Re: Quantum Phase Compactification via Spacetime Expansion

From: Russell E. Rierson (analog57_at_yahoo.com)
Date: 07/24/04 

Date: 24 Jul 2004 12:42:01 -0700

Question:

Let T be a metric space with distance function r(x,y) expressing the
definitive predication that involves T with the real numbers, R.
Therefore the juxtaposition of left and right hemispheres resonates in
perfect accordance with the proposition that T and R are embedded
simultaneously in the full structure of manifold M. Ergo we pass on to
an enlargement *M of M, whereby the non-standard metric space is
diffeomorphism invariant.
So if f(x) is a homeomorphism from T onto S, then for every point p in
T, does f(u(p) = u(f(p)) ?

The metric space has distance function r(x,y), definitively
characterized by involvement with the real numbers, R, such that the
metric space and R are embedded simultaneously in the full structure
of manifold M. A topological space consists of sets of points which
are defined[in this case] to be the intersections of cotangent
bundles.

We move on to functions of a complex variable in the 2-D Euclidean
space Z, utilizing the algebraic structure of complex numbers, where
points of Z may be regarded as pairs of real numbers, R. Formal
statements concerning Z, are also expressable as statements about R.

Complex-valued functions e.g. w = f(z), can also be represented as
binary, or quaternary relations, these sets of complex valued
polynomials then have elements that map Z into Z. Every complex valued
polynomial subset p(z), has natural number coefficients a_k, or the
"kth" coefficient of p(z)..

So, for non-zero polynomials, we take into account the number of zeros
OF the polynomial O[p(z)] with O ranging over the natural numbers such
that O[p(z), R ] = v, where v is a finite natural number.
Thus if G is a metric group and E is a topological group, such, that
an open neighborhood U, of the identity is a metric space, with a
compliant metric, specifiable to the topology of U, then the distance
r(a,b) between any two points (a,b) in U, then in terms of the
distance u(E) = J, consists of the points a of U with the fact that
r(a,E) is an infinitesimal.
f(x) is a homeomorphism from T onto S, then for every point p in T,
f(u(p) = u(f(p))

If f(u(v)) = u(f(v)) the physics for a circle of radius R, is
equivalent to the physics for a circle of radius 1/R.

A point without another "reference" does not exist; the complement of
a
thing distinguishes it from the thing itself. What is the dynamic of
space-time? Is it a ratio? S/T = c ?

When space is taken as a measure of length, space/time is the speed of
light in vacuum for a photon of light:

space/time = c

Where, length = perception of separation between two reference points.

E = mc^2

E/momentum = E/p = c

energy/momentum = space/time

What is the EPR "superluminal?" connection? A shortcut through
configuration space? Phase space?

A point can be defined as an "infinitesimal". The Topological spaces
are defined as being diffeomorphism invariant. Intersecting cotangent
bundles[manifolds] are the set of all possible configurations of a
system, i.e. they describe the phase space of the system.

Potential infinity is defined as a limit via Newton's calculus, while
actual infinity is a Cantorian Cardinal number, which is a Platonic
form, which is also a type of potential.

[abstract representation]--->[semantic mapping]--->[represented
system]

[axiomatic]--->[Isomorphism]<---[Induction]

An abstract representation is exactly that, "abstract". It is not a
space, or time, but is instead a product of consciousness, or a
mental construct; topologically it is equivalent to a "point". The
abstract description contains the concrete topology. Likewise, the
concrete contains the abstract.

A duality?

A point/intersection/overlap contains an infinite expanse of space and
time?

Language describes the universe, because the universe is isomorphic to
a description on some level, and reality can only refer to itself,
because, there is nothing outside of ..."total existence" which
becomes equivalent to a self referential system, which must be a self
aware system. Since descriptions make distinctions, or references to
other entities, and distinctions are tautologically logical, [A or
~A], reality is logical, in that its contents can be described by a
language. The contents within reality are distinctive entities,
individually different from the others, yet consisting of the same
foundational substance. [<-[->[<-[U]->]<-]->]

Universe, U(T) = 0 .

On one level of stratification, two photons are separate. On another
level, of stratification, the photons have zero separation.
Instantaneous communication between two objects, separated by a
distance interval, is equivalent to zero separation[zero boundary]
between the two objects.

According to the book "Gravitation", chapter 15, geometry of spacetime
gives instructions to matter telling matter to follow the straightest
path, which is a geodesic. Matter in turn, tells spacetime geometry
how to curve in such a way, as to guarantee the conservation of
momentum and energy. The Einstein tensor[geometric
feature-description] is also conserved in this relationship between
matter and the spacetime geometry. Eli Cartan's "boundary of a
boundary equals zero."

A quote from the book "The Expanding Universe" by Sir Arthur
Eddington:
quote:
-------------------------------------------------------------------------------
All change is relative. The universe is expanding relatively to our
common standards; our common standards are shrinking relatively to the
size of the universe. The theory of the "expanding universe" might
also be called the theory of the "shrinking atom" .
--------------------------------------------------------------------------------

Gravity exists because the information[event] density of space-time is
increasing. This creates a "pressure force" where processed space,
compresses mass-energy, and mass-energy reacts by compressing space.
The process is "time", which becomes dilated due to the increased
information density of massive objects. Stephen Hawking's excellent
book, "Universe in a Nutshell", explains holography as a phenomenon of
interference of wave patterns. Light from a laser is split into two
separate beams, one bounces off the object and gets reflected onto a
photo-sensitized plate. The other beam is reflected into a lens and
collides with the reflected light of the object. When a laser is shone
through the developed plate, a fully three dimensional image of the
original object is created. According to conventional theories, the
surface area of the horizon surrounding a black hole, measures its
entropy, where entropy is defined as a measure of the number of
internal states that the black hole can be in without looking
different to an outside observer, who can only measure mass, rotation
and charge. This leads to another theory which states that the maximum
entropy of any closed region of space can never exceed one quarter of
the area of the circumscribing surface, with the entropy being the
measure of the total information contained by the system. So the
theorists came to realize that the information associated with all
phenomena in the three dimensional world, can be stored on its two
dimensional boundary, like a holographic image.

Very interesting: http://www.space.com/scienceastronomy/time_theory_030806.html

quote:
--------------------------------------------------------------------------------
"There isn't a precise instant underlying an object's motion," he
said. "And as its position is constantly changing over time -- and as
such, never determined -- it also doesn't have a determined position
at any time."
--------------------------------------------------------------------------------
Heisenberg uncertainty: DxDp >= hbar/2
As the observed expansion of the universe continues, and, as the mean
temperature of the universe continues to approach absolute zero, could
the universe transform into a condition analogous to a "Bose Einstein
condensate"?

Is it possible to develop warp drive technology? or more correctly,
warp drive ..."trek" nology?

http://arxiv.org/abs/gr-qc/0204012

http://www.wired.com/news/technology/0,1282,20152,00.html

Here are some more writings on the plausibility of interstellar travel
via warp drive:

http://www.astro.cf.ac.uk/groups/relativity/papers/abstracts/miguel94a.html

http://arxiv.org/PS_cache/gr-qc/pdf/0107/0107097.pdf

Frame Dragging:

http://msnbc.msn.com/id/3077887/

Impulse Gravity Generator Based on Charged
Y Ba2Cu3O7 Superconductor with Composite Crystal Structure:

http://superconductors.org/gravity.pdf

Back on topic:

When y(x,t) is the transverse displacement of a vibrating string,
y(x,t) can be determined by the wave equation

[@^2y/@t^2] = v^2[ @^2 y/@x^2] , where @ denotes the partial
derivative symbol.

Standing waves can be set up in an n-dimensional string, analogous to
that in a violin string. The form of the standing wave becomes y(x,t)
= psi(x) sin (wt)

Two sinusoidal travelling waves with the same amplitude and wavelength
moving in opposite directions on a string, become resonating "standing
waves":

y(x,t) = y1(x,t) + y2(x,t) = Asin(kx - wt) + Asin(kx + wt) =
[2Asin(kx)]*cos(wt).

As the entropy continues to increase in the universe, and if the
universe is a closed system, the entropy may be considered to be the
result of a "damping force". This damping force may also be one
possible solution to the dark matter enigma.

Solve the Schwarzschild solution for the entire universe. Since the
universe can be postulated to be a closed system with "nothing"
outside the universe. The condition of "nothingness" leads one to ask
"What are the properties of nothingness?" Of course there are no
measurable properties, but nothingness in itself must be a type of
massless solid. A condition that has no distance - metric scales. In
other words, there is no outside to the universe, no measurable border
between something and nothing.

Nothing then becomes analogous to a perfectly symmetrical pressure
force on the surface of existence.

---->|U|<----

U stands for universe. So it becomes reasonable to assume that the
entire universe is analogous to that which is inside the event horizon
of a black hole. The universe becomes a quantum superposition of
states, collapsing under the crushing force of "nothingness".

Analytically continue the Schwarzschild solution to the imaginary
values of the time variable. The Schwarzschild solution becomes
periodic in the imaginary time direction.

All waves would then be standing waves in the closed universe. A
Schrodinger wave equation in one dimension is of the form:

d^2 psi/dx^2 + 2m/hbar^2 [E - U(x)] psi(x)

U(x) is the potential energy and E is the total energy.

psi(x) is the wave function for a state in which the energy E is
constant in time. Such states are called stationary states. Certain
definite vibration frequencies are allowed multiples of fundamental
wavelengths

lambda = h/p

|psi(x)|^2 dx, is the probability of finding a particle(universe) in a
certain state between the region x and x+dx

psi^2 = psi psi* . When psi is complex, psi* is the complex conjugate
of psi. psi^2 (x) is the probability density.

So the goal must be to derive a nonlinear Schrodinger equation that
gives local the invariances that we currently observe! A global
nonlinear, locally invariant system ...?

http://lsec.cc.ac.cn/~ttang/MMref/references/budd99.pdf

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