Science > Physics > Quantum Gravity Via Expansion-Contraction 29.1: Planes and Plane Waves
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
29 Oct 2006 01:17:42 AM |
| Object: |
Quantum Gravity Via Expansion-Contraction 29.1: Planes and Plane Waves |
Doctorow
Probable Influence/Causation (PI) or z = z(x, y) has the planar form:
1) z = 1 + y - x, 0 < = y < = x < = 1
The partial derivatives Dx(z), Dy(z) of z respectively with regard to x
and y are:
2) Dx(z) = -1
3) Dy(z) = 1
Since z increases with y and decreases with x, the partial derivatives
preserve the direction of increase and at the same time do not "reverse
Causation" in the sense for example that Dx regarded as the x-Causation
of z does not increase with x, unlike the case for example with z =
x^(1/2).
In Section 29.0 (about 4 or 5 pages back on my terminal screen), I
pointed out a similar fact with regard to the ordinary derivative
rather than the partial derivative, but the same idea applies here with
partial derivatives.
But how can a plane or section of a plane as in (1) incorporate
(Probable) Causation in physics, which is so 3-4 dimensional oriented?
The Holographic Principle of 't Hooft tells us one good reason, and
another reason comes from the surprising direction of Rigged Hilbert
Space theory as for example in the papers and books of A. Bohm et al at
U. Texas Austin, in which the plane wave among other things played a
prominent role in generalizing Hilbert Space to Rigged HS, Nested HS,
lattices of Rigged and Nested HSs and BSs (Banach spaces), etc.
Of course, the tangent plane to a 3-dimensional object or a
3-dimensional space curve plays a central role in physics in and out of
General Relativity, not to mention differential geometry and so on.
Again, I suggest comparing the simplicity of the above equations and
their meaning with the "oppositeness" of partial derivatives of the
Euclidean distance-function/metric, whose square root is EXACTLY the
wrong choice in terms of the fact that Dx(x^(1/2)) = (1/2)x^(-1/2),
etc. Recall from earlier threads that PI is a one-sided partial
inverse of Euclidean type distance-functions, with PI embodying
proximity or "nearness" rather than distance-function/metric "farness".
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 29.1: Planes and Plane Waves |
29 Oct 2006 01:40:45 AM |
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From Osher Doctorow
Physicists and mathematicians were arguably too fast to give up the
notion of force as a fundamental concept outside particle physics and
some parts of condensed matter physics, and this has a curious
relationship to the Riccati Differential equation result that
linear/planar and exponential solutions preserve Causation.
According to Sir Isaac Newton, in the absence of a force a particle
will (in my terminology) "continue what it was doing before", which by
integrating F = ma for constant m yields of course linear motion or
else rest) in the x direction in modern Cartesian terminology.
When Relativity generalized Newtonian physics in a certain direction,
the linearity seemed contrived in view of the arbitrariness of
coordinate systems (although it was not so contrived in regard to
tangent planes), but had the Newtonians persisted with their viewpoint
as at least a Nonconformist alternative into the modern era, they would
have noticed that via the Riccati Differential equation the linearity
goes deeper than the Euclidean coordinate system and in fact becomes a
more general "exponential and/or linear" rule:
1) y = exp(kt) and y = at + b preserve Birkhoff (lst (partial)
derivative) Causation
This fails for the Euclidean metric/distance-function and for power
laws y = t^k where k is in (0, 1) and for a large variety of other
functions.
Osher Doctorow
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