Posted by Uncle Al on Jun 13, 2003 at 20:32
(68.5.243.16)
The Equivalence Principle postulates that all local test masses fall identically in vacuum regardless of composition or internal structure. General Relativity immediately follows. Test mass opposite geometric parity has never been evaluated.
A vigorous debate has ensued here about the mathematical validity of parity the symmetry vs. parity the property. We've had an epiphany! It looks like there are unexpectedly deep geometric connections. We can now duplicate 6 CPU-days of 2 GHz Linux computation by 20 seconds in a calculator.
I. THE PROPERTY
The existence of a symmetry operator implies the existence of a conserved observable. Given G is the Hermitian generator of nontrivial unitary operator U (e.g., parity), then if U commutes with Hamiltonian H so does G: [H,G]=0. If U commutes with H it is a symmetry and a conserved quantity. Any system that is initially in an eigenstate of U evolves over time to other eigenstates having the same eigenvalue.
U psi = c psi =>
U exp(-itH) psi = exp(-itH) U psi [U commutes with H]
= exp(-itH) c psi
= c exp(-itH) psi
so exp(-itH) psi is again an eigenstate of U, with the same eigenvalue c. Discrete symmetries also give conserved quantities in classical mechanics (e.g., bifurcation theory of dynamical systems). Parity the symmetry is coupled to parity the property.
II. THE THEORY
*All* gravitation theories are either symmetric (metric, General Relativity) or anti-symmetric (affine, teleparallel) to parity transformation. There are no prior observations constraining gravitation to be "gerade" or "ungerade" toward parity. Unpaired spin Eotvos experiments have 99.9+% of their rest mass inert (nuclei plus non-valence electrons). Macroscopic shapes are shown below to be similarly poor. Composition test masses have 99+% of their rest mass inert vs. the opposed property. Parity pair test masses have 99.9+% of their rest mass active.
III. THE QUANTITATIVE MEASURE
CHI is the continuous function normalized parity divergence of a lattice. CHI is calculated ab inito from lattice coordinates alone with no semi-empirical constraints. CHI=0 is an achiral array (superposable parity inversions). CHI=1 is theoretically perfect parity divergence.
Documentation
IV. THE DATA Space group P3121 vs. P3221 single crystal alpha-quartz has a parity emergence scale of 0.113 nm3. Fractional nanometers. Explicitly calculating CHI for quartz from 3.6 A radius (14 atoms) to 100,000 A radius (3.3357x10 log(1-CHI) = -2log(radius) + 0.523167
CHI is the connection between eigenvalues, special functions, and their representation theory with solid angles and exponentials of fractions of pi. The intercept is now the solid angle subtended by the vertex angle of a polyhedron (the suppliment of its dihedral angle) defined by the helix, log(1-CHI)= -2log(radius) + [(180-alpha)(pi)/60] - pi
The explicit calculation predicts a 110.01 degree angle. Subtended solid angle was tested against explicit calculation for quartz, tellurium, distorted tellurium unit cells, and berlinite aluminum phosphate that all express rigorous QCM COR=1 regardless of atom labeling or connectivity. The number of atoms contained within a sphere varies as the cube of the radius log(1-CHI)= (-2/3)log(N) + constant The more parity-active atoms that are packed within a spherical radius, the faster CHI asymptotically approaches a perfect 1. EVERY atom in a quartz test mass is parity active to 0.113 nm3 volume. A crystal is self-similar. Everything adds in-phase. The advantage over machine screws is non-linearly immense. V. THE TEST MASSES Quartz crystal quality at all scales from atomic lattice to macroscopic devices is tightly linked to one commercially tabulated infrared absorption ratio, EIA Standard 477-1. Parity Eotvos experiment test mass raw crystals in Grade A quartz, left- vs. right-handed, cost about as much as two quality propane barbecues. Machine, polish, lay down 10-100 nm of titanium immediately followed by gold. VI. THE CONCLUSION Gravitation is pure geometry. It is naturally challenged with geometric test masses. Do it my way and the Equivalence Principle falls (depending on whether spacetime is in truth metric curvature V4 or affine torsion A4). --
Software download
History of theory
M. Petitjean, "On the Root Mean Square Quantitative Chirality and Quantitative Symmetry Measures" J. Math. Phys. 40[9] 4587-4595 (1999) (source code has been updated)
3-cm quartz test mass has CHI = 1 - 1.48x10-16
Eotvos balance rotor
("alpha" is smallest vertex angle in the helix. O-Si-O is 110.51 degrees)
log(1-CHI) = -2log(radius) + 0.496895
3-cm quartz test mass has CHI = 1 - 1.40x10-16
Uncle Al
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)
Re: Parity Eotvos Experiment update Uncle Al 14/6 11:04
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