Re: Parity Eotvos Experiment update

Posted by Uncle Al on Jun 14, 2003 at 11:04
(68.5.243.16)

Re: Parity Eotvos Experiment update (Dogrock)

All gravitational theories are geometries. That is the whole of it. Mass (or pure geometry - the Weyl tensor) defines the shape of spacetime. Spacetime tells mass how to move. That's gravitation.

A self-consistent axiomatic geometry contains no mistakes. Don't bother looking for any. Euclid claimed plane geometry derived from five axioms but made additional assumptions in his proofs. David Hilbert in "Grundlagen der Geometrie" (1899) gave a complete description of Euclidean geometry using 21 axioms (five being exclusive to 3-D). An axiomatic system is only as strong as its axioms.

Any internally consistent mathematical model can be trivially falsified by observation (economics, psychology). This is obviously *not* the case for General Relativity.

Tests of GR
GPS System

An internally consistent axiomatic mathematical model can be falsified by counterdemonstration of a postulate. Euclid's Fifth Postulate (e.g., Playfaire's Axiom: only one line can be drawn parallel to a given line through a point not on that line) does not accommodate hyperbolic and elliptic geometries (and five other simply-connected geometric 3-manifolds with compact quotients). Euclid is not generally applicable (surveying, navigation; the Shroud of Turin is a trivial forgery).

General Relativity postulates the Equivalence Principle (in three stages of increasing generality): Basically, all local test masses fall identically in vacuum regradless of composition and internal structure; inertial and gravitational masses are fundamentally indistinguishable. Any two test masses that measureably reproducibly violate the EP irreparably falsify GR.

I. THE PROPERTY

Is parity the mathematical symmetry coupled to parity the physical property? One normally uses Noether's theorem for proof. Parity is unique in that the mechanism of Noether's theorem does not obtain and cannot be approximated. This previously posted section demonstrates that the coupling is rigorously valid.

II. THE THEORY

A body of math, when all signs of its variables are reversed (parity inversion), behaves in one of two ways: It gives the same answer (even parity, scalar, axial vector, tensor, or gerade). It gives the answer with its sign reversed (odd parity, pseudoscalar, polar vector, pseudotensor, or ungerade). Newton and all metric theories of gravitation (General Relativity) are gerade. A large body of gravitation theory including affine and teleparallel treaments can be ungerade.

Contrasted parity pair test mass experiments that empirically violate the EP falsify all gerade theories including General Relativity. Nobody has ever looked.

III. THE QUANTITATIVE MEASURE

CHI expresses geometric parity of a crystal lattice on a continuous scale of zero (achiral, unhanded, parity inversions exactly superpose) to one (perfect parity divergence). Parity is chirality along all three coordinate axes.

CHI is peer-reviewed, published, and accepted as rigorously valid mathematics. It requires no semi-empirical constraints. Its only input is list of lattice coordinates, or a function or algorithm for generating them.

IV. THE DATA

There are exactly and only 230 possible ways of constructing a self-similar 3-D crystal lattice, the space groups. 65 space groups are chiral. Of those, 22 space groups are 11 parity pairs. Of those, 6 space groups or 3 parity pairs mathematically qualify to potentially generate CHI=1 (perfect parity divergence).

The alpha-quartz crystal structure in space groups P3121 and P3221 qualifies. This includes alpha-quartz (silicon dioxide), alpha-germanium dioxide, grey selenium, tellurium, langasite, berlinite AlPO4, AlAsO4, GePO4, etc.

QCM is the rigorous calculation of CHI. It begins with a graph theory analysis of the lattice and outputs symmetry correlations, COR. It then rigorously calculates CHI. A few thousand tellurium atoms require 2 CPU-weeks in a mainframe. Theory allows a lattice with QCM COR=1 for increasing radii to be calculated without a bunch of footnotes. A personal computer can crunch a billion tellurium atoms/second through FastCHI.

CHI is asymptotic to a limiting value with increasing lattice radius. Since crystal lattices are quantized vs. radius (atom or empty space), there is a large standard deviation during calculation and a curve must be fit. Log(1-CHI) vs log(radius) is a straight line with theory-demanded slope -2 and an arbitrary intercept. FastCHI speed is inversely proportional to crystallographic unit cell population (3 in tellurium, 9 in quartz, 18 in berlinite). Getting a tight CHI fit can require weeks of PC CPU time and quadrillions of atoms.

To everybody's astonishment, a prodigy mathematican noted CHI is the connection between eigenvalues, special functions, and their representation theory with solid angles and exponentials of fractions of pi. Quintillions of floating point operations become a few punches into a hand calculator

log(1-CHI)= -2log(radius) + [(180-alpha)(pi)/60] - pi
"alpha" is smallest vertex angle in the helix, in degrees. If you are clever you can see a conversion from degrees to radians hidden in there.

Given a crystal structure and standard software to display angles, the intercept can be calculated within ~5% of its fit FastCHI value in under 20 seconds. We suspect the complete model is more accurate than the number crunching.

The process may be much deeper than that. Tellurium's crystal structure has four significant figures (the last one being weak for technical reasons). It gives a fit within 6% extreme vs. average. Quartz is determined to six sig figs. Its fit is within 3%. Do we have a deep insight into how matter must fit together - without any quantum mechanics at all?

V. THE TEST MASSES

More than 1000 tonnes/year of cultured single crystal quartz are commercially grown. About 0.1% of it is left-handed for optical applications, etc. Single crystals can be as large as 20 kg. Crystal quality at all length scales is determined by the ratio of optical absorbance at two near-infrared frequencies, EIA Standard 477-1.

VI. THE CONCLUSION

Fabricate 3-cm diameter and long single crystal alpha-quartz test masses and mount them in an Eotvos torsion balance, space group P3121 opposing P3221 like this.

CHI for 3-cm diameter alpha-quartz parity pair test masses is calculated to be less than a perfect CH=1 by no more than 1.5x10-14. A decimal point followed by 13 nines is adequately close to 1.0.

400+ years of composition measurements show perfect null output within experimental error, supporting the Equivalence Principle. I expect the parity Eotvos experiment to give a net output, difference/average for test mass weight vectors over 90 days continous observation, of 3 parts-per-trillion. Any reproducible parity Eotvos experiment output reduces General Relativity from valid theory to convenient approximation. Einstein goes down, Weitzenboeck is now the hero.

Given the way the universe works, I fully expect two cases of "e before i" to be displaced by a bedeviling "umlaut-o."

Somebody should look. Our major roadblock now appears to be the sin of researcher pride - Not Invented Here. I expect that to be displaced by greed.

--
Uncle Al
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)

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