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Principal G-bundle and "bundle of bases" (footnote)
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."
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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), let's 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 doesn't 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.
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), let's 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 doesn't 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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