Re: resolve to perpendicular components, because they are independent

Hi Fred

FrediFizzx wrote:
> "Ken S. Tucker" <dynamics@xxxxxxxxxxxx> wrote in message
> news:1137871786.842707.317150@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> |
> | FrediFizzx wrote:
> | > "Ken S. Tucker" <dynamics@xxxxxxxxxxxx> wrote in message
> | > news:1137804799.742330.45120@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> | > |
> | > | FrediFizzx wrote:
> | > | > "Ken S. Tucker" <dynamics@xxxxxxxxxxxx> wrote in message
> | > | > news:1137800854.404588.75250@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> | > | > |
> | > | > | Timo Nieminen wrote:
> | > | > | > On Fri, 20 Jan 2006, Ken S. Tucker wrote:
> | > | > | >
> | > | > | > > I find nonorthogonal axes easier than orthogonal,
> | > | > | >
> | > | > | > Then you must be some kind of bizarre freak of nature!!!
> | > | > |
> | > | > | Not really, as in Chess, solving problems in mathematical
> | > | > | physics consists of keeping your options open, to be
> | > | > | closed by physical principle, and certainly not by an aprior
> | > | > | preceived convenience. It's well known "orthogonality" is
> | > | > | at best an approximation in a g-field, but Reimann and his
> | > | > | "gang" evolved quite a nice "tensor" analysis notation that
> | > | > | is easier to use than clunky "ijk" unit vectors.
> | > | > |
> | > | > | > > indeed a Curl
> | > | > | > > becomes A_u,v - A_v,u (== &A_u/&x^v - &A_v/&x^u), because
> | > | > | > > manipulating equations in tensors is streamlined by
> notation.
> | > | > | >
> | > | > | > Can't you just do that with orthogonal metrics too? (Mixing
> | > | > covariant and
> | > | > | > contravariant is just a naughty little trick to hide the
> metric
> | > | > tensor!)
> | > | > |
> | > | > | If your intrinsic dimensionality differs from an integer, i.e
> | > | > | let n= intrinsic dimensionality =2.9, then how the heck do
> | > | > | you expect to squeeze 3 orthogonals into that?
> | > | >
> | > | > Hmm... I wonder if that would apply to what Lisa Randall is
> calling
> | > | > "Warped Passages"?
> | > |
> | > | LOL, ok, how about a link, Randall is super-pop, so
> | > | I know you're not jokin...
> | >
> | > "Discretizing Gravity in Warped Spacetime"
> | > http://www.arxiv.org/abs/hep-th/0507102
> | >
> | > I haven't read this yet but maybe it has something. I was mainly
> | > referring to something she was saying in her new book (did you get
> it
> | > yet? ;-) ). I didn't make the connection at the time I was reading
> it
> | > until you brought this up (forgot what you call it) again.
> | > FrediFizzx
> |
> | Here's an interesting quicky...
> |
> | http://en.wikipedia.org/wiki/Fractional_calculus
> |
> | that demo's a departure from our usual "integer" thinking, we
> | commonly apply to both calculus and so to dimensionality.
> |
> | Recall that when we integrate a line like "x" by
> |
> | $ x dx = x^2/2 == area
> |
> | we go from 1D "x" to 2D "x^2" , but what the link above shows
> | is that integration (and differentiation) can be a continuous thing,
> | and so can dimensionality.
> |
> | Is that where we're going?
>
> Yep, I am really thinking that this is what she is talking about with
> "warped" spacetime. Now what is that particular name you had for this?

I refer to that as "partial interdimensional transformations",
basically lifting the requirement for an integer in the tensor
indices.
Recall that conventionally a spacetime tensor would
have components like A_u == A_0, A_1, A_2, A_3 , but
when we do the tensor calculus the number u=4 is not
required until we substitute a 4D CS into a specific
physical application.
We also know that the "nonorthogonality" (warp) of spacetime
depends on the strength of the g-field.
An example is the hypothetical "event horizon" where both
time and 1 spatial dimension vanish altogether in that extreme
circumstance.

>>From the point of view of experimental mathematics there
is a physical basis to consider fractional dimensionality just
as we may consider,

http://en.wikipedia.org/wiki/Fractional_calculus

to show fractional derivatives and integrals is rational
mathematics.

So if we start with some vector (or tensor) "A" we can
manipulate the components "A_n" without specifing n
to be an integer. But why would we want to?

My reasoning is based on a sort of General Covariance,
where the laws of nature are independent of preconceived
dimensionality. If I was an electron, the laws of nature
would apply to me, but I would not need a wrist watch,
because an electron is totally stable in time, and wouldn't
work :-).

Suppose for example we have preconceived laws of nature
cast in 4D and find difficulties of applying GR inside a sub-
atomic particle. Would we dismiss GR or question the idea
that our macroscopic 4D view of dimensionality is true at all
scales?
So formulating the laws of physics independent of fixed
dimensionality is reasonable.

Regards
Ken S. Tucker















































>
> FrediFizzx
>
> http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.pdf
> or postscript
> http://www.vacuum-physics.com/QVC/quantum_vacuum_charge.ps
>
> http://www.vacuum-physics.com

.

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