Principal G-bundle and "bundle of bases" (footnote)


by marcus
Tags: bundle of bases, footnote, gbundle, principal
marcus
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#1
Aug11-03, 04:24 PM
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The following came up in Lethe's forms thread, but I'll separate it out and add to it to use as a footnote in Gravitivity thread---group action, differential forms, bundles all come up in a classical treatment of gravity.

A differential geometry book (Bishop and Crittenden) that I happened to pick up defines a "principal bundle" as a triple (P, G, M) where P and M are smooth manifolds and G is a Lie group
(1) G acts freely on P, GxP --> P (they choose a right action, it could be left)

(2) M is the quotient space of P mod equivalence by G
the projection map is ξ:P --> M
G acts transitively on the fiber ξ-1(m) over any point m in M

(3) P is locally trivial. that means that around any point m in M there is a neighborhood U ( picture a disk) such that the part of P that is over U ( picture a cylinder over the disk), namely
ξ-1(U), is diffeomorphic to the cartesian product
U x G ( picture a second cylinder U x G, with U a disk and G a vertical line).

The diffeomorphism ξ-1(U) --> UxG takes a point p to (ξ(p), FU(p)) and this map FU: ξ-1(U) --> G satisfies an equation FU(gp) = gFU(p).

The equation says you can do the group action first and then do F, or you can do F first and then do the group action, same result. In other words F "commutes with the group action."
---------------------------
Example---the "bundle of bases":

Let M be a smooth manifold and B(M) be the set of N+1-tuples
(m, e1,...,eN) where m ε M and {e1,...,eN] is a basis of the tangent space at m.

Let ξ: B(M) --> M be the projection ξ(m, e1,...,eN) = m

GL(N, R) acts on the left on B(M), lets define the action,

g ε GL(N, R) can be viewed as a matrix (gij)
so just let its rows specify linear combinations of {e1,...,eN}
and you have a new basis {...Σgijej...)
I see no ambiguity here.
(m, e1,...,eN) --> (m, Σg1jej,...,ΣgNjej)

So far there has been no arbitrary choice. RN has a natural basis (1,0,...), (0,1,0,...) and so on and the matrix is with respect to that but we can apply it to the {e} basis of the tangent space

Now to make this a principal bundle we need to be able to coordinatize it and to say, for any coordinate patch neighborhood U of a point m, what the locally trivializing maps
FU are.
FU: ξ-1(U) --> G

So suppose we have coordinates x1,..,xN defined in a patch U around a point m
and suppose the point m' is in the neighborhood U
and (m', f1,...,fN) is in the fiber ξ-1(m') over m'

Now {f1,...,fN} is a basis of the tangent space
we have to get a NxN matrix in the group G, this being what
FU does. Well define
FU(m', f1,...,fN) = (gij) = (dxjfi)

this provides a coordinate system for ξ-1(U)
the system says, map it by (ξ, FU) into UxG
and then use the x coords in U and the standard RN basis coords to give the matrix. This way you get N + NxN numbers. But it doesnt even look bad. No sweat. Bundles must be good language.
----------

Now I am thinking that if we just took Minkowski space in place of
RN and the Lorentz group in standard matrix form for our G in place of GL(N,R), we could have a bundle-of-bases as a principal G-bundle over a 4D manifold M. The bundle would consist of 5-tuples (m, e0,...,e3) where the e0,...,e3 are a basis of the TANGENT space...there are some details to fill in and maybe this will not work as is. I would like to see if this model can apply to streamline what rovelli is saying. Be back later.
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marcus
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Aug12-03, 07:40 PM
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Some philosophy, or at least opinion.

I think a good bit of the motivation for inventing the formalism of differential forms, principal bundle, associated bundle, is to get away from writing so many subscripts and superscripts.
You get through with much of the 'scripts business once and for all when you define certain structures-----which are then supposed be powerful laborsaving devices and streamline later exposition.

But what if you want your book to be read both by physicists who like indices and differential geometers who like bundles and clean notation? You might find yourself writing indices and talking bundles.

The formulas would have lots of sub and superscripts, but when you describe them and discuss them you use some modern concepts that ought, if used consistently, streamline the notation.

It is one possible way to make sure you communicate to both audiences.

I am trying to understand the concepts well enough that I can rewrite his equations with the same letters---e for the frame, "tetrad" " vierbein" "gravitational field"-----ω for the connection---but without indices or at least with few indices.
It is going slowly.

I have to define a "reduction" of the structure group---or a reduction of the principal bundle: which is how you shrink down from GL(4) to SO(1,3) and from the bundle of bases to the bundle of frames. And I have to define an "associated" bundle which is how you substitute Minkowski space for the the fiber, instead of the structure group.

Is it worth it? just to get rid of a few dozen subscripts and superscripts? I dont know---a bit discouraged at the moment.
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Aug13-03, 12:51 PM
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for me the heart of rovelli's book is pages 43,44
where he is explaining the form of the gravitational field

ultimately my opinion of the book will depend, as much as anything else, on my understanding of that passage

he talks about the cloud of galaxies and the idea of "local inertial frame" at some spacetime point (some "event")

even tho our galaxy may be falling towards the virgo cluster or towards the great attractor or some other damn thing, and is doubtless accelerating in some direction, yet nevertheless our galaxy has a "local inertial frame"----and in fact it has a whole bunch, take any one inertial frame and apply a lorentz matrix Λ to it.

now a naive question is, why does he write the local inertial
frame as a linear map from the tangent space to R4?
(actually calling it M but as a set that's what it is)

Why doesnt he write the local inertial frame as a map from tangent space to itself?

Why does he introduce this copy of M so that the gravitational field can be a 1-form with values in M instead of some more familiar object like a 1-form with values in the tangent space?

We are used to working with a manifold and the tangent space (and cotangent space etc) at a point. Familiar structures are built up from these materials. Why is this unfamiliar feature introduced?

Well, I think the explanation may be that it is a mathematically "natural" way to establish a Lorentz group action on the tangent space-----to set up a PROXY so to speak, on which the lorentz matrices Λ already know how to act, and to let the gravitational field map into that proxy.

And it just so happens that there is a mathematically natural construction called an "associated bundle" which can set you up with a copy of M at each point of the manifold, and do it in short order---in one line of code so to speak.

The shock comes at the top of page 44. Remember that he has done everything from page 21 thru 43 by comparatively unsophisticated means---spelling everything out with lots of indices.
Up thru page 43 he has not (as far as I remember) said the words principal bundle or associated bundle.

He has been using "internal" indices which all physicists seem familiar with and which should have warned us what was coming
but we still dont realize what is in store for us and then, on page 44, he says.

"Well obviously all this stuff we have been writing with a lot of indices since page 21 TRANSFORMS LIKE sections of an associated bundle of a principal G bundle where G is the lorentz group."

The physicists have understood him perfectly well from page 21 thru 43, and the mathematicians have been yawning and grumbling. Then suddenly on page 44 he says something that some of the physicists may not even understand, which is like a tidbit thrown out to placate the mathematicians.

And it is just a sentence or two so anybody who does not understand it can skip right over---the development does not depend on the fact that all these index-bristling tensor-animals
transform like they live in bundle-land.


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