Science > Physics > Solitons vs Leptons vs Bosons vs Gravity 2: Randall-Sundrum and Vibrations
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Science > Physics |
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"OsherD" |
| Date: |
16 Jul 2005 01:24:53 AM |
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Solitons vs Leptons vs Bosons vs Gravity 2: Randall-Sundrum and Vibrations |
From Osher Doctorow
I typed this about 10 minutes ago, but it didn't appear, so I'll
summarize it.
It's rather easy to get a 5th dimension now that we know the
Kaluza-Klein and Randall-Sundrum ideas, but there are three interesting
questions:
A. What happens to Hilbert Space in the process?
B. What happens to vibrations of strings/branes in the process?
C. Is the 5th dimension a new mass dimension, a gravitation dimension,
a curved "vibration" dimension, or a combination of the above?
I'll try to begin with A next time.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Solitons vs Leptons vs Bosons vs Gravity 2: Randall-Sundrum and Vibrations |
16 Jul 2005 01:57:04 AM |
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From Osher Doctorow
The answer to A seems quite easy: Hilbert Space (HS) changes to Banach
Space (BS), which is a generalization of HS that doesn't necessarily
have an inner product. A pre-Hilbert space is a "HS without
completeness".
Hewitt and Stromberg (1965) in Real and Complex Analysis,
Springer-Verlag: N.Y. defined real or complex BS's as respectively real
or complex normed linear spaces compelte in the metric //x - y// with
the sup norm:
1) //f// = sup{/f(x): x in X}, X nonvoid set
provided that sup is finite. This norm is also called the uniform
norm.
They then defined abstract Banach Spaces as linear transformation
spaces between two of the first type Banach Spaces above, A and B,
using a rather interesting norm for bounded linear transformations T:
2) //T// = sup{/T(x)/: x in A, //x// < = 1}
which is called the operator norm. If even only B is a Banach space
and A is a normed linear space over the reals or complex fields, then
the space of such bounded linear transformations with the operator
norm and pointwise linear operations is a Banach Space.
Now in the Kaluza-Klein and Randall-Sundrum pictures, the 5th dimension
is "off our brane" and so can be regarded as "far away" in a sense. We
could define the 5th dimension by requiring that //T - S// > = k for
some constant k > 0 if T and S are respectively restricted to "on our
brane" vs "off our brane" arguments or even general arguments since we
expect a certain subset of bounded linear transformations between
branes or between brane and non-brane bulk to be "further apart" than
internal brane bounded linear transformations.
The intuitive idea here is defining a dimension not as an orthogonal or
some other directional (sub)space but as a "far away" object in a
norm-distance sense, although both types of dimension seem to be
useful.
So question A seems to be basically answerable.
Notice that //T - S// is just based on the main PI operation,
subtraction. That //x// < = 1 just as with probabilities being
restricted to [0, 1] (excluding of course "negative probabilities"
which don't exist) is an added bonus, although it isn't required for
the idea.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Solitons vs Leptons vs Bosons vs Gravity 2: Randall-Sundrum and Vibrations |
16 Jul 2005 02:21:49 AM |
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From Osher Doctorow
Remember that dimension also has a meaning in dimensional analysis
which isn't quite "orthogonal" or "at some other angle" or "far away"
in some sense but is more reminiscent of chemical elements which
compounds are composed of. Curiously enough, all three notions seem to
be coming together now in inquiring about gravitation.
To examine question B, I'm tempted to say that since strings and branes
can vibrate, then they do vibrate in the 5th dimension and that this
vibration constitutes a push/pull!
So the 5th dimension isn't present just in order to give gravitation
"somewhere to be" or "somewhere to be concentrated in", but rather
gravitation acts through the 5th dimension in such a way that if we
have two objects 1 and 2 in our brane, object 1 push/pulls object 2
through a gravitational vibration in the 5th dimension. The whole 5th
dimension is the causation between objects 1 and 2 (and other objects).
Remember that in my previous threads, I isolated 3 length dimensions, 3
time dimensions, 3 mass dimensions, and 1 or 2 force dimensions. The
second time dimension is time as causation (since causes precede
effects in time), and this is what appears to be involved in the 5th
dimension. So it is T2. But also in those previous threads, it turned
out that mass M1 is mass as inertia, mass M2 is mass as energy, and
mass M3 is mass as curvature. So T2 and M3 appear to be involved in
the 5th dimension - which may not be "only" a 5th dimension but rather
a 5th and 6th dimension. Since force appears to be involved in a
quite explicit scenario, dimension F1 or F2 of force seems to be 7th
dimension involved.
So the answer to B appears to be that 3+1 branes vibrate in the 5th or
5th through 7th dimensions and thereby transmit a gravitational force.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Solitons vs Leptons vs Bosons vs Gravity 2: Randall-Sundrum and Vibrations |
16 Jul 2005 02:28:13 AM |
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From Osher Doctorow
Happily, I seem to have answered both B and C above simultaneously.
I keep coming back to Hitchiker's Guide to the Galaxy (I really shall
have to look up the biography of the author of that book, assuming that
it's not my own of course), but the fact is that I'm tempted to say
like the ponderous Aliens: Next Question.
Next question :>)
Osher Doctorow
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