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Science Forum Index » Physics - Research Forum » Octonions, Electrodynamics and Gravity?
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| Rick |
 Posted: Tue Feb 26, 2008 12:01 pm |
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There are two fundamentally different forms octonion algebra can take.
They could be called Left Octonion Algebra and Right Octonion Algebra.
They are not isomorphic algebras.
However, this does not matter, since all meaningful physical results
derived from an octonion representation are algebraic invariants,
meaning the results will not care which of the 16 possible ways to
roll out the algebra is used.
Octonion algebra provides a suitable rank-1 hypercomplex basis for the
classical Electric and Magnetic field types based on their intrinsic
multiplication properties. When the algebra is used to describe the
Electrodynamic fields, there are additional rotational and
irrotational fields indicated. Perhaps the irrotational field that
rides along with the magnetic field in three of the algebraic units is
the gravitational field.
Using Electrodynamics as a roadmap, the form for an 8-current density
can be produced, and a suitable Action Function can be found in the
invariant portion of the Octonion product of field and 8-current. All
expected forms from Electrodynamics for work and force are found
within this Action Function.
The "Law of Algebraic Invariance" can be used to write the Action
Function in an integrable form. This identity can be thought of as the
divergence of the Octonion Stress-Energy-Momentum "tensor". Quotes
because the form is not your grandfather's tensor, it belongs to the
algebra of Octonions. Integration of this identity over the spatial 7-
volume produces the Octonion Conservation of Energy and Momentum.
The 8-potentials can be restricted to only expected functionals from
Electrodynamics. When this is done, the Octonion Stress-Energy-
Momentum "tensor" is identical to that of Electrodynamics, and its
divergence duplicates the work-force from Electrodynamics.
The derivation relies only on the full definition of the algebra, and
the belief that reality does not care which representation of the
algebra is used. Absolutely nothing else is inserted.
For more details, see www.octospace.com
Rick |
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| Rick |
| Posted: Thu Mar 06, 2008 1:25 pm |
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Hello.....echo.....echo....echo
Is anyone out there?....echo...echo....echo
If I have not yet gotten your attention, here are a few more tidbits.
The seventh Octonion unit might seem to some to be a second nose on
the face. It does not fit well
to a dual xyz physical spatial representation. As I have proposed, the
Octonion permutation (123) is
one-to-one with xyz, and the units 5, 6 and 7 are one-to-one with xyz,
yet (567) is not a permutation.
It takes three permutations to cover units 5, 6 and 7. By the way,
magnetic induction would have it
no other way. Likely no "mass related induction", hence a (123) cover
of gravitation.
Ignoring the second nose unit 4, the non-4 permutations are
(123) (761)
(572)
(653)
I have purposely drawn them out in this fashion to point out what will
always be the case when the
Octonion algebra is changed up by negation of particular permutations.
If 1, 2 and 3 are "right
handed", then 5, 6 and 7 are in "left handed" permutations, and 1, 2
and 3 "left" demands 5, 6 and 7
"right".
Now about that second nose. If we allow the unit 4 stuff to form an
analytic pair in the complex
system, the Cauchy-Riemann equations will give us
D0(A0) = D4(A4)
D0(A4) = - D4(A0)
Also, we have the harmonic relationships
D0^2(A0) + D4^2(A0) = 0
D0^2(A4) + D4^2(A4) = 0
The seventh irrotational field component is thus always 0. It does not
exist.
The Analogous Lorentz Condition partitions into the sum of 2 4D
looking terms
+D0(A0) + D1(A1) + D2(A2) + D3(A3)
+D0(A0) + D5(A5) + D6(A6) + D7(A7)
The scalar on the 8-current partitions into the sum of 2 4D looking
terms
-D0^2(A0) + D1^2(A0) + D2^2(A0) + D3^2(A0)
-D0^2(A0) + D5^2(A0) + D6^2(A0) + D7^2(A0)
This would be the sum of mass density and charge density. Very
symmetric looking here.
If we were to surround the two densities with a physical xyz volume,
the surface normals
for {123} and {567} would point in opposite physical directions due to
their opposite chirality.
For like signed source densities, their fields point in opposite
directions. The resultant central
forces are simply the scalar product of density with field. So what do
we have? Like charges
repel and like mass attracts. Sounds familiar.
I believe Octonion Algebra is a suitable mathematical basis for a
Unified Field Theory. I realize that
acceptance of this fact may be slow in coming due to the fact that you
just can't do this stuff on
the chalkboard or with pencil and paper. It takes a computer to do all
the tedious and repetitive work.
I have developed my own symbolic algebra software over the past
decade, which I have used to
develop and prove the ideas I have presented. I have passed the link
to my web site to the SAGE
people, hopefully they can integrate enough of my ideas to give you
all something to get comfortable
with. Maybe someone else could point the MAPLE, etc. people in this
direction:
www.octospace.com
Rick Lockyer |
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| Uncle Al |
| Posted: Fri Mar 07, 2008 7:11 am |
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Rick wrote:
Quote:
Hello.....echo.....echo....echo
Is anyone out there?....echo...echo....echo
[snip]
Quote: I believe Octonion Algebra is a suitable mathematical basis for a
Unified Field Theory.
[snip]
Explanation is not nearly enough. A new theory validated by
prediction and observation is still not enough - though convenience
vs. prior efforts helps. All the fun rests in a three part process,
1) DISJOINT prediction - you must disagree with existing theory.
2) Observation.
3) Somebody is falsified.
Predicting light falls with twice the Newtonian value and accurately
modeling Mercury's perihelion shift are good validations. What do you
have that metric gravitation and quantum field theory do not - and is
testable?
Given a plate of ham and eggs, the chicken made a contribution but the
pig made a committment.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2 |
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| Rock Brentwood |
| Posted: Thu Apr 03, 2008 7:06 am |
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On Mar 7, 12:11 pm, Uncle Al <Uncle...@hate.spam.net> wrote:
Quote: Predicting light falls with twice the Newtonian value and accurately
modeling Mercury's perihelion shift are good validations. What do you
have that metric gravitation and quantum field theory do not - and is
testable?
As I promised a couple years back, I'd look into all matters related
to torsion, metric affine gravity, teleparallel gravity, parity-
violation, etc. in greater depth and straighten everything out that
relate to all these things.
Well, here's the clarification. The complete article is found under
http://federation.g3z.com/Physics/index.htm#Teleparallel (PDF, 86k;
click on "PDF")
For sake of brevity, the mathematical part of the review is dealt with
more fully there.
On the general matter:
You're got issues mixed up and confused; and are drawing all the
boundaries wrong and making the wrong distinctions, calling things
mutually exclusive that are actually orthogonal and mixing up levels
of description, confusing geometry with dynamics. Until recently, I
did too. The conclusion is the last sentence of linked article.
First, parity violation has nothing per se to do with teleparallel
gravity and, as a feature, is completely orthogonal with respect to
all the dynamics considered here. They all have or easily allow for a
parity violating sector, Neither its presence or absence is an
essential ingredient in any of the cases considered here. The most
notable example is Ashtekhar, which has the parity violating term with
the Immirzi coefficient. Not as well-known in Ashtekhar is that adding
a certain quadratic term for torsion can also restore parity even with
a non-zero Immirzi coefficient. So, question of the parity violating
sector is completely orthogonal with respect to the rest of the
formalism.
Two, "metric-affine" has nothing per se to do with "teleparallel" or
"Einstein-Cartan". Rather, it is a general framework in which these
and other geometries reside. The geometries, in turn, are general
frameworks in which different dynamics reside.
Three, the specific model(s) you're trying to advance (Hayashi, et.
al.) actually have nothing per se to do with parity violation; and (in
fact) not much at all to do with the specific issues you're trying to
advance. The Hayashi family of models may admit a parity-violating
sector (and if it doesn't, it's easy to add one in), but it is not an
essential ingredient of the dynamics and is certainly not the
distinguishing feature.
Four, there is a distinction between a geometry and the action
principle or dynamics on it. Any of the dynamics on any of the
geometries can be written equivalently in any of the others; e.g.
Hayashi's 3 parameter model is easily expressible in Riemannian
geometry or Einstein-Cartan geometry. There are a zillion GR-
equivalent action principles; as well as numerous other (like
Ashtekhar) which are equivalent for GR-vacuua. They can all be written
in any of the other geometries. The most that's generally required is
adding external fields in the more restrictive geometries to offset
what the less restrictive geometries have built into them (e.g.
torsion or contorsion). The only real issue that ever arises in this
context is the question of what "energy positivity" in one geometry
means in another (e.g. Einstein-Cartan energy positivity is not
equivalent to energy positivity in Riemannian geometry; "Einstein
Cartan Theory", arXiv:gr-qc/0606062 v1 discusses this in the context
of cosmological solutions).
Most important of all: teleparallel geometry is not more general than
Riemannian geometry. Their respective structures are equivalent. The
geometries and structures are:
Metric-Affine metric + connection + frame = frame + torsion +
metric covariant derivative
Einstein-Cartan metric + connection = frame + connection = frame +
structure coefficients + torsion
Riemannian metric = frame + structure coefficients
Weitzenböck frame + torsion = frame + structure coefficients
Five, none of the dynamics are distinguished specifically as "The
Theory of General Relativity", per se. Historically, the first
dynamics was Einstein-Hilbert, but that had long ago been superseded
by the Palantini action, which is formulated in Einstein-Cartan
geometries. So, even the designation of "GR = Einstein-Hilbert" is no
longer applicable. And even Palantini was essentially superseded by
Ashtekhar, the development which was just the final act in Dirac's
programme to write down the canonical Hamiltonian formulation for the
constrained systems that each of the dynamics defines. Palantini
became Asktekhar as the last step in implementing Dirac's recipe.
There have been plenty of other models that generalize the action
(Carmelli's models of GR as a Yang-Mills-inspired actions; Penrose's
similar developments). They all fall under the umbrella of GR and are
each orthogonal to all the issues you're raising. In stellar dynamics,
for instance, it's common to use a spinor-based or Penrose formalism.
And, six: even for the Hayashi family, there are much better ways to
limit the parameters than what terrestrial experiments can provide.
The spherically symmetric solution for the 3-parameter family can be
worked out ("Metric Affine Approach to Teleparallel Gravity", gr-qc/
0212080). Only the parameter combination matching the Einstein-Hilbert
or Palantini action (the "GR-equivalent") yields black holes. There
are plenty of objects out there that look and behave a lot like black
holes. Notwithstanding their actual nature, their mere presence puts
severe restrictions on the parameter space much greater than any
terrestrial experiment can hope to match. One only need match with the
behaviors of the celestial objects.
The remainder of the article is the more detailed review
1. Metric Affine Geometry
2. Einstein-Cartan Geometry
2.1. Einstein-Cartan Geometries with Generalized Connections
3. Riemannian Geometry
4. Weitzenboeck (or Teleparallel) Geometry |
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| Uncle Al |
| Posted: Fri Apr 04, 2008 9:44 am |
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Rock Brentwood wrote:
Quote:
On Mar 7, 12:11 pm, Uncle Al <Uncle...@hate.spam.net> wrote:
Predicting light falls with twice the Newtonian value and accurately
modeling Mercury's perihelion shift are good validations. What do you
have that metric gravitation and quantum field theory do not - and is
testable?
As I promised a couple years back, I'd look into all matters related
to torsion, metric affine gravity, teleparallel gravity, parity-
violation, etc. in greater depth and straighten everything out that
relate to all these things.
Well, here's the clarification. The complete article is found under
http://federation.g3z.com/Physics/index.htm#Teleparallel (PDF, 86k;
click on "PDF")
For sake of brevity, the mathematical part of the review is dealt with
more fully there.
On the general matter:
You're got issues mixed up and confused; and are drawing all the
boundaries wrong and making the wrong distinctions, calling things
mutually exclusive that are actually orthogonal and mixing up levels
of description, confusing geometry with dynamics. Until recently, I
did too. The conclusion is the last sentence of linked article.
First, parity violation has nothing per se to do with teleparallel
gravity and, as a feature, is completely orthogonal with respect to
all the dynamics considered here. They all have or easily allow for a
parity violating sector, Neither its presence or absence is an
essential ingredient in any of the cases considered here. The most
notable example is Ashtekhar, which has the parity violating term with
the Immirzi coefficient. Not as well-known in Ashtekhar is that adding
a certain quadratic term for torsion can also restore parity even with
a non-zero Immirzi coefficient. So, question of the parity violating
sector is completely orthogonal with respect to the rest of the
formalism.
[snip]
Do left and right shoes locally vacuum free fall identically? You may
erect any theoretical framework you like. The question can only be
answered by experiment. "It doesn't matter how beautiful your theory
is, it doesn't matter how smart you are. If it doesn't agree with
experiment, it's wrong," Richard Feynman.
1) The vacuum is isotropic (e.g., metric gravitation): chemically
identical opposite parity mass distributions fit into it with
identical energies. They locally vacuum free fall along identical
minimum action geodesic trajectories. Opposite parity mass
distributions are gravitationally indistinguishable from themselves
and from any mass distributions.
2) The vacuum is chiral (e.g., teleparallel gravitation):
chemically identical opposite parity mass distributions fit into it
with different energies (left and right shoes on a left foot). They
locally vacuum free fall along divergent minimum action teleparallels
(diastereotopic trajectories. Try walking a straight line with your
eyes closed wearing two left shoes). Opposite parity mass
distributions are gravitationally distinguishable from each other.
Only one parity mass distribution vacuum free falls as do achiral mass
distributions (left shoe or sock on a left foot).
The vacuum is isotropic to massless EM to 16 significant figures. No
vacuum dichroism, birefringence, gyrotropy or any other interaction is
observed over multi-billion lightyear paths. The massed sector can
have different rules without contradiction.
3) Composition and spin Eotvos experiments null to 3x10^(-14)
difference/average,
<http://www.npl.washington.edu/eotwash/publications/pdf/schlamminger08.pdf>
Phys. Rev. Lett. 100, 041101 (2008)
<http://www.npl.washington.edu/eotwash/publications/pdf/prl97-021603.pdf>
Phys. Rev. Lett. 97, 021603 (2006)
Does a parity Eotvos experiment null when opposing single crystal test
masses of space group P3(1)21 (right-handed screw axes) and P3(2)21
(left-handed screw axes) alpha-quartz? The question can only be
answered by experiment. On New Years Day 1957 what everybody knew to
be true was not observed - and for the same reason,
http://physics.nist.gov/GenInt/Parity/cover.html
4) Chemically identical opposite parity mass distributions insert
into a chiral vacuum background with different energies (left and
right shoes tried on a left foot). If they melt into identical
achiral (not racemic) fluid the initial energy difference can be
measured by differential scanning calorimetry,
http://www.mazepath.com/uncleal/shoes.png
Benzil is achiral molten, gas phase, and in solution. It crystallizes
in space group P3(1)21 or P3(2)21,
http://www.mazepath.com/uncleal/benzil.png
stereogram
Benzil melts at 95 C, below the boiling point of water. A 10^(-13)
Eotvos experiment mass/mass divergence is an 8% enthalpy of fusion
energy/mass divergence. A new DSC is 0.1% precise. This parity
calorimetry experiment is 80X as sensitive as any Eotvos experiment
for that alone. Are the /_\H(fusion) of benzil single crystals in
space group P3(1)21 and P3(2)21 identical at all times of day (signal
source in a composition Eotvos experiment)? The questions can only be
answered by experiment.
Folks have quantitatively tested the Equivalence Principle with
composition and whatnot since Stevin and Galileo in the late 1500s.
Every test has nulled within experimental error. NOBODY has
challenged spacetime geometry with test mass geometry. If an EP
violation exists, that is where it resides.
Somebody should look. Theory will thereafter predict what was
observed.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2 |
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| Uncle Al |
| Posted: Fri Apr 04, 2008 9:44 am |
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Guest
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Rock Brentwood wrote:
Quote:
On Mar 7, 12:11 pm, Uncle Al <Uncle...@hate.spam.net> wrote:
Predicting light falls with twice the Newtonian value and accurately
modeling Mercury's perihelion shift are good validations. What do you
have that metric gravitation and quantum field theory do not - and is
testable?
As I promised a couple years back, I'd look into all matters related
to torsion, metric affine gravity, teleparallel gravity, parity-
violation, etc. in greater depth and straighten everything out that
relate to all these things.
Well, here's the clarification. The complete article is found under
http://federation.g3z.com/Physics/index.htm#Teleparallel (PDF, 86k;
click on "PDF")
For sake of brevity, the mathematical part of the review is dealt with
more fully there.
On the general matter:
You're got issues mixed up and confused; and are drawing all the
boundaries wrong and making the wrong distinctions, calling things
mutually exclusive that are actually orthogonal and mixing up levels
of description, confusing geometry with dynamics. Until recently, I
did too. The conclusion is the last sentence of linked article.
[snip]
Is there any error within Euclidean geometry? No.
Can Euclid accurately navigate the Earth's surface? No.
Elliptic, plane, plus hyperbolic geometry - complete set? No.
There are eight primary geometries of 3-space,
E^3, S^3, H^3, S^2 x R, H^2 x R, SL_2, Nil, and Sol
WP Thurston, "Three-dimensional geometry and topology," Vol. 1.
Princeton Mathematical Press, Princeton, NJ, 1997
GP Scott, "The geometries of 3-manifolds," Bull. Lond. Math. Soc.
15(5) 401-487 (1983)
WP Thurston, "Three-dimensional manifolds, Kleinian groups and
hyperbolic geometry," Bull. Amer. Math. Soc. 6 357-381 (1982)
WP Thurston, "The geometry and topology of three-manifolds,"
Princeton Math. Dept., 1979
Mathematics is not an empirical science. Mathematics only requires
self-consistency. To demand that a mathematical model of observed
reality is valid independent of experimental falsification of
predictions is not merely hubris, it is wrong.
When nothing orthodox has empirically succeeded since the late 1500s,
one is faced with a simple choice:
1) Repeat to more decimal places what has already failed, or
2) Try something different external to the failures.
(1) will get you easy grant funding - precedent!
(2) is a tough row to hoe - the risk of failure.
That is madness. Consider the risk of (2) success.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2 |
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