Re: resolve to perpendicular components, because they are independent

"Ken S. Tucker" <dynamics@xxxxxxxxxxxx> wrote in message
news:1137921340.012633.216030@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
| 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.

Ok thanks. Maybe we can call this PIT? ;-) As a way to remember it.

| 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.

Sure. I think Randall is applying the warp factor to a 5th dimension.
However she is using branes that are separated. I wonder if what she is
doing would work with the branes intersecting somewhat? IOW, the 5th
dimension "distance" instead of a separation distance would be an
intersection distance.

| >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?

Do you mean; why would we want to specify an integer? If so, I would
imagine that it makes calculations easier? ;-)

| 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.

Hmm... I thought that using tensors already took care of that. So you
are saying that using integer indices in tensor notation doesn't really
"match" nature? I guess it is due to the fact that we are imposing a
Lorentzian signature.

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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