If you read Poundstone's book,
you'll learn why.
The next book takes more persistence to read:
3) Avner Ash and Robert Gross, Fearless Symmetry: Exposing the Hidden
Patterns of Numbers, Princeton U. Press, Princeton, 2006.
The authors do a creditable job of what might at first seem utterly
impossible: explaining heavy-duty modern number theory to ordinary
mortals. The formal prerequisites are little more than high school
algebra, and the style is expository, but anyone except an expert
will need to stop and think at times.
They start by explaining modular arithmetic - you know, stuff like
adding and multiplying "mod 7". Then they tackle groups, and
permutations, since the main theme of the book is symmetry. Then
they move on to algebraic varieties, in a simple no-nonsense style
cleverly adapted from Grothendieck's later work (without terrifying
the reader by mentioning this fact).
Next they tackle some serious number theory: quadratic reciprocity,
Galois groups, and elliptic curves. Then they describe more general
forms of reciprocity, leading up to a taste of the Langlands program.
They conclude with a sketch of how Fermat's last theorem was proved.
These days mathematical physicists are all excited about a variant
of the Langlands program: the so-called "geometric" Langlands program,
which is related to string theory. Drinfeld has been running a
seminar on this at Chicago for years, but that's not what got the
physicists interested - it's these papers by Witten that did it:
4) Anton Kapustin and Edward Witten, Electric-magnetic duality and
the geometric Langlands program, 225 pages, available as hep-th/0604151.
5) Sergei Gukov and Edward Witten, Gauge theory, ramification, and
the geometric Langlands program, 160 pages, available as hep-th/0612073.
So, if you're trying to learn this geometric Langlands stuff, and
you want to fit it into the grand landscape of mathematics, the book
Fearless Symmetry could be a fun way to learn some the math underlying
the ordinary Langlands stuff.
I started girding myself for a discussion of the Langlands program in
"week217", "week218" and "week221", but then I got distracted. I'll
get back to it someday, but right now I'm in the mood for lighter
stuff... so let me tell you a bit about the first stars.
The story starts around 380,000 years after the Big Bang, when
the hot hydrogen and helium forming our Universe cooled down to
3000 kelvin - just cool enough for the electrons to stick to
the atomic nuclei instead of zipping around on their own.
When the electrons in a gas are hot enough for some to zip around on
their own, we say the gas is "ionized". When a *lot* of them are
zipping around, we call it a "plasma". Because charged particles
interact with the electromagnetic field, light doesn't pass through
plasma cleanly: it keeps getting absorbed and re-emitted.
So, before our story started, you couldn't see very far: it would be
like trying to look through a wall of fire. But, around 380,000 years
after the Big Bang, the gas became transparent!
What would this have looked like? Nobody ever seems to say.
So, I'll just guess, and hope some expert corrects me.
Back when the gas filling the Universe was 5000 kelvin in temperature,
just a bit cooler than the surface of the Sun, everything was yellow.
You couldn't see far at all: you would have been blinded by a yellow
glare.
But when it cooled to 4000 kelvin in temperature, the Universe became
orange.
And when it cooled to 3000 kelvin, the Universe became red.
And when it cooled a tiny bit further, it became infrared. As
far as visible light goes, the Universe became transparent!
This would happen everywhere more or less at once. But since light
takes time to travel, you'd see a transparent sphere around you, expanding
outwards at the speed of light, with reddish walls.
It's been sort of like this ever since.
So, when we look far away with our best telescopes, we look back in
time to the time when the Universe became transparent - but no
further. We're surrounded by a distant, ancient wall of fire.
It's now about 13.3 billion light-years away - or 13.3 billion
year back in time, if you prefer. And, it's receding at a rate of
one light-year per year.
But by now, the light from this wall of fire has been severely
redshifted. In other words, it's been stretched along with the
expansion of the Universe - stretched by a factor of 1100, in fact!
So, what had been the hot infrared glow of 3000-kelvin plasma
is now a feeble microwave glow corresponding to an icy temperature
of 2.7 kelvin. This is the famous "cosmic microwave background
radiation".
But let's go back in time....
From the moment the hot gas became transparent to the time when
the first stars formed, the Universe was dark except for the dimming
infrared glow of that distant wall of fire. This era is called the
"Dark Ages".
During the Dark Ages, gas cooled down and clumped under its own
gravity - apparently with a lot of help from cold dark matter of
some unknown sort. Without postulating this matter, nobody can
figure out how galaxies formed as soon as they did.
As befits their name, the Dark Ages are still shrouded in mystery.
There are a lot of unanswered questions besides the nature of dark
matter. Which formed first - individual stars, or galaxies? And,
when did the Dark Ages end?
It's currently believed that the first stars formed sometime between
150 million and 1 billion years after the Big Bang.
At the later end of that range, the Universe could have gotten quite
cold before starlight warmed up the interstellar gas and reionized it.
There's even a spooky theory that the Universe was full of hydrogen
snowflakes near the end of the Dark Ages - see "week196" for more on
this, and a timeline of the earlier history of the Universe.
But, the current best guess, based on data from the Wilkinson Microwave
Anisotopy Probe, says that reionization happened 400 million years
after the Big Bang:
6) Marcelo A. Alvarez, Paul R. Shapiro, Kyungjin Ahn and Ilian T. Iliev,
Implications of WMAP 3 year data for the sources of reionization,
Astrophys. J. 644 (2006), L101-L104. Also available as astro-ph/0604447.
This would be too early for hydrogen snow, since my rough calculation says
the microwave background radiation was 30 kelvin then, while hydrogen
freezes at 14 kelvin.
What were the first stars like? Without heavier elements to catalyze
nuclear fusion, they could have been larger than current-day stars:
perhaps hundreds of times the size of our Sun! These so-called
Population III stars have not actually been seen. But, it's possible
that we've finally caught a glimpse of them, not individually but
in a sort of statistical sense:
7) A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, New
measurements of cosmic infrared background fluctuations from early
epochs, to appear in Ap. J. Letters. Available as astro-ph/0612445.
8) A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, On the
nature of the sources of the cosmic infrared background, to appear
in Ap. J. Letters. Available as astro-ph/0612447.
Using delicate techniques to carefully sift through the *infrared*
(not microwave) background radiation, the authors claim to find
radiation not accounted for by previously known sources. Assuming
the standard cosmological scenario, the sources of this radiation
date back to less than 1 billion years after the Big Bang, and were
individually much brighter than current-day stars.
Here's a picture of their data:
9) NASA / JPL-Caltech / A. Kashlinsky, Infrared background light from
first stars, http://www.spitzer.caltech.edu/Media/releases/ssc2005-22/
On top is a photograph taken by the Spitzer Space Telescope: a 10-hour
infrared exposure of a tiny patch of sky, 6 x 12 arcminutes across,
chosen for having a bare minimum of foreground stars, galaxies and
dust. (For comparison, the Moon is 30 arcminutes across.) On the
bottom is the same picture with known sources of infrared subtracted.
What's left may be the severely redshifted light from early stars!
Or, it may not. In the following news story, Ned Wright of UCLA
said, "I'm very skeptical of this result. I think it's wrong.
I think what they're seeing is incompletely subtracted residuals
from nearby sources."
10) Dinesh Ramde, Associated Press, Hints of early stars may have
been found,
http://www.usatoday.com/tech/science/space/2005-11-02-early-stars_x.htm
So, we'll have to see how it goes....
But in the meantime, we can think about mathematical physics.
My student Derek Wise is graduating this year, and he's doing his
thesis on Cartan geometry, MacDowell-Mansouri gravity and BF theory.
Let me say a little about this paper of his:
11) Derek Wise, MacDowell-Mansouri gravity and Cartan geometry,
available as gr-qc/0611154.
Elie Cartan is one of the most influential of 20th-century geometers.
At one point he had an intense correspondence with Einstein on
general relativity. His "Cartan geometry" idea is an approach to
the concept of parallel transport that predates the widely used
Ehresmann approach (connections on principal bundles). It
simultaneously generalizes Riemannian geometry and Klein's Erlangen
program (see "week213"), in which geometries are described by their
symmetry groups:
EUCLIDEAN GEOMETRY --------------> KLEIN GEOMETRY
| |
| |
| |
| |
v v
RIEMANNIAN GEOMETRY --------------> CARTAN GEOMETRY
Given all this, it's somewhat surprising how few physicists know
about Cartan geometry!
Recognizing this, Derek explains Cartan geometry from scratch before
showing how it underlies the so-called MacDowell-Mansouri approach
to general relativity. This plays an important role both in
supergravity and Freidel and Starodubtsev's work on quantum gravity
(see "week235") - but until now, it's always seemed like a "trick".
What's the basic idea? Derek explains it all very clearly, so I'll
just provide a quick sketch. Cartan describes the geometry of a lumpy
bumpy space by saying what it would be like to roll a nice homogeneous
"model space" on it. Homogeneous spaces are what Klein studied; now
Cartan takes this idea and runs with it... or maybe we should say he
*rolls* with it!
For example, we could study the geometry of a lumpy bumpy surface by
rolling a *plane* on it. If our surface is itself a plane, this rolling
motion is trivial, and we say the surface is "flat" in the sense of
Cartan geometry. But in general, the rolling motion is interesting
and serves to probe the geometry of the surface.
Alternatively, we could study the geometry of the same surface by
rolling a *sphere* on it. Derek illustrates this with a picture of
a hamster crawling around in a plastic "hamster ball", which is
something you can actually buy for your pet hamster to let it
explore your house without escaping or getting in trouble.
(I've read about falling cats in papers on gauge theory, but this
is the first mathematical physics paper I've read containing the
word "hamster".)
If our surface is itself a sphere of the same radius, this rolling
motion is trivial, and we say the surface is flat in the sense of
Cartan geometry - but now it's a different sense than when we used
a plane as our "model geometry"!
Which model geometry should we use in a given problem? It depends
on which one best approximates the lumpy bumpy space we're studying!
The ordinary formulation of general relativity fits into this
framework, with a little work. Two well-known mathematical gadgets
called the "Lorentz connection" and "coframe field" fit together to
describe what would happen if we rolled a copy of Minkowski spacetime
over the lumpy bumpy spacetime we live in.
That's great if Minkowski spacetime is the best homogeneous
approximation to the spacetime we live in. But nowadays we think
the cosmological constant is nonzero, so the Universe is expanding
in a roughly exponential way. This makes another model geometry,
"deSitter spacetime", the best one to use!
So, if we know Cartan geometry, we can use that... and we get something
called the MacDowell-Mansouri formulation of gravity. Or, if we don't
want our spacetime to have lumps and bumps - if we want it to look
locally just like the Klein model geometry - we can use a different
theory, a topological field theory called BF theory (see "week232").
In short, the passage from a topological field theory describing a
"locally homogeneous" spacetime to full-fledged gravity with all its
lumps and bumps is nicely understood in terms of how Cartan's approach
to geometry generalizes Klein's!
For more details, you'll just have to read Derek's paper. You might
also try these:
12) Michel Biesunski, Inside the coconut: the Einstein-Cartan
discussion on distant parallelism, in Einstein and the History
of General Relativity, eds. D. Howard and J. Stachel, Birkhauser,
Boston, 1989.
This describes the correspondence between Cartan and Einstein.
I believe this centered, not on Cartan geometry per se, but on
the "teleparallel" formulation of gravity (see "week176"). But,
they're somewhat related.
13) Richard W. Sharpe, Differential Geometry: Cartan's Generalization
of Klein's Erlangen Program, Springer-Verlag, New York, 1997.
This is the main textbook on Cartan geometry. But, it's probably
best to read a few chapters of Derek's paper first, since the
key ideas are presented more intuitively.
My friend the geometer and analyst Rafe Mazzeo, whom I recently saw
at Stanford, told me that Cartan geometry was all the rage these days.
I'm embarrassed to say I hadn't known this! I think the kinds of
Cartan geometry being intensively studied are related to conformal
geometry, CR structures and stuff like that...
Merry Christmas!
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Quote of the Week:
"The Universe has as many different centers as there are living
beings in it." - Alexander Solzhenitsyn
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Addenda: I thank Chris Weed for catching typos. For more discussion,
go to the n-Category Cafe:
http://golem.ph.utexas.edu/category/2006/12/this_weeks_finds_in_mathematic_3.html
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