Re: Geometric Algebra
- From: stebla@xxxxxxxxxxxx (Stephen Blake)
- Date: Mon, 4 Apr 2005 09:34:39 +0000 (UTC)
> Igor Khavkine <igor.kh@xxxxxxxxx> wrote
>Ok, if you explicitly start with an embedding into a vector space, then
>all the derivatives that you talk about are well defined. However, I
>must point out that if you do that all your "non-tensorial" representations
>will explicitly depend on the embedding. In this picture it is a little
>hard to talk about diffeomorphisms of the embedded manifold into itself,
>since its points don't have an identity independent of where they are
>embedded. I would say that the transformations g are "change of
>embedding" transformations rather than diffeomorphism transformations.
>If you are ok with that, then everything is fine.
If Alice's manifold is A and Bob's manifold is B and A and B are both
embedded in the same (say) vector space, then I understand a
diffeomorphism as a smooth 1-1 map g:A->B. However, there is nothing to
stop Bob's manifold also being manifold A as well, then we can have a
diffeomorphism g:A->A so that Alice has a point p in A then Bob sees
this point as gp which is also in A, but gp and p are different points
in the single manifold A. These different points are just the different
viewpoints of Alice and Bob of the same event. In the spr thread,
http://groups.google.co.uk/groups?hl=en&lr=&selm=c845ea%24rtu%241%40glue.ucr.edu&rnum=6
John Baez took an active transformation as g:A->A, but in section 3.1 of
"The large scale structure of space-time" Hawking and Ellis take g:A->B
as the effect of an active transformation. I don't think it matters
whose definition you choose.
>In fact, classical differential geometry, as it was in the time of
>Gauss and others, started out by studying parametrizations of various
>curves and surfaces in R^n. However, one would like to give these
>manifolds some identity independent from their embedding or
>parametrization. That's what the apparatus of modern differential
>geometry was developed for.
Yes, I agree that the idea of a manifold on its own without an embedding
space is an attractive idea because the embedding space is an extra
structure, and if there is no need for it, then it should be thrown
away.
So, let's get rid of the embedding space. I can go along with that,
because I've just used the embedding space as an intellectual crutch to
help me understand why differential forms is crafted in the way that it
is.
Now, our everyday notion of objects makes us to want to think of the
manifold and its fields as an invariant geometric object which is "out
there". For example, you argued that modern differential geometry was
developed to give "some identity" to the manifold. However, the picture
is more subtle than this because, in fact, we are forced to consider the
manifold and its fields as an equivalence class of manifolds and their
fields related by diffeomorphisms. This was illustrated by the example
of the 1-d circle manifolds that we discussed in my previous post in
this thread; all the circle manifolds and their velocity fields and
accelerations represent the same physical situation. There is not one
circle manifold but a bunch of circle manifolds. I said that each circle
manifold represents a particular observer's view of the physical
situation. However, I think that the diffeomorphism group goes well
beyond representing the viewpoints of all observers. I think it also
represents a bunch of viewpoints which do not correspond to the
viewpoints of physical observers; these extra viewpoints are just
meaningless mathematical viewpoints with no physical content.
To see this, consider a physical situation in which Alice measures a
field on some initial data hypersurface in her manifold, and then she
integrates a field equation, which deterministically fixes the field
throughout her manifold. Suppose the field is Q and the field equation
is something like L(Q)=0 where L is some sort of operator which acts on
the field. Now suppose we transform to Bob's viewpoint with an element f
of the diffeomorphism group. The operator will transform as
L(.)->L'(.)=TfL(Tf^{-1}.) and the field will transform as Q->Q'=TfQ
where Tf is the rep. On physical grounds, Bob's field equation L'Q'=0
should also be deterministic in the sense that it uniquely fixes Bob's
field given the boundary conditions on his version of the initial data
hypersurface. The element f of the diffeomorphism group is a
transformation between the viewpoints of two physical observers Alice
and Bob. Now suppose we use an element g of the diffeomorphism group
which goes g:A->M where A is Alice's manifold and M is the manifold
shown in the ascii sketch below.
/
p
A /
A /
h------<
M \
M\
gp
\
In this sketch h is Alice's initial data hypersurface. The
diffeomorphism g is the identity near to the initial data hypersurface,
but it smoothly separates the two manifolds further out from the
hypersurface. The sketch also shows Alice's point p on her manifold A
becomes gp on manifold M in the region where the diffeomeorphism
separates the two manifolds. The field equation on M is
TgL(Tg^{-1}Q'')=0 where Q'' is the field on M. However, this field
equation does not fix the field uniquely. For, if we solve the equation
and find the solution Q'' we know that we can apply the diffeomorphism
g^{-1} to the solution on M and get Alice's field Q. But Alice's field Q
and the field Q'' are two solutions obtained from the same boundary
conditions. Consequently, I don't think that the diffeomorphism g:A->M
represents the change to a view point of a physical observer because if
an observer has a deterministic field equation, one would expect the
field equation to remain deterministic for another physical observer.
The diffeomorphism g must represent a mathematical transformation which
has no physical content. This is my version of the kind of argument that
Kretschmann made against Einstein concerning the lack of physical
content of the diffeomorphism group in GR.
So, presumably, there are elements of the diffeomorphism group which are
physically meaningless. Now, at the start of this argument, we threw
away the embedding space on the grounds that it was a spurious element
of the theory. Therefore, by the same token, we should throw away these
physically meaningless transformations. In other words, we should work
with a subgroup of the diffeomorphism group which represents the
transformations between physical observers. Of course, as soon as one no
longer cares about the diffeomorphism group, then the reason to study
differential forms evaporates because everything in the theory of
differential forms is set up to transform under a rep of the
diffeomorphism group. Once we give up the diffeomorphism group we
typically have to work with (say) the Poincare group or the de Sitter
group O(4,1) which represent the transformations between viewpoints of
physical observers. Differential forms aren't needed in this situation
and issue of an embedding space doesn't arise.
Stephen http://homepage.ntlworld.com/stebla
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