Stephen Paul King (email@example.com)
Thu, 02 Dec 1999 23:58:18 -0500
I am attempting to get a discussion going about how to model the
interactions of LSs. Here we have a discussion about connections,
parallel transport and fiber bundles that might help:
On 30 Nov 1999 10:59:35 -0600, in sci.physics.research
firstname.lastname@example.org (John Baez) wrote:
In article <email@example.com>,
Oz <Oz@upthorpe.demon.co.uk> wrote:
>In article <firstname.lastname@example.org>, John Baez <email@example.com>
>>Remember those Roman chaps who
>>march around carrying javelins tangent to the surface of the
>>earth - i.e., tangent vectors? Suppose one of them is standing on the
>>north pole and the other is standing somewhere else - say, the equator -
>>and you ask them if their tangent vectors are the same. They'll say
>>"Of course not! They couldn't possibly be the same because they live
>>in completely different vector spaces."
>This concept that 'everything is local' takes a bit of time to hammer
>thouroughly into one's subconcious and it takes even longer to burn all
>traces of globalism at the stake.
Yes! As you probably remember, Ted and I started this painful process
when you first started asking questions about cosmology, a long time
ago. Lots of your questions simply didn't make sense. The reason
was usually that you were comparing quantities at different points of
spacetime without specifying how to carry one over to the other to
compare them. You would get very upset when we told you this didn't
make sense. You probably thought we were nitpicking. Now you probably
see how important it is that "everything is local". All the forces in
nature are forms of CURVATURE, and curvature is just another word for:
it matters which path you use to carry something from here to there.
Alas, you are still struggling against your old bad habits a bit -
otherwise you would never have DREAMT of suggesting that tangent vectors
at two different points of space might count as "the same". But at
you are aware of your bad habits, aware of the problems they cause,
and willing to battle against them. You should like the concept of
"tangent bundle", because it's precisely designed to help us battle
"unwitting globalism". Each point in a manifold has its own tangent
All these tangent spaces taken together form the tangent bundle! But
we never mix up a vector in *one* tangent space with a vector in
We bundle them together - but maintain their individuality!
More generally, the concept of "fiber bundle" is the tool we use
to talk about more general quantities "at a point" without succumbing
to the temptation to treating quantities at different points as "the
This is why fiber bundles are so important in modern physics - they
provide a language that helps us remember that "everything is local"!
The simplest example of a fiber bundle is a tangent bundle: the
"fibers" of this bundle are the tangent spaces. You should probably
master this example before moving on to others, because once
you do, you'll see all the others are very similar.
>Now I have to get another viewpoint of vector spaces in this sort of
>context. I think I had it once, but it was less generalised.
Hmm, I don't know what you mean... maybe you're referring to vector
bundles? A vector bundle is basically a fiber bundle where all the
fibers are vector spaces. The classic example is, again, the tangent
bundle of a manifold.
>So (wearily) I suppose (I really shouldn't do this) you superpose paths
>and connections onto this to get some way of describing how a vector
>behaves as it moves over M?
Yeah, basically that's right. If your tangent bundle has a connection,
you can take any tangent vector at some point p and "parallel transport"
it along a path from p to q. As you do, you get a path in the tangent
bundle!!! Visualize that, please. We have a path in the original
manifold M and now we've "lifted it" - that's the usual jargon - to
a path in the tangent bundle TM.
Wow, we're actually doing differential geometry! Great!
You might take a peek at that nasty section in my book where I
first introduce the concept of fiber bundles... it might make
a little bit more sense now.
On 1 Dec 1999 10:40:46 -0800, in sci.physics.research
firstname.lastname@example.org (John Baez) wrote:
In article <email@example.com>,
Daryl McCullough <firstname.lastname@example.org> wrote:
>This may be a stupid question, but is it correct to say that
>the "phase space" of classical physics is actually one of these
Exactly right! And this is the most important use of bundles
in physics, apart from general relativity and gauge theory.
If the configuration space - the space of "positions" - is
a manifold M, the phase space - the space of "positions and
momenta" - is the cotangent bundle T*M.
Why not the tangent bundle? Well, it turns out that while
velocity is best thought of as a tangent vector, momentum
is best thought of as a cotangent vector. One reason is
that there's a natural way to define the Poisson brackets
of smooth functions on a cotangent bundle - but not on a
>I always thought of phase space in Euclidean
>terms---you start with N-dimensional position space, and you
>combine it with N-dimensional momentum space to get 2N-dimensional
>phase space. But this picture leads you to think that it makes
>sense to independently vary position and momentum---that it makes
>sense to say that two points have the same momentum, but different
>positions just as it makes sense to say that two points have the
>same position, but different momenta. But comparing momenta
>at two different places probably makes no sense, in general.
As long as configuration space is good old Euclidean space,
it does make sense to compare momenta at two different
places, so there is a symmetry between position and momentum.
But when configuration space is an arbitrary manifold, it
no longer makes sense to compare momenta at different positions,
so this symmetry is lost, and the cotangent bundle picture
becomes essential. It's even nice in the case of Euclidean
A simple example of a configuration space that's not Euclidean
space: the spinning top.
(Btw, I still want to answer your question on Hawking radiation,
but it sorta sunk under a pile of more easily answered questions!)
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