Quantum geometrically, spacetime becomes quantized or "sliced up" but
that could be what nature really does, in the "fractal" sense.
According to relativity, an objects position and momentum can only be
defined with respect to a frame of reference, i.e. another object. Yet
the universe as a whole has no frame of reference outside of itself,
so how can its momentum be defined? It can only be defined with
reference to itself. Worldlines fill up spacetime and the criss
crossing of world lines mark events beyond the need for coordinate
systems or coordinates. Points in spacetime are given the name
"events" so there is a coordinate independence.

The geometric view of physics means that the laws of physics are the
same in every Lorentz reference system. Local Lorentz invariance. But
since the universe has no exterior reference frame, and it must refer
to itself, its world line intersects with itself. This
quantized-evolution of spacetime dictated by GR and QM, means that the
world line of the past intersects with the world lines of the present,
for the universe.

The total space-time and energy of the universe is represented by an
equation. the equation is self contained. A closed loop? Then there
can be no "outside" source.

G - T = 0

The Casimir effect is symmetric.

Conservation of energy is symmetric.

Yet Lorentz invariance and conservation of energy, might be violated
below the Planck scales. Perfect symmetry does not quite exist.

Stephen Hawking states that the Weyl tensor can and will be small but
can never reach zero, because this would be a direct violation of the
Heisenberg uncertainty principle.

Two different solutions to Einstein's field equations, represent the
two types of curvature, Ricci and Weyl. The Kerr vacuum solution,
models space-time outside a spherical rotating body such as a star,
and it has a zero Ricci curvature, with a nonzero Weyl curvature at
each event in the space-time. The general solution of Einstein`s
equations with the energy-momentum tensor of an ideal dust in a
Friedman universe, which models the universe on a very large scale,
has a zero Weyl curvature but a nonzero Ricci curvature at each event
in the space-time.

The Weyl tensor becomes analogous to the electromagnetic field tensor
F_ab, which can be seen as an antisymmetric 4x4 matrix with 6
independent components at each event in the space-time. Basically, the
Einstein field equation can represented as a set of equations
analogous to Maxwell's field equations, being a set of non-linear
gravitational field equations.

The negentropic effects of consciousness/self awareness are described
via a set of nonlinear partial differential equations.

The basic feature of cognitive functions appears to be the simulation
of possible future scenarios via a self modeling process. Memory
corresponds to a simulation of past to present to future plans, with
predictions to the simulation of ever further future scenarios. The
thermodynamic arrow of time shows how time gives a reliable simulation
of the past via repeatable experiments/memories, such, that it is so
reliable that one speaks of actual memories rather than predictions of
the past. The possible nonuniqueness of the classical spacetime and
related classical nondeterminism suggests a possible origin for the
simulatory aspects of consciousness built in to the geometric
structure of space-time.

The compactified dimensions could hold a tremendous amout of energy?
not unlike compressed springs: giving a potential energy, that can be
converted to kinetic energy.

A higher-dimensional universe was compactified into that of the
4-dimensional universe we now observe. The Kaluza compactified extra
dimensions appear as the diagonal term of 4-dimensional
energy-momentum tensor. Therefore, the average density of the
4-dimensional universe is affected accordingly.


The large extra dimensions that are felt only by gravity can reveal
themselves through the emission of gravitational Kaluza-Klein states.
The emission is a way of describing graviton "evaporation". Moreover,
because of the relatively large size of the extra dimension, the mass
difference between one Kaluza-Klein state and the next is very small.
There is therefore a huge number of such Kaluza-Klein excitations
below the new fundamental scale of gravity.

A typical process might involve a proton and antiproton colliding to
produce a single spray or jet of particles plus a graviton, which is
emitted into the compactified dimensional landscape.

Since the energy of the graviton would be lost from the macroscopic
4-D world, the experimental signature for such a process, would be an
excess of collisions with one jet and a "missing" energy, above and
beyond the expectations of the Standard Model.

Due to the stronger gravitational interactions at short distances,
there is a slight possibility that microscopic black holes can be
produced. Such small black holes would probably quickly evaporate and
not be dangerous. They would resemble exotic particles that decayed
very quickly. Hence gravity is also virtual.

What if it is possible to "feed" such a microscopic black hole with
photons? - electrons?, thus extending its lifetime?

Now imagine using two of these micro black holes to form a symmetric
hyperdimensional "warp-shell" over a space-craft, thus enabling its
Alcubierre warp drive.

In the 4-dimensional point of view, a graviton propagating in the
extra compactified dimensions is equivalent to a tower of an infinite
number of Kaluza-Klein mass states via a 4-dimensional Friedmann
cosmology that evolves and compactifies from a (4+D)-dimensional
Kaluza-Klein universe.

After compactification of the 6 extra dimensions, the 4-dimensional
density parameter appearing in the dimensional compactification is
negative. Thus there is an inflationary transition, with the
matter-dilaton coupling quickly becoming exponentially small. Then the
universe undergoes a polar-type superluminal expansion, and the
density parameter of the post-inflationary asymptotically flat
universe becomes a positive quantity. Omega = 1


The symmetric forms of free energy of a pre-bang universe has the
inevitable breaking of their symmetry, during the first "Creation
Event" - the "Big Bang".

The description of any entity inside the real universe can only be
with reference to other things in the universe. Space is then
relational, and the universe, self referential. For example, if an
object/event has a momentum, that momentum can only be explained with
respect to another object/event within the universe. Space then
becomes an aspect of the relationships between things in reality.

If the universe is a causally closed system, the "information" or
entangled quantum states cannot leak out of the closed system. So the
"event" density of entangled quantum states, continually increases, as
the entropy must always increase. While to us, it is interpreted as
entropy or lost information, it is actually recombined information, to
the universe.


The present moment is created and recreated constantly - analogous to
continually opposing/juxtaposing reflective mirror images… originating
deep in quantum phase space. The Heisenberg uncertainty relation
provides both a resolution boundary and the invariant relational
fabric for a translation between quantum[Planck scale] space and
experiential reality. It is the quantum T-dual compactification that
provides the Heisenberg resolution boudaries for
experiential[perceptual] reality. Unstable or chaotic states at a
given level are always "compactified" (stabilized and bounded by
eigenstates) into 6 higher dimensions condensing to the next level of
"event density".

Since relativity explains that there is no preferred frame of
reference, the ether becomes superfluous; consequently, the metric of
space-time must be defined by related events, such, that there is no
space-time if there are no events. Time is thus a sequence of events,
with each "event", having its own measure of location, and its own
measure of time, with reference to other events. Space becomes an
event density-probability distribution.


Theoretical physicist Richard Feynman derived the "sum over histories"
interpretation of quantum mechanics, where a system does not have a
single history, but it has every possible history, and each history
has its own probability amplitude. A probability distribution of
histories. For example, an electron travels from point A to point B by
every possible route at once. Each possible route or "path"
corresponds to a history.

The amplitude for each history defines the probability of that
particular path being followed. The number involves the "action"
associated with the history-path, which seems to determine that the
path taken, will be the history closest to the "classical" trajectory,
in accordance with the natural law: conservation of energy.

Stephen Hawking explains that when we apply the Feynman sum over
histories to particles moving in a background of spacetime, we must
also include histories[waveforms] in which the particle travels
backwards in time. This generates the space-time/event-density fractal
resonance.


Gravity becomes a refractive/compression effect, as light cones are
rotated near a massive object:

Topological metric 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. Waves are ripples in a basic medium. Einstein
explains that the ether is unecessary as a medium, so the ripples are
vibrations of space itself which are actually the overlapping of event
densities/conic sections. As the ripples overlap/intersect with each
other, it becomes a domino effect with the ripples continually
increasing in density. Very similar to taking a penny and doubling it
as a sequence.

2, 4, 8, 16, 32, 64, 128, 256, ... 2^n

Since the ripples are increasing in density they are defined by
density gradients. A compression force corresponding to the Shannon
entropy of the system.

Resonating standing waves/waveforms. The past collapses/condenses/
compactifies in 6 dimensions to the present moment, while the future
is an expanding uncertainty, in four space-time dimensions.

T-duality, if true, has interesting consequences. There has been a
long conceptual struggle, by theorists, to understand reality at the
extremely small scales near the Planck length at 10^35 meters. The
supposition has always been that the laws of nature break down at the
extreme micro scales. T-duality basically suggests that that at the
Planck scales, the universe looks just the same as it does at large
scales. One may even imagine that if the universe were to shrink to
less than the Planck length, it would simultaneously transform as an
expanding space-time of macroscopic dimensions.


When four of the 10 dimensions compactify, or "curl up" and the
five-brane wraps around them, the latter ends up as a one-dimensional
object described as a solotonic string in six-dimensional space-time.
In addition, a fundamental string in 10 dimensions remains fundamental
even in six dimensions. So the concept of duality between strings and
five-branes gives another interesting conjecture, which is a duality
between a solitonic string and a fundamental string.


When the six-dimensional space-time is reduced to four dimensions, via
the compactification of two dimensions: the fundamental string and the
solitonic string each inherit a T-duality. Consequently, the T-duality
of the solitonic string is just the S-duality of the fundamental
string, and vice versa; an S-duality transformation maps states with
coupling constant g in one theory to states with coupling constant 1/g
in the dual theory. It exchanges the electric and magnetic fields, and
the electrically charged particles with magnetic monopoles.


Where the interchange of charges in one picture is just the inversion
of length in the dual picture, is named the Duality of Dualities by
string theory. It is puts the previously shaky S-duality on as firm a
footing as the well-established T-duality. In addition, it predicts
that the strength with which objects interact, i.e. their charges,
corresponds to the size of the invisible dimensions. What is charge in
one universe of radius 1/R may be size in its dual universe of radius
R.

Reverberating standing waves double with every overlap and the total
space-time is finite.

The Universal Geometric Set:

2^0 = 1

2^1 == [.......]

2^2 == [[.....]]

2^3 == [[[[.]]]]

2^N == [[[[[[[[...2^N...]]]]]]]]


If the waves become "compressed", they become smaller and smaller
with each overlap/intersection.

The reverse temporal - transactional feedback gives a fractal
universe!

GR postulates a continuous space-time that is background independent
and QM is a theory of discrete particles interacting within a fixed
background space.

What if the geometry is actually fractal in nature?, a discrete, yet
causally connected space-time that is background independent.
Gravitons would then be an aspect of the fractal geometry of
space-time? GR is a nonlinear theory. Fractals are nonlinear.


A network of fractals would be continuous in the sense of being
connected and relational. Similar to the way floor tiles appear to be
discrete sections with definable boundaries, yet still a continuously
connected network, relating to the others.

Waves are a probability distribution of particles, each with its own
probability of position and momentum.

Gravitons could be a connected network of relativistic quantum
operators.

Since mathematical existence is defined by David Hilbert as "freedom
from contradiction" It holds that, if, mathematical existence is equal
to physical existence, then physical existence is also freedom from
contradiction. That is to say, physical phenomena[events] are
constrained by an intrinsic, logical self-consistency.


[1.] Mathematics is a meta language.

[2.] Language is descriptive.

[3.] Language must be free of contradiction. Mathematics is also
defined as a descriptive system that has "freedom from contradiction".

[4.] Mathematics describes physical existence/processes/events.

[5.] Observation is a physical process.

[6.] Mathematics describes observations.

[7.]A description of an observation must be free of
contradiction-following from [3.]

[8.] Observation must be free of contradiction.

[8.] A description is an abstract representation of a physical system.
The description must be as exact as possible.

[9.] An exact description implies equivalence between abstract
structures and physical systems.

[10.] If the exact description exists, then physical existence is a
meta-language. A self descriptive entity, free of contradiction. The
universe is equivalent to its[exact] description.



The universally distributive laws of nature are explained in terms of
symmetry. Also, Cantor's "alephs" could be explained as cardinal
identities, akin to "qualia" from which, the elements(subsets) of our
universe can be derived.

I suppose we could say that the completed infinities of Georg Cantor
are distributive in nature, similar to the way that a set of "red"
objects has the distributive property of redness. Predicates like
"red" are numbers in the sense that they interact algebraically
according to the laws of Boolean algebra. Take one object away from
the set of red objects and the distributive number "red" still
describes the set. The distributive identity "natural number" or "real
number" describes an entire collection of individual elements.

The Cantorian alephs can be set into a one to one correspondence with
a proper subset of of themselves. So we see that these infinite
Cantorian alephs are really distributive.

Yet, if we have a finite set of X objects, the cardinal number X does
not really distribute over its individual subsets. Take anything away
from the set and the number X ceases to describe it.

Symmetry is analogous to a self evident truth and is distributive via
the laws of nature, and is distributed over the entire set called
universe. Symmetry appears to be a stratification of Cantorian alephs
with varying degrees of freedom. More freedom is greater symmetry with
the higher infinity-alephs. So the highest aleph, the
"absolute-infinity" distributes over the entire set called universe
and gives it "identity".


On a flat planar surface, the three angles of an equilateral triangle
sum to 180 degrees. On the curved surface of a sphere, the three
angles add up to more than 180 degrees. The two types of surfaces are
not equivalent.

There is a rotational invariance for the triangle, that seems to hold
for both types of surface though.

ABC = BCA = CAB

Does this rotational invariance hold for all geometries?


Aristotle's law of excluded middle is really an invariance principle.

A V ~A

[T|F] = [F|T] = T

This is also a rotational invariance.


The laws of physics become the laws of geometry. Certain invariants
hold, which are analogous to the "absolutes".

Space becomes a tranformation. Two objects with relative velocity
will have a relative measure that transforms into the other. In
effect, the separation does not exist in an extrinsic sense.
Equilateral triangle rotation: ABC = BCA = CAB... Then it is realized
that an absolute spatial separation cannot exist, therefore, the EPR
paradox cannot actually exist. Distance interval, which is a property
of space, is a type of dynamic relation. So, relativity is really a
theory of invariants. Space is a set of invariance principles which,
has a boundary that is zero. Yet, with the self inclusive manifold,
information[structure-complexity] is increasing as a function of time.

Information is also a type of relation, in that certain invariants
must hold.. So to describe tautologies of logic e.g. X or ~X , as
absolute truths would not be a complete definition. A tautology is an
invariance principle. A rule that transforms according to a choice of
truth value, which is an invariant, in that it is always true. Yes,
the force called gravity can be elucidated as a geometric effect, a
"non-Euclidean geometry", where spacetime becomes anisotropic and
inhomogeneous in the presence of mass-energy. Then the question
becomes "what is space?" ..."What is time?" Space is relational. Time
is the manifold changes OF ...space.

Heisenberg Uncertainty: DxDp = hbar/2

The relation becomes totally unpredictable below the Planck length. So
yes, space could be described as a self similar relation which is
generated by the quantum uncertainty and forms analogous Penrosian
"spin networks".

The curvature of spacetime could be represented as a Gaussian
distribution? If mathematics only is an approximation of reality, then
the mathematics of probability corresponds "exactly" with reality. The
Riemann tensor explains how a tangent vector, parallel translated
around a tiny parallellogram is changed. So, to say that spacetime is
"curved" means how much a tangent vector changes during parallel
transport around a loop. The translation of an infinitesimal tangent
vector along a geodesic. So the probability distribution should agree
exactly with Einstein's relativity.

The gravitational field, described by the metric of spacetime g_uv ,
is generated by the stress-energy tensor T_uv of matter. Various field
equations relating g_uv to T_uv have been proposed. The most
succsessful have been the Einstein field equations, which are of
course, the foundation of general relativity.

G_uv == R_uv - 1/2 g_uv R = 8pi T_uv

where R_uv and R are the Ricci tensor and scalar curvature derived
from the metric g_uv , and G_uv is the Einstein tensor.

The equations are non-linear, since the left hand side is not a linear
function of the metric.


Mathematically speaking, the existence of a "dual" vector space,
abstractly reflects the relationship between row vectors (1×n) and
column vectors (n×1). The construction can also take place for
infinite-dimensional spaces and gives rise to important ways of
looking at different distributions and Hilbert space. The use of the
dual space can be a characteristic of functional analysis. It is also
built into the Fourier transform.


Because the tangent space and the cotangent space at a given point are
both real vector spaces of the same dimension, they are isomorphic to
each other. But they are not "naturally isomorphic", since, for an
arbitrary tangent covector, there is no canonical tangent vector
associated with it. With the introduction of a symplectic form, the
additional structure gives a "natural isomorphism". Consequently, the
distinction between the tangent space and the cotangent space must be
maintained. Many definitions are more natural on one space than on the
other. All the cotangent spaces of a manifold can be "glued together"
to form a new differentiable manifold of twice the dimension, the
cotangent bundle of the manifold.


Geometric quantization using tensor fields on fiber bundles generally
have holonomy[parallel translation on a piecewise smooth path] and
noncommutative properties, from which, a geometric quantization
algebra can be constructed, equivalent to a geometric quantization
approach to Q.M.. The metaplectic[a double covering group of the
(linear) symplectic group.] structure, used to define geometric
quantization, is given on a bundle. General relativity can be put into
the Ashtekar variables with the Sen connection, interpreted in terms
of the metapletic covering.

By quantizing spacetime geometry, it seems that the
wavefunctions/waveforms aren't based on a background space. The
wavefunction space, can be thought of as the space of square-
integrable wavefunctions over classical configuration space.

In ordinary quantum mechanics, configuration space is space itself
{i.e.,to describe the configuration of a particle, location in space
is specified}. In general relativity, there is a more general kind of
configuration space: taken to be the space of 3-metrics {"superspace",
not to be confused with supersymmetric space} in the geometrodynamics
formulation,{or the space of connections of an appropriate gauge
group)in the Ashtekar/loop formulation. So the wavefunctions will be
functions over these abstract spaces, not space itself-- the
wavefunction/algorithm defines "space itself".



The gravity tensor should be able to rotate into the electromagnetic
tensor and the electromagnetic tensor should be able to rotate into
the gravity tensor.

Time
^
|
|
|
|--------------space



G
^
|
|
|
|--------------EM


Compression waves become a self embedding of surface integrals? This
gives continuously increasing density gradients, as matter-energy is
sequentially re-embedded from previous computations/iterations.

If the universe is closed, the "information" or entangled quantum
states cannot leak out of the closed system. So the density of
entangled quantum states, continually increases, as the entropy must
always increase. While to us, it is interpreted as entropy or lost
information, it is actually recombined information, to the universe.

Shannon entropy.


What is needed is a tensor equation which is parallel to "wave"
equations described in terms of a covariant d'Alembertian operator...
An alternative description for the general relativistic space-time
continuum that allows for covariant "compressional" waves, rather than
allowing only "transverse" waves.

TIME
^
|
|
|----------------- SPACE


COVARIANCE
^
|
|
|---------------- CONTRAVARIANCE


COMPRESSION SCALAR WAVES
^
|
|
|--------------- TRANSVERSE VECTORIAL WAVES


Quantum entities are described as probability distributions, which are
attributes of an underlying phase space, where the
properties-attributes such as "spin" and "charge" are not the
attributes of individual particles, but they are universally
distributive entities, being the attributes of a "coherent wave
function". It is this wave-distribution property that then "decoheres"
into the ostensible "wave function collapse", as waves become
localized particles that are "in phase" creating
standing-spherical-wave resonances, which are condensations of space
itself. The continual collapse-condensation of space into
matter-energy is the continual "change", i.e. the property called
"time". The spherical waves, or probability distributions are
represented by the Schrodinger wave function, "psi".

The continual intersection and collapse of probability distributions,
also known as quantum phase entanglement, is a continual increasing of
the "total" combined information of the universal wavefunction itself.
Information density. With more information, more complex structures
can be created.

Quantum mechanics leads us to the realization that all matter-energy
can be explained in terms of "waves" or probability distributions
containing information. In a confined region(i.e. a closed universe or
a black hole) the waves exist as STANDING WAVES In a closed system,
the entropy never decreases.

The analogy with black holes is an interesting one but if there is
nothing outside the universe, then it cannot be radiating energy
outside itself as black holes are explained to be. So the amount of
information i.e. "quantum states" in the universe is increasing. We
see it as entropy, but to an information processor with huge
computational capabilities, it is compressible information.

The information density of the universe is increasing. The increase of
information density is analogous to a pressure gradient.

[density 1]---[density 2]---[density 3]--- ... ---[density n]

"Russell E. Rierson" wrote:

Quantum geometrically, spacetime becomes quantized or "sliced up" but
that could be what nature really does, in the "fractal" sense.
According to relativity, an objects position and momentum can only be
defined with respect to a frame of reference, i.e. another object. Yet
the universe as a whole has no frame of reference outside of itself,
so how can its momentum be defined?

Learn the difference between linear and angular momentum.
Acceleration is an absolute measurement (Machian arguments have
no empirical support). There is something about spin
(mathematical meaning, not a whirling lump per se) that is
profoundly fundamental.

It can only be defined with
reference to itself.

Bull****. Chirality is the classical example of self-referential
property. Chir1ality as a propertry exists in comparison with
its non-superposable mirror-image. Chirality only requires a
causal and orientable spacetime manifold. Chirality then can be
defined in terms of coordinate-free Hodge duality. This is
equivalent to chirality defined in terms of a pseudoscalar field,
e.g., the Levi-Civita tensor. Pseudoscalar particles such as the
pion exist; the weak force violates parity conservation by
exhibiting left-handed chirality only. No coordinates or
reference frames are required to establish this - only the
properties of the vector triple product that is a
coordinate-invariant concept.

Worldlines fill up spacetime and the criss
crossing of world lines mark events beyond the need for coordinate
systems or coordinates. Points in spacetime are given the name
"events" so there is a coordinate independence.

The geometric view of physics means that the laws of physics are the
same in every Lorentz reference system. Local Lorentz invariance. But
since the universe has no exterior reference frame, and it must refer
to itself, its world line intersects with itself. This
quantized-evolution of spacetime dictated by GR and QM, means that the
world line of the past intersects with the world lines of the present,
for the universe.

The total space-time and energy of the universe is represented by an
equation. the equation is self contained. A closed loop? Then there
can be no "outside" source.

G - T = 0

The Casimir effect is symmetric.

Conservation of energy is symmetric.

Yet Lorentz invariance and conservation of energy, might be violated
below the Planck scales. Perfect symmetry does not quite exist.

[snip 530 lines of unsupported silliness]

http://www.hep.upenn.edu/~max/toe.html

So? So what?

--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz.pdf

"Russell E. Rierson" wrote in message
om...
Quantum geometrically, spacetime becomes quantized or "sliced up" but
that could be what nature really does, in the "fractal" sense.
According to relativity, an objects position and momentum can only be
defined with respect to a frame of reference, i.e. another object. Yet
the universe as a whole has no frame of reference outside of itself,
so how can its momentum be defined? It can only be defined with
reference to itself. Worldlines fill up spacetime and the criss
crossing of world lines mark events beyond the need for coordinate
systems or coordinates. Points in spacetime are given the name
"events" so there is a coordinate independence.

The geometric view of physics means that the laws of physics are the
same in every Lorentz reference system. Local Lorentz invariance. But
since the universe has no exterior reference frame, and it must refer
to itself, its world line intersects with itself. This
quantized-evolution of spacetime dictated by GR and QM, means that the
world line of the past intersects with the world lines of the present,
for the universe.

The total space-time and energy of the universe is represented by an
equation. the equation is self contained. A closed loop? Then there
can be no "outside" source.

G - T = 0

The Casimir effect is symmetric.

Conservation of energy is symmetric.

Yet Lorentz invariance and conservation of energy, might be violated
below the Planck scales. Perfect symmetry does not quite exist.

Stephen Hawking states that the Weyl tensor can and will be small but
can never reach zero, because this would be a direct violation of the
Heisenberg uncertainty principle.

Two different solutions to Einstein's field equations, represent the
two types of curvature, Ricci and Weyl. The Kerr vacuum solution,
models space-time outside a spherical rotating body such as a star,
and it has a zero Ricci curvature, with a nonzero Weyl curvature at
each event in the space-time. The general solution of Einstein`s
equations with the energy-momentum tensor of an ideal dust in a
Friedman universe, which models the universe on a very large scale,
has a zero Weyl curvature but a nonzero Ricci curvature at each event
in the space-time.

The Weyl tensor becomes analogous to the electromagnetic field tensor
F_ab, which can be seen as an antisymmetric 4x4 matrix with 6
independent components at each event in the space-time. Basically, the
Einstein field equation can represented as a set of equations
analogous to Maxwell's field equations, being a set of non-linear
gravitational field equations.

The negentropic effects of consciousness/self awareness are described
via a set of nonlinear partial differential equations.

The basic feature of cognitive functions appears to be the simulation
of possible future scenarios via a self modeling process. Memory
corresponds to a simulation of past to present to future plans, with
predictions to the simulation of ever further future scenarios. The
thermodynamic arrow of time shows how time gives a reliable simulation
of the past via repeatable experiments/memories, such, that it is so
reliable that one speaks of actual memories rather than predictions of
the past. The possible nonuniqueness of the classical spacetime and
related classical nondeterminism suggests a possible origin for the
simulatory aspects of consciousness built in to the geometric
structure of space-time.

The compactified dimensions could hold a tremendous amout of energy?
not unlike compressed springs: giving a potential energy, that can be
converted to kinetic energy.

A higher-dimensional universe was compactified into that of the
4-dimensional universe we now observe. The Kaluza compactified extra
dimensions appear as the diagonal term of 4-dimensional
energy-momentum tensor. Therefore, the average density of the
4-dimensional universe is affected accordingly.


The large extra dimensions that are felt only by gravity can reveal
themselves through the emission of gravitational Kaluza-Klein states.
The emission is a way of describing graviton "evaporation". Moreover,
because of the relatively large size of the extra dimension, the mass
difference between one Kaluza-Klein state and the next is very small.
There is therefore a huge number of such Kaluza-Klein excitations
below the new fundamental scale of gravity.

A typical process might involve a proton and antiproton colliding to
produce a single spray or jet of particles plus a graviton, which is
emitted into the compactified dimensional landscape.

Since the energy of the graviton would be lost from the macroscopic
4-D world, the experimental signature for such a process, would be an
excess of collisions with one jet and a "missing" energy, above and
beyond the expectations of the Standard Model.

Due to the stronger gravitational interactions at short distances,
there is a slight possibility that microscopic black holes can be
produced. Such small black holes would probably quickly evaporate and
not be dangerous. They would resemble exotic particles that decayed
very quickly. Hence gravity is also virtual.

What if it is possible to "feed" such a microscopic black hole with
photons? - electrons?, thus extending its lifetime?

Now imagine using two of these micro black holes to form a symmetric
hyperdimensional "warp-shell" over a space-craft, thus enabling its
Alcubierre warp drive.

In the 4-dimensional point of view, a graviton propagating in the
extra compactified dimensions is equivalent to a tower of an infinite
number of Kaluza-Klein mass states via a 4-dimensional Friedmann
cosmology that evolves and compactifies from a (4+D)-dimensional
Kaluza-Klein universe.

After compactification of the 6 extra dimensions, the 4-dimensional
density parameter appearing in the dimensional compactification is
negative. Thus there is an inflationary transition, with the
matter-dilaton coupling quickly becoming exponentially small. Then the
universe undergoes a polar-type superluminal expansion, and the
density parameter of the post-inflationary asymptotically flat
universe becomes a positive quantity. Omega = 1


The symmetric forms of free energy of a pre-bang universe has the
inevitable breaking of their symmetry, during the first "Creation
Event" - the "Big Bang".

The description of any entity inside the real universe can only be
with reference to other things in the universe. Space is then
relational, and the universe, self referential. For example, if an
object/event has a momentum, that momentum can only be explained with
respect to another object/event within the universe. Space then
becomes an aspect of the relationships between things in reality.

If the universe is a causally closed system, the "information" or
entangled quantum states cannot leak out of the closed system. So the
"event" density of entangled quantum states, continually increases, as
the entropy must always increase. While to us, it is interpreted as
entropy or lost information, it is actually recombined information, to
the universe.


The present moment is created and recreated constantly - analogous to
continually opposing/juxtaposing reflective mirror images… originating
deep in quantum phase space. The Heisenberg uncertainty relation
provides both a resolution boundary and the invariant relational
fabric for a translation between quantum[Planck scale] space and
experiential reality. It is the quantum T-dual compactification that
provides the Heisenberg resolution boudaries for
experiential[perceptual] reality. Unstable or chaotic states at a
given level are always "compactified" (stabilized and bounded by
eigenstates) into 6 higher dimensions condensing to the next level of
"event density".

Since relativity explains that there is no preferred frame of
reference, the ether becomes superfluous; consequently, the metric of
space-time must be defined by related events, such, that there is no
space-time if there are no events. Time is thus a sequence of events,
with each "event", having its own measure of location, and its own
measure of time, with reference to other events. Space becomes an
event density-probability distribution.


Theoretical physicist Richard Feynman derived the "sum over histories"
interpretation of quantum mechanics, where a system does not have a
single history, but it has every possible history, and each history
has its own probability amplitude. A probability distribution of
histories. For example, an electron travels from point A to point B by
every possible route at once. Each possible route or "path"
corresponds to a history.

The amplitude for each history defines the probability of that
particular path being followed. The number involves the "action"
associated with the history-path, which seems to determine that the
path taken, will be the history closest to the "classical" trajectory,
in accordance with the natural law: conservation of energy.

Stephen Hawking explains that when we apply the Feynman sum over
histories to particles moving in a background of spacetime, we must
also include histories[waveforms] in which the particle travels
backwards in time. This generates the space-time/event-density fractal
resonance.


Gravity becomes a refractive/compression effect, as light cones are
rotated near a massive object:

Topological metric 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. Waves are ripples in a basic medium. Einstein
explains that the ether is unecessary as a medium, so the ripples are
vibrations of space itself which are actually the overlapping of event
densities/conic sections. As the ripples overlap/intersect with each
other, it becomes a domino effect with the ripples continually
increasing in density. Very similar to taking a penny and doubling it
as a sequence.

2, 4, 8, 16, 32, 64, 128, 256, ... 2^n

Since the ripples are increasing in density they are defined by
density gradients. A compression force corresponding to the Shannon
entropy of the system.

Resonating standing waves/waveforms. The past collapses/condenses/
compactifies in 6 dimensions to the present moment, while the future
is an expanding uncertainty, in four space-time dimensions.

T-duality, if true, has interesting consequences. There has been a
long conceptual struggle, by theorists, to understand reality at the
extremely small scales near the Planck length at 10^35 meters. The
supposition has always been that the laws of nature break down at the
extreme micro scales. T-duality basically suggests that that at the
Planck scales, the universe looks just the same as it does at large
scales. One may even imagine that if the universe were to shrink to
less than the Planck length, it would simultaneously transform as an
expanding space-time of macroscopic dimensions.


When four of the 10 dimensions compactify, or "curl up" and the
five-brane wraps around them, the latter ends up as a one-dimensional
object described as a solotonic string in six-dimensional space-time.
In addition, a fundamental string in 10 dimensions remains fundamental
even in six dimensions. So the concept of duality between strings and
five-branes gives another interesting conjecture, which is a duality
between a solitonic string and a fundamental string.


When the six-dimensional space-time is reduced to four dimensions, via
the compactification of two dimensions: the fundamental string and the
solitonic string each inherit a T-duality. Consequently, the T-duality
of the solitonic string is just the S-duality of the fundamental
string, and vice versa; an S-duality transformation maps states with
coupling constant g in one theory to states with coupling constant 1/g
in the dual theory. It exchanges the electric and magnetic fields, and
the electrically charged particles with magnetic monopoles.


Where the interchange of charges in one picture is just the inversion
of length in the dual picture, is named the Duality of Dualities by
string theory. It is puts the previously shaky S-duality on as firm a
footing as the well-established T-duality. In addition, it predicts
that the strength with which objects interact, i.e. their charges,
corresponds to the size of the invisible dimensions. What is charge in
one universe of radius 1/R may be size in its dual universe of radius
R.

Reverberating standing waves double with every overlap and the total
space-time is finite.

The Universal Geometric Set:

2^0 = 1

2^1 == [.......]

2^2 == [[.....]]

2^3 == [[[[.]]]]

2^N == [[[[[[[[...2^N...]]]]]]]]


If the waves become "compressed", they become smaller and smaller
with each overlap/intersection.

The reverse temporal - transactional feedback gives a fractal
universe!

GR postulates a continuous space-time that is background independent
and QM is a theory of discrete particles interacting within a fixed
background space.

What if the geometry is actually fractal in nature?, a discrete, yet
causally connected space-time that is background independent.
Gravitons would then be an aspect of the fractal geometry of
space-time? GR is a nonlinear theory. Fractals are nonlinear.


A network of fractals would be continuous in the sense of being
connected and relational. Similar to the way floor tiles appear to be
discrete sections with definable boundaries, yet still a continuously
connected network, relating to the others.

Waves are a probability distribution of particles, each with its own
probability of position and momentum.

Gravitons could be a connected network of relativistic quantum
operators.

Since mathematical existence is defined by David Hilbert as "freedom
from contradiction" It holds that, if, mathematical existence is equal
to physical existence, then physical existence is also freedom from
contradiction. That is to say, physical phenomena[events] are
constrained by an intrinsic, logical self-consistency.


[1.] Mathematics is a meta language.

[2.] Language is descriptive.

[3.] Language must be free of contradiction. Mathematics is also
defined as a descriptive system that has "freedom from contradiction".

[4.] Mathematics describes physical existence/processes/events.

[5.] Observation is a physical process.

[6.] Mathematics describes observations.

[7.]A description of an observation must be free of
contradiction-following from [3.]

[8.] Observation must be free of contradiction.

[8.] A description is an abstract representation of a physical system.
The description must be as exact as possible.

[9.] An exact description implies equivalence between abstract
structures and physical systems.

[10.] If the exact description exists, then physical existence is a
meta-language. A self descriptive entity, free of contradiction. The
universe is equivalent to its[exact] description.



The universally distributive laws of nature are explained in terms of
symmetry. Also, Cantor's "alephs" could be explained as cardinal
identities, akin to "qualia" from which, the elements(subsets) of our
universe can be derived.

I suppose we could say that the completed infinities of Georg Cantor
are distributive in nature, similar to the way that a set of "red"
objects has the distributive property of redness. Predicates like
"red" are numbers in the sense that they interact algebraically
according to the laws of Boolean algebra. Take one object away from
the set of red objects and the distributive number "red" still
describes the set. The distributive identity "natural number" or "real
number" describes an entire collection of individual elements.

The Cantorian alephs can be set into a one to one correspondence with
a proper subset of of themselves. So we see that these infinite
Cantorian alephs are really distributive.

Yet, if we have a finite set of X objects, the cardinal number X does
not really distribute over its individual subsets. Take anything away
from the set and the number X ceases to describe it.

Symmetry is analogous to a self evident truth and is distributive via
the laws of nature, and is distributed over the entire set called
universe. Symmetry appears to be a stratification of Cantorian alephs
with varying degrees of freedom. More freedom is greater symmetry with
the higher infinity-alephs. So the highest aleph, the
"absolute-infinity" distributes over the entire set called universe
and gives it "identity".


On a flat planar surface, the three angles of an equilateral triangle
sum to 180 degrees. On the curved surface of a sphere, the three
angles add up to more than 180 degrees. The two types of surfaces are
not equivalent.

There is a rotational invariance for the triangle, that seems to hold
for both types of surface though.

ABC = BCA = CAB

Does this rotational invariance hold for all geometries?


Aristotle's law of excluded middle is really an invariance principle.

A V ~A

[T|F] = [F|T] = T

This is also a rotational invariance.


The laws of physics become the laws of geometry. Certain invariants
hold, which are analogous to the "absolutes".

Space becomes a tranformation. Two objects with relative velocity
will have a relative measure that transforms into the other. In
effect, the separation does not exist in an extrinsic sense.
Equilateral triangle rotation: ABC = BCA = CAB... Then it is realized
that an absolute spatial separation cannot exist, therefore, the EPR
paradox cannot actually exist. Distance interval, which is a property
of space, is a type of dynamic relation. So, relativity is really a
theory of invariants. Space is a set of invariance principles which,
has a boundary that is zero. Yet, with the self inclusive manifold,
information[structure-complexity] is increasing as a function of time.

Information is also a type of relation, in that certain invariants
must hold.. So to describe tautologies of logic e.g. X or ~X , as
absolute truths would not be a complete definition. A tautology is an
invariance principle. A rule that transforms according to a choice of
truth value, which is an invariant, in that it is always true. Yes,
the force called gravity can be elucidated as a geometric effect, a
"non-Euclidean geometry", where spacetime becomes anisotropic and
inhomogeneous in the presence of mass-energy. Then the question
becomes "what is space?" ..."What is time?" Space is relational. Time
is the manifold changes OF ...space.

Heisenberg Uncertainty: DxDp = hbar/2

The relation becomes totally unpredictable below the Planck length. So
yes, space could be described as a self similar relation which is
generated by the quantum uncertainty and forms analogous Penrosian
"spin networks".

The curvature of spacetime could be represented as a Gaussian
distribution? If mathematics only is an approximation of reality, then
the mathematics of probability corresponds "exactly" with reality. The
Riemann tensor explains how a tangent vector, parallel translated
around a tiny parallellogram is changed. So, to say that spacetime is
"curved" means how much a tangent vector changes during parallel
transport around a loop. The translation of an infinitesimal tangent
vector along a geodesic. So the probability distribution should agree
exactly with Einstein's relativity.

The gravitational field, described by the metric of spacetime g_uv ,
is generated by the stress-energy tensor T_uv of matter. Various field
equations relating g_uv to T_uv have been proposed. The most
succsessful have been the Einstein field equations, which are of
course, the foundation of general relativity.

G_uv == R_uv - 1/2 g_uv R = 8pi T_uv

where R_uv and R are the Ricci tensor and scalar curvature derived
from the metric g_uv , and G_uv is the Einstein tensor.

The equations are non-linear, since the left hand side is not a linear
function of the metric.


Mathematically speaking, the existence of a "dual" vector space,
abstractly reflects the relationship between row vectors (1×n) and
column vectors (n×1). The construction can also take place for
infinite-dimensional spaces and gives rise to important ways of
looking at different distributions and Hilbert space. The use of the
dual space can be a characteristic of functional analysis. It is also
built into the Fourier transform.


Because the tangent space and the cotangent space at a given point are
both real vector spaces of the same dimension, they are isomorphic to
each other. But they are not "naturally isomorphic", since, for an
arbitrary tangent covector, there is no canonical tangent vector
associated with it. With the introduction of a symplectic form, the
additional structure gives a "natural isomorphism". Consequently, the
distinction between the tangent space and the cotangent space must be
maintained. Many definitions are more natural on one space than on the
other. All the cotangent spaces of a manifold can be "glued together"
to form a new differentiable manifold of twice the dimension, the
cotangent bundle of the manifold.


Geometric quantization using tensor fields on fiber bundles generally
have holonomy[parallel translation on a piecewise smooth path] and
noncommutative properties, from which, a geometric quantization
algebra can be constructed, equivalent to a geometric quantization
approach to Q.M.. The metaplectic[a double covering group of the
(linear) symplectic group.] structure, used to define geometric
quantization, is given on a bundle. General relativity can be put into
the Ashtekar variables with the Sen connection, interpreted in terms
of the metapletic covering.

By quantizing spacetime geometry, it seems that the
wavefunctions/waveforms aren't based on a background space. The
wavefunction space, can be thought of as the space of square-
integrable wavefunctions over classical configuration space.

In ordinary quantum mechanics, configuration space is space itself
{i.e.,to describe the configuration of a particle, location in space
is specified}. In general relativity, there is a more general kind of
configuration space: taken to be the space of 3-metrics {"superspace",
not to be confused with supersymmetric space} in the geometrodynamics
formulation,{or the space of connections of an appropriate gauge
group)in the Ashtekar/loop formulation. So the wavefunctions will be
functions over these abstract spaces, not space itself-- the
wavefunction/algorithm defines "space itself".



The gravity tensor should be able to rotate into the electromagnetic
tensor and the electromagnetic tensor should be able to rotate into
the gravity tensor.

Time
^
|
|
|
|--------------space



G
^
|
|
|
|--------------EM


Compression waves become a self embedding of surface integrals? This
gives continuously increasing density gradients, as matter-energy is
sequentially re-embedded from previous computations/iterations.

If the universe is closed, the "information" or entangled quantum
states cannot leak out of the closed system. So the density of
entangled quantum states, continually increases, as the entropy must
always increase. While to us, it is interpreted as entropy or lost
information, it is actually recombined information, to the universe.

Shannon entropy.


What is needed is a tensor equation which is parallel to "wave"
equations described in terms of a covariant d'Alembertian operator...
An alternative description for the general relativistic space-time
continuum that allows for covariant "compressional" waves, rather than
allowing only "transverse" waves.

TIME
^
|
|
|----------------- SPACE


COVARIANCE
^
|
|
|---------------- CONTRAVARIANCE


COMPRESSION SCALAR WAVES
^
|
|
|--------------- TRANSVERSE VECTORIAL WAVES


Quantum entities are described as probability distributions, which are
attributes of an underlying phase space, where the
properties-attributes such as "spin" and "charge" are not the
attributes of individual particles, but they are universally
distributive entities, being the attributes of a "coherent wave
function". It is this wave-distribution property that then "decoheres"
into the ostensible "wave function collapse", as waves become
localized particles that are "in phase" creating
standing-spherical-wave resonances, which are condensations of space
itself. The continual collapse-condensation of space into
matter-energy is the continual "change", i.e. the property called
"time". The spherical waves, or probability distributions are
represented by the Schrodinger wave function, "psi".

The continual intersection and collapse of probability distributions,
also known as quantum phase entanglement, is a continual increasing of
the "total" combined information of the universal wavefunction itself.
Information density. With more information, more complex structures
can be created.

Quantum mechanics leads us to the realization that all matter-energy
can be explained in terms of "waves" or probability distributions
containing information. In a confined region(i.e. a closed universe or
a black hole) the waves exist as STANDING WAVES In a closed system,
the entropy never decreases.

The analogy with black holes is an interesting one but if there is
nothing outside the universe, then it cannot be radiating energy
outside itself as black holes are explained to be. So the amount of
information i.e. "quantum states" in the universe is increasing. We
see it as entropy, but to an information processor with huge
computational capabilities, it is compressible information.

The information density of the universe is increasing. The increase of
information density is analogous to a pressure gradient.

[density 1]---[density 2]---[density 3]--- ... ---[density n]


Does that come with fries??

RJ Pease


---
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Russell E. Rierson wrote:
Quantum geometrically, spacetime becomes quantized or "sliced up" but
that could be what nature really does, in the "fractal" sense.
According to relativity, an objects position and momentum can only be
defined with respect to a frame of reference, i.e. another object. Yet
the universe as a whole has no frame of reference outside of itself,
so how can its momentum be defined? It can only be defined with
reference to itself. Worldlines fill up spacetime and the criss
crossing of world lines mark events beyond the need for coordinate
systems or coordinates. Points in spacetime are given the name
"events" so there is a coordinate independence.


[major snippage]


The analogy with black holes is an interesting one but if there is
nothing outside the universe, then it cannot be radiating energy
outside itself as black holes are explained to be. So the amount of
information i.e. "quantum states" in the universe is increasing. We
see it as entropy, but to an information processor with huge
computational capabilities, it is compressible information.

The information density of the universe is increasing. The increase of
information density is analogous to a pressure gradient.

[density 1]---[density 2]---[density 3]--- ... ---[density n]


And therefore what?

I kept looking for a punch line but never found it.

sci.physics sure has some strange denizens.

Uncle Al wrote in message

Learn the difference between linear and angular momentum.
Acceleration is an absolute measurement (Machian arguments have
no empirical support). There is something about spin
(mathematical meaning, not a whirling lump per se) that is
profoundly fundamental.



According to Mach's principle, a single particle in a bleak void of a
universe, has ZERO inertial mass. An infinitesimally small force
accelerates it - energy is zero because inertial mass is zero, so that
potential energy, mgh and kinetic energy, 1/2 mv2 are each zero.

Einstein emphasized that a single particle in a void could not have
inertia, for there can be no inertia of matter against space, only
inertia against other matter.


Matter-energy is a form of "event density" where the distribution of
matter-energy determines the geometry of space-time. If there is no
matter energy, that is to say "events", then there is no geometry.
Therefore a body in an otherwise empty universe would have no
intertial properties. Even though Einstein's theory of general
relativity can describe universes that contain no matter, possibly
leading one to believe that GR is not relational - because there can
be space without matter, and there are no material relations that
serve to define space, it is incorrect to make such an assumption. GR
is a theory of fields, and fields vary continuously over space,
forming networks of interactive relations[events].


Einstein believed Mach's Principle should be incorporated into the
theory of relativity.

Mach's Principle requires that inertia of mass and consequently, the
potential energy of inertial mass must be generated by the rest of the
Universe.

G = (gravitational constant)

M = (mass of the observable Universe)

R = (Hubble radius)

the relation:

GM /R= c2,

where c is the velocity of light.





Chirality is the classical example of self-referential
property. Chir1ality as a propertry exists in comparison with
its non-superposable mirror-image. Chirality only requires a
causal and orientable spacetime manifold. Chirality then can be
defined in terms of coordinate-free Hodge duality. This is
equivalent to chirality defined in terms of a pseudoscalar field,
e.g., the Levi-Civita tensor. Pseudoscalar particles such as the
pion exist; the weak force violates parity conservation by
exhibiting left-handed chirality only. No coordinates or
reference frames are required to establish this - only the
properties of the vector triple product that is a
coordinate-invariant concept.


According to the CPT theorem, via the lifting of the degeneracy of
mirror-image chiral molecules by the parity-violating weak neutral
current interaction, the strict enantiomer of a chiral molecule having
an energy "identical" with the original, is that with the opposite and
invariant configuration but composed of antiparticles, that is to say,
the CP enantiomer and extended analysis, proves that this strict
degeneracy holds even if CP is violated. CP violation in particle
physics, is analogous to chemical catalysis, since the CP-violating
force changes the rates of particle-antiparticle processes without
affecting the equilibrium thermodynamics.


Einstein's equations are conventionally expressed in terms of
variables with a "chirality", such, that a circle drawn in the
clockwise direction would look different from one drawn in the
opposite direction.

Einstein's equations consequently break into simpler pieces with the
Ashtekar variables. From the Ashtekar vantage, space-time is
quantized.

The metric of spacetime can be defined by the criss crossing of
worldlines. Coordinate independence.

hmmm I am not a rocket scientist but to me you speak of things that are not
practical and impossible to verify.

you saying "According to Mach's principle, a single particle in a bleak void
of a
universe" is enough for me to go *YAWN* as it is not a possible scenario so
why even discuss it?

--

hawktooie
Loogie out
"Russell E. Rierson" wrote in message
m...
Uncle Al wrote in message

Learn the difference between linear and angular momentum.
Acceleration is an absolute measurement (Machian arguments have
no empirical support). There is something about spin
(mathematical meaning, not a whirling lump per se) that is
profoundly fundamental.



According to Mach's principle, a single particle in a bleak void of a
universe, has ZERO inertial mass. An infinitesimally small force
accelerates it - energy is zero because inertial mass is zero, so that
potential energy, mgh and kinetic energy, 1/2 mv2 are each zero.

Einstein emphasized that a single particle in a void could not have
inertia, for there can be no inertia of matter against space, only
inertia against other matter.


Matter-energy is a form of "event density" where the distribution of
matter-energy determines the geometry of space-time. If there is no
matter energy, that is to say "events", then there is no geometry.
Therefore a body in an otherwise empty universe would have no
intertial properties. Even though Einstein's theory of general
relativity can describe universes that contain no matter, possibly
leading one to believe that GR is not relational - because there can
be space without matter, and there are no material relations that
serve to define space, it is incorrect to make such an assumption. GR
is a theory of fields, and fields vary continuously over space,
forming networks of interactive relations[events].


Einstein believed Mach's Principle should be incorporated into the
theory of relativity.

Mach's Principle requires that inertia of mass and consequently, the
potential energy of inertial mass must be generated by the rest of the
Universe.

G = (gravitational constant)

M = (mass of the observable Universe)

R = (Hubble radius)

the relation:

GM /R= c2,

where c is the velocity of light.





Chirality is the classical example of self-referential
property. Chir1ality as a propertry exists in comparison with
its non-superposable mirror-image. Chirality only requires a
causal and orientable spacetime manifold. Chirality then can be
defined in terms of coordinate-free Hodge duality. This is
equivalent to chirality defined in terms of a pseudoscalar field,
e.g., the Levi-Civita tensor. Pseudoscalar particles such as the
pion exist; the weak force violates parity conservation by
exhibiting left-handed chirality only. No coordinates or
reference frames are required to establish this - only the
properties of the vector triple product that is a
coordinate-invariant concept.


According to the CPT theorem, via the lifting of the degeneracy of
mirror-image chiral molecules by the parity-violating weak neutral
current interaction, the strict enantiomer of a chiral molecule having
an energy "identical" with the original, is that with the opposite and
invariant configuration but composed of antiparticles, that is to say,
the CP enantiomer and extended analysis, proves that this strict
degeneracy holds even if CP is violated. CP violation in particle
physics, is analogous to chemical catalysis, since the CP-violating
force changes the rates of particle-antiparticle processes without
affecting the equilibrium thermodynamics.


Einstein's equations are conventionally expressed in terms of
variables with a "chirality", such, that a circle drawn in the
clockwise direction would look different from one drawn in the
opposite direction.

Einstein's equations consequently break into simpler pieces with the
Ashtekar variables. From the Ashtekar vantage, space-time is
quantized.

The metric of spacetime can be defined by the criss crossing of
worldlines. Coordinate independence.

A metric space (M, d) is a set M equipped with a function

d : M × M --R satisfying the following axioms for all x, y, and z in
M:

d(x, y) = 0

d(x, x) = 0

if d(x, y) = 0 then x = y (identity of indiscernibles)


d(x, y) = d(y, x) (symmetry)

d(x, z) = d(x, y) + d(y, z) (triangle inequality)


A metric field can be defined by the primary substratum of events.
Thus the intrinsic geometrical structure of spacetime is predicated on
the pseudo-Riemannian spaces via the affine relationships — all
physical events are fully reducible to manifestations of the
substratum i. e. the metric field.


A covering map is a continuous surjective map p : C -- X, with C and
X being topological spaces, which has the following property: to every
x in X there exists an open neighborhood U such that p^ -1(U) is a
union of mutually disjoint open sets S_i (where i ranges over some
index set I) such that p restricted to S_i yields a homeomorphism from
S_i to U for every i in I.


Basically, a collection of subsets of a given space is a cover (or
covering) of that space if the union of the collection is the whole
space. Also, a space is "countably compact" if every countable open
cover has a finite subcover.

A cover K is a refinement of a cover L if every member of K is a
subset of some member of L.


Let's assume we are working in a complete metric space.

If so, we are in a topological space which is a set of points with a
definition of a neighborhood of a point, also with a notion of
distance ala a "metric", and that in the said space, Cauchy sequences
have limits, where a Cauchy sequence is defined as one such that
d(x_n,x_m) epsilon can be satisfied for any epsilon by specifying an
N such that n,m N. So we are probably talking about compact spaces,
or compact manifolds such that a compact complete metric space is one
in which all sequences have subsequences that converge.

Stochastically speaking, gravity is must be taken beyond the limits of
classical reality, where the mean value of the stress energy tensor of
quantum fields also has fluctuations as a source of stochastic
Einsteinian vacuum equations. Such is the necessary foundation for
neo-classical gravity for the viability of inflationary cosmology
based on the vacuum energy dominated phase. Metric fluctuations and
spacetime foams form a chaotic substrate.

Yes, the shortest distance between two points is a straight line;
energy is conserved.


Form a space-time triangle with vertices A,B,C, of three lines - two
null and one spacelike line, such that the triangle "spans" the
timelike plane.

Because of the one-to-oneness of the mapping, the image of the
triangle with the vertices A,B,C , "spans" the transformed plane as
different points are mapped onto different points. Therefore, the
three lines forming the transformed triangle must be coplanar. In
general, the images of all lines lying in a time-light plane must be
coplanar. Thus, timelike planes map into planes.

Any time light-cone line, is the intersection of two time-light
planes. Since timelike planes are mapped onto planes, they intersect
into a line. Thus, any timelike line is mapped into a line.

Basically, all three types of lines - lightlike(null), spacelike, and
timelike, are mapped onto lines.


The uncertainty principle and gravity are related to the same
mathematical properties. The proof of the uncertainty relation
involves the Cauchy Schwartz inequality. The triangle inequality
follows from the Cauchy Schwartz.

--------------------------------------------------------------------------------