Extremal parity pair test masses will measurably violate the Equivalence Principle in Eotvos experiments.
©2005 Alan M. Schwartz. All Rights Reserved.
organiker'at'lycos.com
PACS 04.80.Cc, 11.30.Er
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Revised 03 July 2007

Novel Equivalence Principle Tests

Metric gravitation theories[1] (General Relativity) postulate the Equivalence Principle (EP): local bodies vacuum free fall along identical (parallel) minimum action trajectories regardless of composition and mass distribution, requiring spacetime curvature. Affine and teleparallel gravitation theories ignore the EP and instead evolve spacetime torsion. The two approaches give otherwise identical predictions. Non-metric gravitation allows EP violations. EP parity violations should be measurable.

Newtonian gravitation only depends upon radial mass distribution (e.g., Green's function). It and metric gravitation (mass as a tensor quantity) are symmetric to parity transformation. (x,y,z) and (-x,-y,-z) give identical answers. Non-metric gravitation (mass as a pseudotensor) can be antisymmetric. (x,y,z) and (-x,-y,-z) give different answers. The proper analysis of spacetime geometry is test mass geometry (chemically identical, opposite parity mass distributions). Quantitative geometric parity divergence is ab inito calculated from atomic coordinates. Dr. Michel Petitjean presents an extensive survey of theory in pure and applied mathematics. A novel parity Eötvös (pronunciation) experiment is proposed in unmodified apparatus using maximally parity-divergent alpha-quartz single crystal test masses. An Equivalence Principle violation >520 times that allowed for opposed composition test masses is predicted.

EXECUTIVE SUMMARY


Introduction
    Table I.    Test Mass Property Magnitudes
    Table II.   Symmetry Groups
    Table III.  Postulated Gravitation Independence
    Table IV.   Spin and Parity Operations
Testing the Equivalence Principle
    Table V.    Equivalence Principle Tests
    Table VI.   Solar Gravitation At Earth Orbit
    Table VII.  Horizontal Acceleration vs. Latitude
Symmetry, Chirality, Parity
     Figure 1.  Chirality compared to Parity
    Table VIII. Paired Enantiomorphic Space Groups
    Table IX.   Comparison Of Gravitation Theories
Optical and Geometric Chiralities
Geometrically Chiral Test Masses
    Table X.    Candidate Chiral Test Masses
    Subtopic    Hemiparity Test Masses
Calculated Geometric Chirality/Parity
    Table XI.   43 Invalid and 16 Deficient Enantiomorphic Sohncke Space Groups
    Table XII.  Six Qualified Enantiomorphic Sohncke Space Groups
    Table XIII. Benzil Differential Parity Enthalpy of Fusion
     Figure 2.  Tellurium Atomic Lattice
     Figure 3.  Tellurium Atomic Lattice with Bonds
    Table XIV.  Calculated Lattice Parity Divergence, Te and alpha-Quartz
     Figure 4.  log(1-CHI) vs. log(radius): Tellurium,      505,000 Å Radius
     Figure 5.  log(1-CHI) vs. log(radius): alpha-Quartz, 1,010,000 Å Radius
     Figure 6.  log(1-CHI) vs. log(radius): Berlinite,      100,000 Å Radius
     Figure 7.  log(1-CHI) vs. log(radius): Distorted Te Unit Cells
    Table XV.   Te Distorted Unit Cells Data
    Table XVI.  Te Distorted Fractional Coordinates
    Table XVII. Te Distorted Fractional Coordinates Data
    Table XVIII.Calculated Lattice Parity Divergence, Te Cylinders
    Table XIX.  Measured Quartz Quality
Green's Function
    Table XX.   CHI Fluctuations at Small Radius Increments
    Table XXI.  Quartz CHI at Small Radii
Gravitational Scaling
    Table XXII. Compactified Dimensions' Effects
Other Chiral Test Masses
    Table XXIII.Mean Atomic Weights
Conclusion
Acknowledgements
References

Introduction

Galileo published the universality of free fall in 1638[2], describing pendulum and inclined plane experiments: "if one could totally remove the resistance of the medium, all substances would fall at equal speeds." Newton's 1687 Principia[3] declared the Equivalence Principle: "This quantity that I mean hereafter under the name of... mass... is known by the weight... for it is proportional to the weight as I have found by experiments on pendulums, very accurately made...." Einstein's Special Relativity[4] reconsidered mass (four-momentum and invariant mass),
mass = sqrt(px2 + py2 + pz2 - pt2);   metric signature (+1,+1,+1,-1), Relativity
mass = sqrt(pt2 - px2 - py2 - pz2);   metric signature (+1,-1,-1,-1), field theory
time, and distance as geometry. General Relativity[5] is a geometric model of gravitation. Einstein's elevator Gedankenexperiment[6] embodies the Weak (Galilean-Newtonian) Equivalence Principle[1]:

  1. Local bodies fall identically, because
  2. Gravitational mass (F=mgGM/r2) is indistinguishable from inertial mass (F=mia),
  3. Regardless of composition,
  4. Regardless of geometry (internal structure),
  5. Regardless of mass;
Inertial reference frames (coordinate systems) have constant relative velocity in a flat spacetime manifold. Accelerating frames with consistent definitions of energy and momentum (or mass and angular momentum) require non-zero spacetime curvature (assuming an asymptotic symmetry group, which obtains: the Bondi-Metzner-Sachs group restricted to the Poincaré subgroup[7]). Local spacetime must have a unique curvature. Local test masses exhibiting non-parallel geodesic trajectories require simultaneous different values of local spacetime curvature. Any paired (sets of) test masses violating the Equivalence Principle empirically falsify metric theories of gravitation at their founding postulate.

The Weak Equivalence Principle extends to the Strong (Einsteinian) Equivalence Principle :

  1. Non-rotating free fall is locally indistinguishable from uniform motion absent gravitation. Linear acceleration relative to an inertial frame in Special Relativity is locally identical to being at rest in a gravitational field. A local reference frame always exists in which gravitation vanishes.
  2. Local Lorentz invariance[8] (absolute velocity does not exist) and position invariance. All local free fall frames are equivalent.
  3. The Strong Equivalence Principle embraces all laws of nature; all reference frames accelerated or not, in a gravitational field or not, rotating or not, anywhere at any time (frame covariance; global diffeomorphism invariance aside from the Big Bang).

The Equivalence Principle demands that gravitational fields contain a local Minkowski (flat) spacetime reference frame (free fall). Gravitational fields cannot have a stress-energy tensor if free fall exists. If no reference frame makes gravitation locally vanish, spacetime curvature is counter-demonstrated as a violated Equivalence Principle.

General Relativity's physical systems are always spatially separable into independent components. Systems of three or more particles require cluster separability (macroscopic locality). When the system is separated into subsystems, the overall mathematical description must reduce to descriptions of the subsystems. This is vital in scattering problems with two or more fragments.

Quantum mechanics allows entangled states (superpositions of product states) that require a fundamental irresolvable connection within readily demonstrated physical systems (two-slit diffraction, the Einstein-Podolsky-Rosen paradox). Macroscopic locality is violated: Measuring the state of one slit in a double slit experiment alters the observed diffraction pattern to single slit patterns (quantum eraser experiments). Relativistic and quantum views are in conflict. Test masses contrasting an emergent phenomenon that cannot be reduced to a point mass equivalent are novel and important tests.

Weak and Strong Equivalence Principles are coupled[9]. Gravitation studies deny non-point geometric[10] consequences. Chirality and parity are emergent phenomena - components collectively evolve discrete symmetry system properties absent and unpredictable from any smaller subset. Geometric parity is an obligatory Equivalence Principle challenge if maximum calculated parity divergence (>99% divergence for parity pair alpha-quartz single crystals) is evaluated as are composition divergences (0.19% divergence in nuclear binding energy/nucleon for magnesium vs. beryllium).

Table I. TEST MASS PROPERTY MAGNITUDES

PropertyFraction of
Rest Mass
rest mass 100%
crystal lattice
atomic geometric parity
  99.9775%  (*Te)
  99.9771%  (*HgS)
  99.9769%  (*PdSbTe)
  99.9730%  (*AlPO4)
  99.9726%  (*SiO2)
  99.9713%  (*benzil)
nuclear binding energy (low Z)     0.76%    (2He4)
neutron versus proton mass     0.14%
electrostatic nuclear repulsion     0.06%
electron mass     0.03%
unpaired spin mass     0.005%  (55Mn**)
nuclear antiparticle exchange     0.00001%
Weak Force interactions     0.0000001%
*(nuclear mass)/(atomic mass), corrected for isotopic abundance
Atomic Data Nucl. Data Tables 59 185 (1995)
**globally aligned undecatiplet

When a set of transformations is closed (any transformation can be expressed as the product of other transformations in the set), the set is called a "symmetry group."

Table II. SYMMETRY GROUPS

GroupTransformationUnmeasurable
Quantities
Conserved
Quantities
Rotation, SO(3) Spatial rotations Absolute angle Angular momentum L
Translation Spacetime
translations
Absolute position Energy E, or Mass M
and Momentum P
Lorentz Spacetime rotations
and Reflections
Absolute uniform
velocity, Orientation
Spacetime interval S,
Parity P, Time reversal T
SL(2,C)
(Homogeneous Lorentz)
Spacetime rotations Absolute uniform
velocity
S (not P or T)
Diffeomorphism
(General Coordinate)
Spacetime curvature
(acceleration)
Absolute
acceleration
Topological invariants*
Poincaré Lorentz plus
Translations
(see above) L, E (or M) and P
U(1) Scalar Phase Shift Absolute phase Electric charge
SU(2) 2-D Phase Shift Absolute 2-D phase Isospin
SU(3) 3-D Phase Shift Absolute 3-D phase Color
©2002 Kenneth R. Koehler. All Rights Reserved.

*Manifolds with multiple differential structures (e.g., 7-D sphere) have multiple, mutually-exclusive equivalence classes of metrics characterized by different, independent definitions of volume (by a factor of the Jacobian under coordinate transformations).

Unitary groups U(1), SU(2) and SU(3) parameter spaces are isomorphic to (in one-to-one correspondence with) the circle, the sphere (a surface) and the "three sphere" (not a ball) respectively. Fields with non-abelian symmetries divide into "electric" (curl-free; e.g., gravitational) and "magnetic" (divergence-free; e.g., inertial) fields as do abelian electromagnetism and the linearized form of Einstein's field equations for weak gravity and slow matter.

Table III. POSTULATED GRAVITATION INDEPENDENCE[11]

ClassInvarianceConserved Quantity
Proper
orthochronous
Lorentz
symmetry
translation in time
  (homogeneity)
energy
translation in space
  (homogeneity)
linear momentum
rotation in space
  (isotropy)
angular momentum
Discrete
symmetry
P, coordinates' inversion spatial parity
C, charge conjugation charge parity
T, time reversal time parity
CPT product of parities
Internal
symmetry

(independent
of spacetime
coordinates)

U(1) gauge transformation electric charge
U(1) gauge transformation lepton generation number
U(1) gauge transformation hypercharge
U(1)Y gauge transformation weak hypercharge
U(2) [not U(1)xSU(2)] electroweak force
SU(2) gauge transformation isospin
SU(2)L gauge transformation weak isospin
PxSU(2) G-parity
SU(3) "winding number" baryon number
SU(3) gauge transformation quark color
SU(3) (approximate) quark flavor
S((U2)xU(3))
[not U(1)xSU(2)xSU(3)]
Standard Model

EP tests exploit external symmetries' observables. Internal symmetries' observables (gauged using fiber bundle theory, e.g., charge conjugation) transform fields amongst themselves leaving physical states (translation, rotation) invariant. A local gauge transformation always exists to make the local gauge-field vanish. Two vector potentials differing only by a gauge transformation give the same field. EP tests opposing properties coupled to internal symmetries are empirical first order default nulls.

Linear and angular momenta, mass-energy, electric charge, and CPT are strongly conserved (black holes). Properties derived from internal symmetries transform fields amongst themselves leaving physical states (translation, rotation) invariant: U(1) symmetry in electromagnetism, U(2) symmetry in electroweak theory, SU(3) in strong force theory. CPT, quark color, baryon number, and lepton generation number are locally conserved. Weak interactions violate parity conservation and violate the remainder. Gravitation has never been challenged with test mass geometric parity.

Parity symmetry obtains when physics is invariant under a discrete transformation that reverses all space coordinates' signs (x,y,z) but ignores time (t). Angular momentum and spin (J = r X p, axial vector or pseudovector), angular velocity, torque, auxiliary magnetic field (H), magnetic dipole moment, and longitudinal and transverse polarizations remain constant under parity. Parity reverses the signs of the position vector (proper or polar vector) of a particle in space (r to -r), velocity (v = dr/dt), linear momentum (p), force (f = dp/dt), electric field (E = -nablaV), and electric dipole moment (sigma·E).

Table IV. SPIN (J) AND PARITY (P) OPERATIONS

JPEven Parity
Object
JPOdd Parity
Object
0+ Scalar (S) 0- Pseudoscalar (P)
1+ Axial vector (A) 1- Polar Vector (V)
2+ Tensor (T) 2- Pseudotensor

Newtonian gravitation requires parity invariant conservative forces. General Relativity models gravitation as a parity invariant rank-2 tensor. Nobody has sought empirical exceptions despite acceptable theoretical loopholes[12] including teleparallel gravitation (below).

External symmetry-derived properties (e.g., parity) act on spacetime. A Poincaré group gauge theory can be equivalent to the Einstein-Cartan theory of gravitation[13]. Einstein-Cartan theory operates in Riemann-Cartan spacetime U4 - a paracompact, Hausdorff, connected, Cinfinity, and oriented four-dimensional manifold on which are defined a local Lorentz metric g and a linear affine connection capital gamma. Curvature and torsion tensors can be obtained from capital gamma on U4:

  1. If the torsion tensor vanishes, Riemann-Cartan spacetime becomes pseudo-Riemannian spacetime, V4 (General Relativity's description of gravitation);
  2. If the curvature tensor vanishes, it becomes Weitzenböck spacetime, A4 (in which the teleparallel gravitational energy-momentum pseudotensor is anti-symmetric to parity transformation);
  3. If both tensors vanish, it becomes Minkowski spacetime, M4.

Quantum field theories (QFT) with hermitian hamiltonians are invariant under the Poincaré group containing spatial reflections. Parity is a spatial reflection and parity is not a QFT symmetry! QFT are invariant under the identity component of the Poincaré group - the subgroup consisting of elements that can be continuous path joined to the Poincaré group identity; only an orthochronous Poincaré group representation. This subgroup excludes parity and time reversal. All hermitian hamiltonians will contain a symmetry and an observable with the properties of parity, even though the Hamiltonian will not be symmetric under space reflection. QFT with non-hermitian Hamiltonians can have real and positive energy spectra with PT invariance, but do not contain parity invariance alone[14]. Metric and quantum field theories of gravitation are tested by the parity Eötvös experiment.

Supersymmetric (SUSY, gauge symmetry plus spacetime symmetry) grand unified theories relating fermions and bosons to each other contain added allowances for symmetry breaking (inserted soft breaking terms into the Lagrangian where they maintain the cancellation of quadratic divergences)[15]. When global symmetry contracts to local symmetry, supergravity (SUGRA) emerges: if one gauges the SUSY transform, because of the SUSY algebra, one inevitably gets a gravitation theory)[16]. Supergravity by itself is not a renormalizable quantum theory.

A gravitational stress-energy (energy-momentum) pseudotensor constructs volume integrals for total gravitational four-momentum and total angular momentum. Teleparallel gravitation can allow a gravitational stress-energy pseudotensor obtained by comparing vectors at different points of spacetime - a coframe field - unlike ordinary General Relativity. (Teleparallel theories wholly equivalent to General Relativity are inoperative here.) When the coframe field changes, the pseudotensor changes (not gauge-invariant; not covariant under general coordinate transformations)[17]. This defines an integral energy-momentum obeying an exact conservation law. It is an observable, and it can be sensitive to parity inversion[18] - "a redistribution of energy between material and gravitational (coframe) fields is possible in principle."

(The difference between the Weitzenböck and Levi-Civita connections is "contorsion" quantifying coframe field twist as measured by the Levi-Civita connection. Contorsion can be expressed as the torsion of the Weitzenböck connection. The Levi-Civita connection can be expressed as the Weitzenböck connection and its torsion. The Ricci scalar curvature can be expressed as the Weitzenböck connection and its torsion. The Lagrangian for General Relativity therefore can be expressed purely in terms of the coframe field - in a way that is not symmetric to parity transformation.)

Testing the Equivalence Principle

A non-geometric model of spacetime is unimaginable given mathematical and physical consequences of symmetries. General Relativity models continuous spacetime, going beyond conformal symmetry (scale independence) to symmetry under all smooth coordinate transformations - general covariance (the stress-energy tensor embodying local energy and momentum) - resisting quantization. General Relativity is invariant under transformations of the diffeomorphism group. General Relativity predicts evolution of an initial system state with arbitrary certainty. Quantum mechanics' observables display discrete states. Heisenberg's Uncertainty Principle limits knowledge about conjugate variables in a system state, disallowing exact prediction of its evolution. Covariance with respect to reflection in space and time is not required by the Poincaré group of Special Relativity or the Einstein group of General Relativity. Anomalies must exist. Single crystal test mass parity imposes self-similar discrete coordinate transformations with gravitational consequences.

Physics seeks Equivalence Principle empirical violation. Tests often examine quantities evolved from internal symmetries via Noether's theorem. Internal symmetries by definition - a local gauge transformation always exists to make the local gauge-field vanish - do not affect spacetime. Studies contrasting baryon number, isospin, hypercharge, lepton vs. baryon number... are automatic null results.

Geometric parity is the only untested physical property arising from an external symmetry. Parity is an absolutely discrete symmetry that cannot be approximated by a Taylor series or a sum of infinitesimals. Noether's theorem with its dependence upon smooth Lie groups is inappropriate.

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.

Upsi = cpsi     then,
Uexp(-itH)psi = exp(-itH) Upsi   [U commutes with H]
                     = exp(-itH) cpsi
                     = 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 geometric parity the property.

Table V. EQUIVALENCE PRINCIPLE TESTS

YearInvestigatorAccuracyMethod
  500? Philoponus[20] "small" Drop Tower
1585 Stevin[19] 5·10-2 Drop Tower
1590? Galileo[2] 2·10-2 Pendulum, Drop Tower
1686 Newton[3]    10-3 Pendulum
1832 Bessel[21] 2·10-5 Pendulum
1910 Southerns[22] 5·10-6 Pendulum
1918 Zeeman[23] 3·10-8 Torsion Balance
1922 Eötvös[24] 5·10-9 Torsion Balance
1923 Potter[25] 3·10-6 Pendulum
1935 Renner[26] 2·10-9 Torsion Balance
1964 Dicke,Roll,Krotkov[27] 3·10-11 Torsion Balance
1972 Braginsky,Panov[28]    10-12 Torsion Balance
1976 Shapiro, et al.[29]    10-12 Lunar Laser Ranging
1981 Keiser,Faller[30] 4·10-11 Fluid Support
1987 Niebauer, et al.[31]    10-10 Drop Tower
1989 Heckel, et al.[32]    10-11 Torsion Balance
1990 Adelberger, et al.[33]    10-12 Torsion Balance
1999 Baeßler, et al.[34] 5·10-13 Torsion Balance
2010? MiniSTEP[35]    10-17 Earth Orbit
Ciufolini & Wheeler Gravitation and Inertia (Princeton University Press: Princeton, 1995) pp. 117-119
einstein.stanford.edu/STEP/information/data/gravityhist2.html

Contrasting compositions classically define[36] "different" test masses : baryon number (N+Z), isopin (N-Z)/2, Yukawa forces[37]; gravitational self-energy[38]; electron spin (Dy6Fe23 and HoFe3[39]; 94% of Alnico 5 magnetic field) versus electron orbital angular momentum (37% of Sm2Co17 magnetic field) - a mole of electrons masses an insignificant 5.4858·10-4 grams; neutrino-antineutrino exchange in different nuclei[40]; and inverse square deviations at small radii (string then M-theory[41]). 400+ years of exquisite tests, notably those derived from Vásárosnaményi Báró Eötvös Loránd[24], null to differential detection limits.

Ferrimagnet Dy6Fe23 has no external magnetic field at -1°C. It has 0.4 unpaired electrons/formula unit, or 97 nanograms of net unpaired spins/g test mass. It is ridiculously dilute. This spin test mass is calculated to display measurable spacetime torsion interaction as a cylinder 20 meters (!!!) in diameter and length weighing some 50,000 metric tonnes (25% more than the largest WWII Essex-class aircraft carrier fully loaded and fueled).

Quantifiable composition divergences arise in quantum mechanics, quantum chromodynamics, and relativistic electronic effects (gold[42]). To the extent that bodies are identical they are equivalent. Mirror-image chiral and parity pair bodies cannot be identical by definition. 65 Sohncke space groups of 230 total[43] describe chiral crystal configurations. 22 Sohncke space groups are eleven enantiomorphic parity pairs. Metric theories of gravitation are falsified at the postulate level if extreme mirror-image chiral or parity pair bodies are not Eötvös experiment nulls.

Starting in 1889 Eötvös' torsion balance tested the Equivalence Principle. Masses gravitate toward Earth's center. Given two (sets of) test masses, one at each end of a fiber-suspended rotor, the horizontal component of centripetal force is directed away from the Earth's spin axis proportional to inertial mass. Net torque twists the fiber if mGM/r2 not equal to ma. An attached mirror reflects light, amplifying angular deviation into an interferometer. Dicke[27] used Earth's free fall in the sun's gravitational field[44],

Table VI. SOLAR GRAVITATION AT EARTH ORBIT

Acceleration,
cm/sec2
Orbital
Position
Date
0.593008 One astronomical unit
0.613307 Geocenter
perihelion
2001-2101 mean value
0.613393 2020, 05 January
0.613214 2098, 05 January
0.613305 Earth/Moon
barycenter
perihelion
2001-2101 mean value
0.613381 2020, 04 January
0.613252 2098, 04 January
0.573700 Geocenter
aphelion
2001-2101 mean value
0.573786 2085, 04 July
0.573626 2019, 04 July
0.573702 Earth/Moon
barycenter
aphelion
2001-2101 mean value
0.573764 2097, 05 July
0.573654 2019, 05 July
Corrected for light-time, aberration, General Relativity in ephemerides,
and Special Relativity in apparent place to eight+ significant figures.

and surface inertial centripetal acceleration (sidereal day, WGS 84; sea level unless noted) of Earth's rotation.

r = (6378136.46)[1-([sin2(lat)]/298.257223563)] meters
a = (3.380199)(cosine[lat])/sqrt[1-(0.006694380)cosine2(lat)] cm/sec2
   horizontal component of a = a[sin(lat)]

Table VII. HORIZONTAL ACCELERATION vs. LAT I TUDE

Resultant,
cm/sec2
Geocentric
Latitude,
degrees
Horizontal
Component,
sin(lat)
Centripetal
Acceleration,
cm/sec2
3.66 45 550 mph east
ground speed
commercial
airliner
9.51 45 1336 mph east
ground speed
Concorde
1.46490 60 0.866025 1.691516
1.58993 55 0.819152 1.940941
1.66673 50 0.766044 2.175761
1.69294 45 0.70711 2.394172
1.69294 44.951894 0.706513 2.396188
1.66770 40 0.642788 2.594484
1.59175 35 0.573576 2.775137
1.46736 30 0.500000 2.934715
1.29826 25 0.422618 3.071958
1.08960 20 0.342020 3.185778
0.84770 15 0.258819 3.275266
0.57993 10 0.173648 3.339705
0.29446   5 0.087156 3.378578
0   0 0 3.391570


g = (978.032677)[1+(0.00193185139)sin2(lat)]/sqrt[1-(0.00669437999)sin2(lat)] cm/sec2
dg/dh = 0.000308766[1-(0.0014665)sin2(lat)] cm/sec2-meter
dg/ds = -0.0008109[cos[(2)(lat)]+(0.0022)cos[(4)(lat)]-(0.0033)cos[2(lat)]sin2(lat)] cm/sec2-km

r = geocentric radius
a = total centripetal acceleration
lat = latitude
g = gravitational acceleration
h = altitude above sea level
s = distance

Any torque is diurnally modulated: For an Eötvös torsion balance with masses aligned east-west, maximum torque occurs at local noon or midnight when Earth's orbit is maximally anti-parallel or parallel to its rotation. Six hours offset torque is minimized. For test masses aligned north-south, torque maxima occur at 0600 hrs and 1800 hrs. The entire apparatus may be rotated with phase-lock detection to decouple noise.

Gravitational versus inertial mass anomaly is the dimensionless Eötvös parameter eta, the difference divided by the average of mg/mi for the two sets of test masses "A" and "B."

eta = 2[(mg/mi)A - (mg/mi)B] / [(mg/mi)A + (mg/mi)B]

Falling mass gravimeters detect 10-9 of ambient gravity[45]. Levitated dual sphere superconducting gravimeters[46] sense 10-11. Eötvös balances discern eta ~ 10-13 where room temperature vibration of component atoms interferes[47] as noise. Cryogenic 2.2°K Eötvös apparatus is being debugged[48]. Drag-shielded free-fall apparatus with eta ~ 10-15 is proposed[49].

Symmetry, Chirality, Parity

Chirality/Parity Cn proper rotation axes (360°/n) do not affect chirality. Sn improper (rotation-reflection, alternating) axes are Cn rotations followed by reflection through a plane normal to the Cn axis (sigmahCn). Any shape lacking an Sn axis (a baseball seam is S4) cannot be superposed upon its mirror image (one spatial coordinate reversed in sign). Mirror planes (sigma, S1) and an inversion point (i, S2) will be absent. Such objects (hands, feet, shoes, screws) are chiral (handed). Chiral mirror images are enantiomers. IUPAC stereochemistry nomenclature.

Eleven parity pairs of enantiomorphic space groups[43] have inverse lattices in which all three spatial coordinates reverse in sign. Unlike chirality, parity changes the space group:

Table VIII. PAIRED ENANTIOMORPHIC SPACE GROUPS

P31 P3112 P3121 P41 P4122 P41212 P4132 P61 P62 P6122 P6222
P32 P3212 P3221 P43 P4322 P43212 P4332 P65 P64 P6522 P6422

Enantiomers are not interconverted by time reversal plus proper spatial rotation in "true chirality"[50]. Quantitative geometric parity can be calculated[51]. Macroscopic chirality is a complex[52] concept. Liaisons between paired chiral bodies (a left shoe on a left or right foot) exhibit different free energies. Parity Eötvös experiments are profound tests of spacetime geometry.

Spacetime events are empirically left-handed: cobalt-60 beta-decay parity violation[53]; K-meson[54] and B-meson[55] decay charge-parity violation. Parity violation[56] is inarguable. Enantiomeric excited nuclei have different energy levels for otherwise identical transitions[57]. All atoms are intrinsically homochiral[58]. Excess matter vs. antimatter requires chiral discrimination[59]. A weakly interacting reverse-parity entire universe is proposed[60]. Linearly polarized massless photons traversing intergalactic space do not show optical rotation[61]. A photon is not a valid inertial reference frame. Massed interactions are different.

Gravitation arises from spacetime curvature (Riemann, metric), torsion (Weitzenböck, affine[13]), or both. Teleparallelism has torsion acting as a force, analogous to electrodynamics' Lorentz force equation, without geodesics. Parity pair Eötvös experiments offer self-similar helical geometry in greater active masses, subtended areas, and volumes than spin-polarized test masses[39,62] whose Eötvös experiments null within experimental error. Non-metric gravitation theories also postulate the Equivalence Principle[63].

Metric theories of gravitation postulate[64] spacetime is a Lorentzian manifold, test particles pursue space-time geodesics (all sufficiently small bodies subject only to gravitational interactions and starting with the same initial positions and velocities follow identical spacetime trajectories), and the Strong Equivalence Principle obtains. Component form physical laws move from flat into curved spacetime by replacement of partial derivatives with covariant derivatives (note factor ordering given non-commutation of covariant derivatives, as classical mechanics transforms into quantum mechanics). Geometry's response to matter distribution, Einstein's field equations, arises from the principle of least action. Equivalence Principle violation annuls spacetime curvature.

Local symmetries create conservation laws through Noether's theorem[65]. A conserved quantity derives from each symmetry commuting with time, and the reverse. A divergence-free current (conserved property) arises if the Lagrangian or the action is invariant under continuous transformation.

  1. To each continuous symmetry of an action there corresponds a conserved quantity because of the Euler-Lagrange equations of the Lagrangian, and the reverse.
  2. To each gauge symmetry of an action there corresponds an identity among Euler-Lagrange equations of the Lagrangian, and the reverse.
A physical system with a Lagrangian invariant with respect to the symmetry transformations of a Lie group has, in the case of a group with a finite (or countably infinite) number of independent infinitesimal generators, a conservation law for each such generator, and certain "dependencies" in the case of a larger infinite number of generators (General Relativity and the Bianchi identities). The reverse is true.

A symmetry can be broken explicitly - a term in the action or equations of motion may not be invariant. A symmetry can be broken anomalously - not all classical theory symmetries exist in the corresponding quantum theory. Quantum field theory anomaly spoils renormalizability. Anomaly absence in the Standard Model is crucial. A symmetry can be broken spontaneously if it is an exact symmetry of the equations of motion but not of a particular solution therein. Noether's theorem holds if the symmetry is not broken explicitly. Conservations can be relaxed in subsystems displaying reduced symmetry (Born scattering approximation, Fermi's golden rule, Snell's law).

A classical field theory conserved quantity does not demand a quantum field theory conserved quantity in kind. Parity Eötvös experiments offer measurable anomalies between General Relativity's continuous symmetries and test masses' discrete symmetries.

Table IX. COMPARISON OF GRAVITATION THEORIES[66]

TheoryMetricOther
Fields
Free
Elements
Status
Newton
(1687) [67]
Nonmetric Potential None Nonrelativistic
theory
Nordstrom
(1913) [66,68]
Minkowski Scalar None Unpredicted
light detection
Einstein
(1915) [67,68]
Dynamic None None Viable
Whitehead
(1922) [69]
Fixed Tensor 1 parameter
[70]
Possibly viable[71]
Belifante-Swihart
(1957) [68]
Nonmetric Tensor K parameter Contradicted by
Eötvös experiments
Brans-Dicke
(1961) [67,68,72]
Generic
Scalar
Dynamic
Scalar
OMEGA parameter Viable for
OMEGA>620
Tensor
(1970) [80]
Dynamic Scalar 2 free
functions
Viable
Ni
(1970) [67,67]
Minkowski Tensor
Vector
Scalar
1 parameter
3 functions
Preferred-frame
effects unobserved
Will-Nordtvedt
(1972) [68]
Dynamic Vector None Viable
Rosen
1973) [68]
Fixed Tensor None Contradicted by
binary pulsar data
Rastall
(1976) [68]
Minkowski Tensor
Vector
None Viable
Variable Mass,VMT
(1977) [68]
Dynamic Scalar 2 parameters Viable for wide
range of parameters
Modified Newtonian
Dynamics, MOND
(1983) [73]
Nonmetric Potential Free Function Nonrelativistic
theory

Gravitation is modeled by loop quantum theory[74], brane/string/M-theory[75], Lorentzian lattice quantum gravity[76]..., though few predictions exist to be tested. M-theory fails to predict the absence of supersymmetry at low energies, the presence of a positive cosmological constant, and the utter absence of a massless scalar field that its component string theories predict in abundance. Loop quantum theory predicts unobserved vacuum dichroism - lightspeed being very slightly smaller for high energy photons.

The Michelson-Morley experiment showed space is isotropic for massless photons to differential 10-8 in 1887 and 1.7·10-15 in 2002[77]. The Higgs field[78] uncouples massless and massed particles. Single crystal silicon polarized interferometry (revolving Earth orbiting the sun) detects 10-2[79] to 10-3[80] relative, ten orders of magnitude too insensitive (Colella-Overhauser-Werner[81] and Bonse-Wroblewski[82] neutron interferometers; Kasevich-Chu[83] atom interferometers).

Optical and Geometric Chiralities

Sculpted shapes are feebly chiral. Chiral crystal lattices approach 1022 unit cells/cm3. Optical chirality measured as rotation of sodium D-line 589.3 nm (in air[84]) plane-polarized light is specific rotation [alpha]D[85], or gyrotropy (circular birefringence, optical activity, optical rotation) in general. No chiroptical method measures solid state geometric chirality or parity.

Gyrotropy is assigned spiraling toward the observer, geometry spirals away from the observer. Dextrorotatory gyrotropy is levorotatory structure (optically right-handed quartz is space group P3221). Distant from optical transitions in ionic crystals[86]:

  1. Smaller pitch (repeat distance) of atomic helices gives larger gyrotropy.
  2. Smaller radial distance from helical axes gives larger gyrotropy.
    (Smaller relative radius or pitch increases geometric parity. Smaller radius and pitch together leave scale invariant geometric parity unaltered)
  3. Larger electronic (Pauling) polarizability gives larger gyrotropy.
  4. Tangential or radial (reverses the contribution) polarizability orientation in the plane normal to the helical axis dramatically alters gyrotropy.

Solid state optical activity theory[87] flags three irreducible tensor components, the last two of which tolerate mirror planes of symmetry (gyrotropy without chirality):

  1. Pseudoscalar, from molecular chirality and persists in disordered solution.
  2. Vector. A property of pyroelectric lattices.
  3. Pseudodeviator, from lattice symmetry. Silver thiogallate[88], AgGaS2 with non-polar achiral tetragonal space group I-42d (#122), has immense optical rotatory power: 522°/millimeter along [100] at 497.4 nm, reversed along [010].

Gyrotropy arises from differing refractive indices (n) for left- and right-circularly polarized light in a medium. Gyrotropy summed over the electromagnetic spectrum vanishes:

  1. Oscillator strength or f-sum rule[89]. The f-values sum to unity for all transitions from a given state;
  2. The Thomas-Reiche-Kuhn sum rule[90]. The photoabsorption cross section integral equals the total number of electrons involved in the collective excitation.
  3. Kramers-Kronig relationship[91]. Refractive index has a real part (n) arising from phase velocity and an imaginary part (k) arising from absorption. Optical rotatory dispersion is the real part and circular dichroism is the imaginary part of one complex spectrum.
  4. The sum over (n-1) across the electromagnetic spectrum is zero. The difference between two (n) for orthogonal polarizations sums to zero overall[92].
Chiroptical methods do not measure geometric parity.

Geometrically Chiral Test Masses

Geometric test masses benefit from small unit cells, heavy atoms and large calculated parity divergences[51]. Eötvös experiment test masses are encapsulated within gold metallization.

Table X. CANDIDATE CHIRAL TEST MASSES

Chiral Test MassTemp,
°K
a-axis,
Å
b-axis,
Å
c-axis,
Å
Volume,
  Å3
Density,
gm/cm3
Space
Group
grey Selenium[94] 300
  85
     2*
  4.3712
  4.304
  4.278
  4.3712
  4.304
  4.278
  4.9539
  4.968
  4.974
    81.975
    79.650
    78.909
  4.798
  4.935
  4.985
P3121
P3221
Tellurium[95]   4.456   4.456   5.921   101.82   6.243 P3121
P3221
neutron diffraction
x-ray diffraction


alpha-SiO2[96]
278
298
  13
  4.9134
  4.9137
  4.9021
  4.9134
  4.9137
  4.9021
  5.4052
  5.4047
  5.3977
  113.01
  113.01
  112.33
  2.649
  2.649
  2.665
P3121
P3221
alpha-Cinnabar, HgS   4.145   4.145   9.496   141.29   8.203 P3121
P3221
Berlinite, AlPO4   4.766
  4.9438
  4.766
  4.9438
 10.724
 10.9498
  210.96
  231.771
  2.880
  2.621
P3121
P3221
Palladium antimonide
 telluride
, PdSbTe[93]
  6.5362   6.5362   6.5362   279.24   8.463 P213
Azatwistanone[97] (model)   6.662  13.36     8.606  (756.60)   1.327 (P21/n)
[4]Triangulane[98] 120   5.798 10.434 11.872  (718.21)   1.112 P212121
[5]Triangulane[99] 110 12.024   5.3156   7.609   443.17   1.096 C2
benzil 294
100
  70
  8.402
  8.356
14.380
  8.402
  8.356
  8.373
13.655
13.375
13.359
  834.81
  808.76
1608.1
  1.255
  1.295
  1.303
P3121
P3221;
P21
*a(T) = [4.277 + (3.215·10-4)T] Å
  c(T) = [4.974 - (0.685·10-4)T] Å

Chiral crystals are complex objects[86]: Quartz (SiO2; no detectable gyrotropy 56.16° from crystallographic [0001]), paratellurite (TeO2; P41212, P43212), (HgS; P3121, P321), langasite[100] (La3Ga5SiO14; P321), berlinite (AlPO4; P3121, P3221), Bi12(Si, Ge, or Ti)O20 (I23), sodium bromate and chlorate (P213) all possess at least one traceable counter-helix to the dominant atomic helices. Space group P213 in QCM often shows good CHI growth, COR=1 and DSI=0 through 1100 atoms contained. Ferroelectric crystals are polarity twinned. Grey selenium must be grown from hot aniline solution[101] or different allotropes obtain; melt growth is disordered. Mechanical trauma deeply disrupts its lattice.

Achiral benzil[102] crystallizes in enantiomorphic space groups. A twinning phase transition[103] occurs at 84°K. Boules obtain by Bridgman-Stockbarger directional solidification[104]. Good crystals require solution growth as hot solid plastically deforms into lattice dislocations. Molecular solids have undesirable large unit cells.

Hemiparity Test Masses

An Eötvös rotor must be balanced around its vertical supending fiber for mass and moments of inertial overall. Petitjean's mathematics demands each test mass have three equal moments of inertia for maximum parity divergence. Full parity Eötvös experiment test masses are then explicit - identical chemical compositions, macroscopically identical forms, single crystal solid convex bodies without surface drilling or internal hollowing, and extremal opposite parity crystal lattices. Solid spheres or solid right cylinders with height equal to (radius)sqrt3 qualify.

Quartz offers a unique opportunity to perform two hemiparity Eötvös experiments, space group P3121 or P3221 quartz against amorphous fused silica. Different densities must be compensated. As all test masses are gold plated, this is straightforward.

dquartz =  2.649 g/cm3
dsilica   =  2.203 g/cm3
dgold   = 19.3 g/cm3

A quartz ball of radius R would be balanced by a fused silica ball of radius (R-r) with a vacuum-plated gold shell of thickness r. The general solution, sphere or cylinder, is

r3 - 3Rr2 + 3R2r - [(dquartz-dsilica)/(dgold-dsilica)]R3 = 0
r3 - 3Rr2 + 3R2r - (0.0260864)R3 = 0
www.1728.com/cubic.htm

For R = 1 centimeter in quartz, r = 87.72 micrometers in gold.

Calculated Geometric Chirality/Parity

Metric theories of gravitation demand continuous symmetries. Test mass discrete symmetries were not considered Equivalence Principle challenges. No ab initio method to calculate parity divergence existed.

Eötvös balances typically employ test mass right cylinders or spheres. A homogeneous isotropic solid sphere is the most symmetric figure in three dimensions. It possesses the following symmetry elements:

  1. the identity element;
  2. one point of inversion (its center);
  3. an infinity of Cn rotation axes each passing through the inversion point;
  4. an infinity of mirror planes each containing the inversion point and a rotation axis;
  5. an infinity of mirror planes each containing the inversion point and normal to a rotation axis;
  6. an infinity of Sn improper axes each passing through the inversion point.
"Infinity" above is at least the number of points on a line. It is infinitely larger than the number of integers, integers being countable. Parity pair spheres suitable for a parity Eötvös experiment Equivalence Principle challenge will contain only the identity element. The positions of their atoms must constitute a lattice that is quantitatively maximally non-superposable with preferably even Cn axes overall absent.

The rigorous abstract mathematics of chirality has been extensively examined and reduced to QCM software. CHI[51] is the quantitative calculation of the geometric parity divergence of N points. CHI is globally minimum for all rotations (R) and translations (t) for all correspondences (P) permitted by the colors and/or graph:

CHI = (d)[Min{P,R,t}D2]/4T
where d is the Euclidean dimension, D2 is the sum of the N squared-distances between the set and its parity inversion for a fixed pairwise correspondence with coincident centers of mass, and T is the geometric inertia of the set. All points are assigned equal mass, though their labels may differ. CHI varies between zero (exactly superposable (x,y,z)-parity inversions) and one (perfect divergence) inclusive. CHI is a continuous function of nuclear coordinates only - independent of translation, scale, and size. One value exists for a given target and its inverse lattice. It detects zero and approach toward zero. It does not require empirical constraints.

CHI is limited by three conditions:

  1. Intrinsic parity divergence. Perfect parity divergence delivers CHI=1 given two other conditions:
  2. Inertial moment disparity. CHI=1 can obtain only if  IX = IY = IZ exactly, and
  3. Inertial moment products. CHI=1 can obtain only if
    Sigmai (IXi*IYi) = 0
    Sigmai (IXi*IZi) = 0
    Sigmai (IYi*IZi) = 0
CHI is the quantitative measure of a parity Eötvös experiment wherein test masses are machined to exquisite equality of inertial moments to minimize perturbations. (2) and (3) do not exactly obtain in a crystal lattice volume at discontinuous atomic scales. Dense calculation of log(1-CHI) vs. radius is quite interesting for quartz, benzil, and PdSbTe. A macroscopic crystal is apparently continuous and homogeneous at scale, allowing arbitrarily close asymptotic approach to (2) and (3).

Parity is a subset of chirality. Lattice chirality is an emergent phenomenon wherein components collectively evolve discrete symmetry system properties absent and unpredicted from any smaller subset. Lattice chirality does not simplify below a microscopic packing unit seamlessly translated along its axes in three dimensions into a self-similar macroscopic crystal. Four possibilities obtain:

  1. Unit cell contents are achiral, crystal is achiral
  2. Unit cell contents are   chiral, crystal is achiral
  3. Unit cell contents are achiral, crystal is   chiral
  4. Unit cell contents are   chiral, crystal is   chiral
(1) and (4) obviously exist. (3) is a crystal whose formula units, when reduced to dimensionless points, remain a volumetric array non-superposable upon its mirror image (translational symmetry included). Tellurium in parity space groups P3121 and P3221 has an achiral single atom formula unit. Optical rotation is not diagnostic of geometric chirality. Silver thiogallate[88], AgGaS2 in non-polar achiral tetragonal space group I-42d, has immense optical rotatory power: 522°/millimeter along [100] at 497.4 nm.

(2) includes local achiral superunits as in La Coupe du Roi[105]. Cut a ball (apple) in from a pole to its equator along a great circle (lines of longitude 180° apart) with a plane. Repeat rotated 90° at the other pole (perpendicular planes). Make two equatorial cuts in the same direction, surface to center radial sectors, from the equatorial edge of one pole cut to the other and again on the opposite side (rotate about the polar axis; cut, skip, cut). The ball (apple) cleaves into identical homochiral halves. Homochiral components can assemble into a zero chirality lattice. Aspects of (2) lessen parity pair divergence.

Aspects of (2) weaken parity Eötvös experiments. If the set of points corresponding to positions occupied by all symmetry copies of the asymmetric unit is intrinsically chiral (3132 4143 6165 6264 screw axes), CHI>zero even if the formula units are achiral. 21 42 63 or no screw axes can allow geometric chirality to decrease with volume even if formula units are intensely chiral. 10 Sohncke space groups become achiral if their unit cell contents are achiral. 6 Sohncke space groups have no screw axes at all. 21 additional Sohncke space groups contain both left and right screw axes. 22 additional Sohncke space groups contain 21 42 63 that are each simultaneously left- and right-handed screws with the same screw vector[106]. These 59 space groups contain conflicting geometries that decrease CHI increase with increasing lattice sample radius.

Table XI. 43 INVALID AND 16 DEFICIENT SOHNCKE SPACE GROUPS

DeficienciesSpace Groups
Zero intrinsic
lattice chirality
P1 P2 P21 C2 P4 P42
I4 P3 P6 P63
21 42 63 screw
axes only
P2221 P21212 P212121 C2221 C222 F222
I222 I212121 P4222 P42212 I422 P6322
Cn axes only P222 P422 P4212 P312 P321 P622
Opposite
sense
helices
I41 I4122 R3 R32 P23 F23
I213 P432 P4232 F432 I23 P213
P4132 F4132 I432 P62
P64
P6222
P6422
I4132
P4332
Same-sense
helices plus 21
P41
P43
P4122
P4322
P41212
P43212
P61
P65
P6122
P6522

The three pairs of remaining enantiomorphic space groups possess unique screw axes[106]. These structures exclude primary conflicting symmetries and support maximal CHI increase with increasing lattice sample radius . A given material's lattice must still be examined with QCM for graph theoretic correspondences beyond the identity correspondence.

Table XII. SIX QUALIFIED ENANTIOMORPHIC SOHNCKE SPACE GROUPS

AcceptableSpace Groups
Same-sense
helices
P31
P32
P3112
P3212
P3121
P3221

10-13 mass anomaly is 2.148 cal/gram. Water hydrogen bond energy[107] is 5.57 kcal/mole (328 cal/gram), carbon-carbon single bond energy is 85 kcal/mole (3500 cal/gram). Test the Equivalence Principle with calorimetry [108]. Parity pair single crystals' heats of combustion vary with geographic orientation, local time of day, and impressed inertial acceleration.

Table XIII. BENZIL DIFFERENTIAL PARITY ENTHALPY OF FUSION

Property[109]Value
Molecular
weight
210.2322 g/mol
Triple
point
94.864°C
Dynamic
melting point
Onset
94.43°C
Meniscus
94.77°C
Melt
95.08°C
Thermodynamic
melting point
Onset
94.55°C
Meniscus
94.72°C
Melt
94.86°C
Enthalpy of fusion
mp = 94.82°C
112.0 J/g
  26.77 cal/g
23.546 kJ/mol
  5.6276 kcal/mol
Enthalpy of fusion
mp = 94.85°C
110.6 J/g
  26.44 cal/g
23.26 kJ/mol
  5.559 kcal/mol
Enthalpy of fusion
mp = 94.86°C
112.0 J/g
  26.76 cal/g
23.54 kJ/mol
  5.626 kcal/mol
Enthalpy of
parity divergence
  26.96 J/g
    6.444 cal/g
  5.668 kJ/mol
  1.355 kcal/mol
Differential enthalpy
of parity divergence
~24% for 3·10-13 g/g parity anomaly
E = (3·10-16 kg)(299,792,458 m/sec)2

Putting unique atoms' fractional coordinates through space group symmetry operations[43] populates the unit cell, the smallest repeated lattice volume. Selenium and tellurium unit cells contain three atoms, but three points define an achiral plane. Moved, the unit cells each contain six half-atoms - the repeating chiral unit as an emergent phenomenon. Selenium and tellurium values would not change given CHI weighted for atomic mass and fractional unit cell occupancy. The goal is the fastest growing lattice CHI from the smallest unit cell with the smallest volume/atom and heaviest atoms in the best parity space group.

The abstract mathematics of quantitative parity divergence only sees points' coordinates for analysis. A 3300 Å3 cube of tellurium lattice with 98 atoms (innermost 20% electron probability ellipsoids shown) appears to be a spatially repeating collection of points. Where is the parity divergence or even chirality?

TELLURIUM ATOMIC LATTICE

Te atomic lattice

Drawing nearest neighbor distances reveals aligned homochiral 3-fold helices. Perception and "common sense" are irrelevant. Rigorous mathematics is the only valid analysis. (Look beyond the screen to relax your eyes' convergence. The two-block image will double, one view from each eye. Increasing distance from the screen eases the deconvergence process. As you go increasingly walleyed the images will move further apart causing the center blocks to overlap. When the bottom crosses merge you can look into the 3-D image.)

TELLURIUM ATOMIC LATTICE WITH BONDS

Te bonded lattice

A macroscopic test mass is an apparently continuous medium. Lattice volumes are quantized by nearest neighbor distances at scale. The number of calculations needed to determine CHI grows as the products of factorials of the number of each color of atom (not each chemical element) given full QCM diagnostics. A lattice is qualified by QCM evaluation of consecutive angstrom increment radii from half the longest unit cell axis length to at least 1000 atoms contained. A QCM 1000-atom run will typically require 10-20 CPU-hours in an RS6000/Power3 given connectivity-optimized HyperChem *.hin file input. It is a quirk of QCM graph theory analysis that 15 points input as *.xyz file format would require more than the age of the universe to calculate - to give the same answer as *.hin format input after a few minutes.

QCM begins with enumeration of graph automorphisms in concentric layers of the array starting at its origin. "Direct symmetry index" DSI, the normalized minimized sum of the N squared-distances between the vertices of the d-set and the permuted d-set, measures the set's similarity to itself. A set having no direct symmetry in a d-dimensional Euclidean space is still not symmetric after immersion in a higher dimensional space while a nonspecific chiral set would become achiral. DSI>0 beyond a few contained unit cells is a disqualification for extremal parity-divergent test masses.

"Correspondences" COR includes the identity element but is not a count of group theory symmetry elements. COR>1 beyond a few contained unit cells is a disqualification for extremal parity-divergent test masses. Assigning different atom labels (SiO2) does not default assign different graph theory point colors. QCM numeric outputs are independent of input file structure format, atom connectivity (if any), and list ordering. All points are assigned unit weight. Mass is mass.

If DSI=0 and COR=1, a FastCHI subset of QCM is valid. FastCHI coding calculates CHI (how a set is not similar to its parity inversion) linearly with the number of atoms (runs in O(n) time and O(1) space). A 2 GHz Linux PC in AMD Athlon hardware (Wintel is 40% slower) can examine ~109 atoms/second. Large quartz radii were run in a cluster of 16 AMD Opteron-848s in 30 days. Tellurium, quartz, and berlinite were given 609, 90,386 (150,000 CPU-hrs; 444 quadrillion atoms), and 571 radius samplings respectively to calculate the following:

Table XIV. LATTICE PARITY DIVERGENCE, Te, alpha-Quartz, and Berlinite

Sphere
Radius, Å
Lattice
CHI
Total
Atoms
Unit
Cells
Volume,
Å3
Unit Cell*
Tellurium
Quartz
Berlinite

0.018926
0.173940
0.174566

6
11
23

1
1
1

102
113
211
Quartz 3.6
Berlinite 5.0
0.843228753630922340
0.928954111975661469
14
46
1.6
2.6
176
539
10 0.953161251843284545
0.989463287292071760
0.922918192326544688
126
340
350
42
38
19
4,189
4,269
4,102
20 0.994970730830562692
0.991987685060543311
0.989437910906527996
987
2,670
2,854
329
297
159
33,510
33,528
33,449
30 0.995341618942362163
0.994391347133448844
0.999136218439710468
3,315
9,014
9,658
1,105
1002
537
113,100
113,200
113,200
40 0.994311225094353860
0.995770871078683400
0.999632392784788830
7,906
21,346
22,888
2,635
2,372
1,272
268,100
268,000
268,200
50 0.997182871713665811
0.997497356338878373
0.997751761704918893
15,428
41,696
44,665
5,143
4,633
2,481
523,600
523,600
523,500
100 0.998674087455959555
0.999455952622276367
0.999643499178059791
123,411
333,526
357,384
4.11·104
3.71·104
1.99·104
4.19·106
4.19·106
4.19·106
500 0.999991634437844450
0.999980456860055180
0.999989604104047446
15,427,878
41,696,416
44,676,376
5.14·106
4.63·106
2.48·106
5.24·108
5.24·108
5.24·108
1,000 0.999993481890267883
0.999997440127579185
0.999998427421108820
123,422,301
333,575,254
357,408,684
4.11·107
3.71·107
1.98·107
4.19·109
4.19·109
4.19·109
2,000 0.999998519283409702
0.999999080483513838
0.999999455994921283
987,378,232
2,668,598,322
2,859,270,260
3.29·108
2.97·108
1.58·108
3.35·1010
3.35·1010
3.35·1010
5,000 0.999999871340933588
0.999999789636193490
0.999999929783070689
15,427,793,636
41,696,845,260
44,676,133,458
5.14·109
4.63·109
2.48·109
5.24·1011
5.24·1011
5.24·1011
10,000 0.999999897747013216
0.999999968983088517
0.999999980194705342
123,422,354,560
333,574,731,752
357,409,097,084
4.11·1010
3.71·1010
1.99·1010
4.19·1012
4.19·1012
4.19·1012
20,000 0.999999983730030373
0.999999992565814146
0.999999990115051648
987,378,792,095
2,668,597,813,994
2,859,272,764,336
3.29·1011
2.97·1011
1.59·1011
3.35·1013
3.35·1013
3.35·1013
100,000 0.999999999546726956
0.999999999648281730
0.999999999778529238
123,422,348,782,767
333,574,726,196,900
357,409,096,311,540
4.11·1013
3.71·1013
1.98·1013
4.19·1015
4.19·1015
4.19·1015
Tellurium
200,000
300,000
400,000
505,000

0.999999999810150121
0.999999999962125567
0.999999999948796222
0.999999999981536820

987,378,790,078,890
3,332,403,416,661,100
7,899,030,320,918,100
15,895,271,169,403,000

3.29·1014
1.11·1015
2.63·1015
5.30·1015

3.35·1016
1.13·1017
2.68·1017
5.39·1017
Quartz
200,000
300,000
400,000
500,000
750,000
1,010,000

0.999999999899102952
0.999999999990628857
0.999999999964449503
0.999999999987239756
0.999999999987958381
0.999999999994484622

2,668,597,809,177,800
9,006,517,605,688,300
21,349,690,977,976,000
41,698,615,192,377,000
140,732,826,273,680,000
343,696,999,447,690,000

2.97·1014
1.00·1015
2.37·1015
4.63·1015
1.56·1016
3.82·1016

3.35·1016
1.13·1017
2.68·1017
5.24·1017
1.77·1018
4.32·1018
*Te unit cells contain six half-atoms; quartz seven atoms and four half-atoms,
berlinite thirteen atoms and ten half-atoms. All lattice points are calculated whole.
(Berlinite, Z. Kristallogr. 192 119 (1990))

log(1-CHI) vs. log(radius): Tellurium to 505,000 Å Radius

Te CHI vs. Radius
FastCHI Programming, Matthew Francey
80-bit BigCHI and statistics package, Tony Lapen
559 CPU-hrs, 1.8 GHz Opteron, Linux, John Edward Scott

log(1-CHI) vs. log(radius): alpha-Quartz to 1,100,000 Å Radius

Quartz CHI vs. Radius
FastCHI Programming, Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen
Parallelized CHIpir, John Edward Scott
115,000 CPU-hrs in a 178-Xeon cluster + 12,000 CPU-hrs in a 16 Opteron-848 cluster + 20,000 PC CPU-hrs

log(1-CHI) vs. log(radius): Berlinite AlPO4 to 100,000 Å Radius

Berlinite CHI vs. Radius
FastCHI Programming, Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen
157.2 CPU-hrs, Wintel, John Hooper

The berlinite unit cell is the alpha-quartz structure with a doubled c-axis, Si-O- vs. Al-O-P-0-. If atoms are taken to be anonymous mass the c-axis doubling vanishes. The smaller the volume/atom the faster CHI vs. radius asymptotically approaches CHI=1, and the less positive is the graph intercept of log(1-CHI) vs. log(radius).

LatticeVolume,
Å3/atom
Intercept
Tellurium  33.939 0.788633
Quartz  12.557 0.546185
Berlinite  11.720 0.506840

Te unit cell axes are a,b=4.456 Å, c=5.921 Å. The a,b-axes or c-axis were given large arbitrary variations and CHI was progressively calculated from 5.1-13 Å radius up to a 10,000 Å radius ball with 390-430 radius samplings for the test cases. (The mathematics of CHI is sensitive to shape but not to scale.)

log(1-CHI) vs. log(radius): Distorted Te Unit Cells

Te CHI in Distorted Lattices
FastCHI Programming by Matthew Francey
80-bit BigCHI upgrade and statistics package, Tony Lapen

Table XV. Te DISTORTED UNIT CELLS DATA

Latticea,b-axes,
Å
c-axis,
Å
Unit Cell
Volume, Å3
Slope InterceptTotal
Atoms
Theory, Te -2 arbitrary
Tellurium 4.456 5.921 101.816 -1.99950 0.786973 1.05·1015
Short a,b 0.913 5.921 4.274 -1.97716 0.000013 2.94·1012
Short c 4.456 1.184 20.360 -2.00775 0.302008 6.17·1011
Long a,b 17.287 5.921 1532.374 -1.97145 1.46827   8.20·109  
Long c 4.456 30.307 521.151 -1.98165 1.38036   2.41·1010

Te unit cell fractional coordinates are -0.2636, 0.0 ,1/3. The symmetry operations of space group P3121 then populate the unit cell with three atoms total (six half-atoms in the native structure). All 27 permutations of an arbitrary 0.389 change were evaluated through all three fractional coordinates added (P), subtracted (M), or unchanged (0). CHI was progressively calculated with 401 samplings from a 9 Å radius ball containing 84-94 atoms to a 10,000 Å radius ball containing 123.4 billion atoms in each test case.

Table XVI. Te DISTORTED FRACTIONAL COORDINATES

Fractional
Coordinate
ChangeValueMark
a/x    none
+0.389
 -0.389
 -0.2626
+0.1254
 -0.6526
0
P
M
b/y    none
+0.389
 -0.389
  0.0
+0.3890
 -0.3890
0
P
M
c/z    none
+0.389
 -0.389
  1/3
+0.722333...
 -0.055666...
0
P
M

Table XVII. Te DISTORTED FRACTIONAL COORDINATES DATA

Change,
a/x,b/y,c/z
SlopeIntercept Change,
a/x,b/y,c/z
SlopeIntercept
P 0 0 -2.01944 0.901199 M 0 0 -2.02110 0.811078
0 P 0 -1.99126 0.720836 0 M 0 -1.99929 0.779414
0 0 P -1.99906 0.798343 0 0 M -1.99987 0.798708
P P 0 -1.99798 0.775438 M M 0 -1.99103 0.718670
P 0 P -1.99552 0.864120 M 0 M -2.02444 0.848134
0 P P -1.99069 0.691879 0 M M -1.99191 0.750860
P M 0 -1.99173 0.759762 M P 0 -2.00055 0.783633
P 0 M -1.99590 0.865412 M 0 P -2.02192 0.844351
0 P M -1.99736 0.705094 0 M P -2.00640 0.783672
P P M -2.00477 0.779102 M M P -1.99885 0.708784
P M P -2.00139 0.764053 M P M -2.00811 0.795166
M P P -2.00692 0.805269 P M M -1.99445 0.741966
P P P -1.99071 0.744485 M M M -1.99097 0.692458
0 0 0
Tellurium
-1.99950 0.786973

Theory predicts an exact -2 slope. The less positive the intercept the more parity divergent is the lattice. Unit cell volume - smaller volume is more divergent - is much more important than atom placement within the unit cell of an acceptable parity pair space group.

Dr. Penelope Smith at Lehigh University notes that CHI is a 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 smallest vertex angle of a polyhedron (the supplement of its dihedral angle) defined by the c-axis helix,

log(1-CHI)= -2[log(radius)] + [(180-alpha) (pi)/60] - pi
(alpha is the smallest vertex angle in the helix. The slightly distorted tetrahedral O-Si-O helix angle is 110.56° vs. 109.47° undistorted)

log(1-CHI) = -2[log(radius)] + 0.494277
3-cm quartz test mass has CHI = 1 - 1.387·10-16, theory
3-cm quartz test mass has CHI = 1 - 1.535·10-16, graph fit

Subtended solid angle was tested against explicit calculation for quartz, tellurium, distorted tellurium unit cells, and berlinite aluminum phosphate (double length c-axis) that all express rigorous QCM DSI=0 COR=1 regardless of atom labeling or connectivity. It works to 10% difference/average values. As the standard deviation of calculated log(1-CHI) is typically 0.997 units, the two routes to fitting CHI give indistinguishable values.

The optimum crystal structure would have the

  1. parity space groups P3(1,2) P3(1,2)12 or P3(1,2)21
  2. smallest unit cell volume,
  3. smallest lattice volume/atom,
  4. unit cell with most equal inertial moments (all atoms weighted equally),
  5. largest helix angle extended along the c-axis,
  6. smallest a,b-plane helical radius,
  7. heaviest atoms.

The parity Eötvös experiment using parity pair single crystal tellurium, cinnabar, or quartz is astounding robust against a real world minor fraction of crystal structural imperfections and impurities. No heavy atom crystal lattice is substantively better than single crystal tellurium, cinnabar, or quartz for achieving maximum parity divergence as Eötvös experiment test masses.

Table XVIII. CALCULATED LATTICE PARITY DIVERGENCE, Te CYLINDERS

Hexagonal
Cylinder
Lattice
CHI
Total
Atoms
Unit
Cells
Volume,
Å3
Inertial
Disparity*
Tellurium   
unit cell positions
  7 helices x   5 atoms
19 helices x   9 atoms
37 helices x 13 atoms
61 helices x 16 atoms
91 helices x 20 atoms
127 helices x 23 atoms  
169 helices x 27 atoms  
217 helices x 30 atoms  
271 helices x 34 atoms  
 
0.018926
0.672986
0.949332
0.971947
0.988757
0.987677
0.986497
0.992871
0.983477
0.997870
 
     6
   35
  171
  481
  976
1820
2921
4563
6510
9214
 
    1
  12
  57
160
325
607
974
1521  
2170  
3071  
 
     102
  1,189
  5,811
16,344
33,164
61,844
99,256
155,051  
220,940  
312,711  
 
64.00  %  
13.55  %  
5.337%
4.132%
0.920%
0.930%
0.547%
0.796%
1.181%
0.226%
*(Imax - Imin)/Imin
Acta Cryst. 21 A46 (1966)

Selenium and tellurium are exceptional for small unit cell 100% heavy atom content forming identical helices. Next-nearest neighbor distances are almost identical (3.436 Å and 3.491 Å respectively), but selenium bonds are shorter (2.373 Å and 2.835 Å). Selenium's fragile lattice is disordered by physical manipulation, and the grey allotrope cannot be obtained as sufficiently large single crystals by solution growth. Tellurium and selenium display identical variation of CHI with spherical radius despite selenium having a 17.2% shorter pitch when scaled to the same helical radius. The mercury sublattice of cinnabar duplicates the tellurium lattice. It does not confer "tellurium breath" nor does it react with gold during vacuum gilding.

Large tellurium crystals obtain by Czochralski growth and annealing under hydrogen (50-200 hours at 320-380° C)[110] which further removes oxygen, sulfur, selenium, arsenic, lead... as volatile hydrides. Physical helicity has the same sense as optical rotation (±55.6°/mm at 5000 nm[95]). Crystals have easy cleavage planes along lattice indices. Typical dislocation densities are 1-3·104/cm3 with a hole concentration of 1014/cm3 (intrinsic p-type semiconductor). A single crystal tellurium right cylinder two centimeters in diameter and long contains 1.83·1015 helices for a summed axial length of 3.66·1013 meters (33.9 light-hours; 3.1 times the solar system's diameter). Each helix has 3.38·107 360° turns. Both elements' helices are remarkably isolated in space (view with CHIME plug-in; hold left mouse click and drag; right mouse click for menu; reduced window eases stereoimage fusion).

A classical Eötvös experiment opposes test masses' nuclear binding energies/baryon. Riley Newman's 2.2°K Eötvös balance[48] will oppose Be and Mg (weighted for natural isotopic abundance[111]). Neutron (939.565330 MeV) and proton (938.271998 MeV) average mass equivalent, weighted for 16 protons and 17.3202 neutrons (magnesium isotopic abundance), is 938.944286 MeV.

Be  = 6.462844 MeV/baryon
Mg = 8.265129 MeV/baryon
[Mg - Be]/[(17.3202n+16p)/33.3202] = 0.1919%
A one centimeter diameter alpha-quartz spherical test mass with 99.9726% parity-active mass calculates to CHI=0.999999999999998619 (1 - 1.381·10-15) of perfect CHI=1 geometric parity divergence. Parity Eötvös experiments offer 521 times larger property divergence than the planned composition experiment.

Cultured quartz is sold as multi-kilogram single crystals of both chiralities. The alpha-quartz lattice (view with CHIME plug-in) contains conflicting helices: 6-fold helices with long axes parallel to the z-axis are of opposite chirality to parallel 12-fold helices; 4-fold helices with the 12-fold helices' chirality form a 60° grid weaving through the xy plane. Care must be taken not to heat quartz near 573° C during working to avoid a first order twinning transition. Quartz is a potentially poorer geometric test mass than tellurium for its conflicting helices and lower average atomic weight. However, quartz is hydrothermally grown to 20 kg single crystals whereas tellurium is a monumental struggle for Czochralski growth to one centimeter single crystals.

Quartz crystal imperfections at all scales degrade its resonant acoustic quality factor Q,

Q = 2pi(energy stored)/(energy lost) each cycle.
Infrared OH-stretch absorption at 3410, 3500 or 3585 cm-1 is used to predict Q at 5 MHz[112], EIA Standard 477-1, JIS C 6704, and IEC 60758, e.g.,
Alpha = (A3500 - A3800)/Y
where A is absorbance at the given wavenumber and Y is the pathlength in centimeters of a Y-cut crystal (z-region material, as opposed to +x, -x, or s). Proton impurities in quartz terminate otherwise continuous ~Si-O~ helices as helix~O-H. Proton uptake varies as crystal growth rate. Slower growth gives higher Q.

Table XIX. MEASURED QUARTZ QUALITY

GradeMinimum Q
at 5 MHz
Alpha A3410A3500 A3585
Aa 3.8·106 0.015 0.075 0.026 0.015
A 3.0·106 0.024-0.033 0.082 0.033 0.024
B 2.4-2.2·106 0.045-0.050 0.100 0.045 0.050
C 1.8·106 0.060-0.069 0.124 0.060 0.069
D 1.4-1.0·106 0.100-0.120 0.145 0.080 0.100
E 1.0-0.5·105 0.160-0.250 0.190 0.120 0.160

Green's Function

A huge number of point masses (atomic nuclei; delta function sources) aggregate to form a body. The far field result is obtained by summing all the point masses of a system, each divided by its distance from a given point. This calculates the gravitational potential at that point. Given information about the source distribution one can then deduce the asymptotic behavior of the solution at distance. Given atomic lattice structure at fractional nanometer scales, one can obtain the gravitation of a test mass at Newtonian centimeter scales

The properties of a potential field depend only on position. The amount of energy gained or lost is independent of what path is taken start to finish. The integral of potential over path to give energy only depends upon the end points. Green's function is an integral kernel that can solve an inhomogeneous differential equation with boundary conditions. Green's function only depends on the distance between the source and the measured field points.

http://www.maths.soton.ac.uk/staff/Andersson/MA361/node46.html The gravitational potentials of configurations (triaxial ellipsoids, spheroids, spheres, disks) in Newtonian gravity, i.e. the potentials derived by integration of the Poisson equation Green's function 1/|r - r'| over the volume of the configuration, are well known. A Green's function solution is unique.

Consider a parity pair of alpha-quartz test masses in the weak field limit (Newtonian). Choose a z-axis position such that a silicon is centered at the same unit cell position in both space groups P3121 and P3221. Spatial distribution of the four oxygens around that silicon then define lattice chirality. Said oxygens are in positions "R" in the R-enantiomorph and "S" in the S-enantiomorph. Green's function G(x,y) has two arguments,

x = position of source
y = position of field potential measurement

Suppose G(x,y) varies in x on an angstrom scale. A function maximum falls on the central silicon. The next function maxima are at position "R" but miss position "S." There would then be a different answer for R- versus S-enantiomorphs. Can this obtain, and on what scale? Laplace's equation describing the behavior of gravitational potential is symmetric to parity inversion,

Laplacian

Laplacian2

Replacing (x,y,z) by (-x,-y,-z) does not change anything. The only component of gravitation is the radial one, and it only sees overall density (identical for enantiomorphic crystals). A gravitational parity anomaly must have non-Newtonian origins.

From general topological arguments,

G(x,y) = K[dist(x,y)](2-N)
where "K" is a scaling constant, "N" is the dimension of space (3 not 4 for traditional Relativity), and "dist(x,y)" is the distance between x,y (ignoring details about timelike components, retarded potentials, etc.). Where would subatomic-scale wiggles arise to allow a gravitational parity anomaly? Self-gravitation would require the nuclear masses to be enormous to give a macroscopic effect. The average nearest neighbor internuclear Si-O distance in quartz is 1.609 Å. Such small wavelength, high frequency components have no basis for origin. (A 0.161 nm photon has energy ~7.7 keV compared to Si-O bond strength of 8.3 eV.)

What are the fluctuation symmetry and radius scale of CHI as radius increases? The symmetry is explicit. A radial increment sufficient to add atoms to the existing solid sphere of lattice will always do it in mirror image along all coordinate axes, space group P3121 versus P3221.

The minimum radius increment that will add atoms to the existing solid sphere of lattice decreases as radius increases. It is remarkably small even at small radii. Green's function requirements for a gravitational parity anomaly - incommensurate structure and a characteristic scale much smaller than atomic lattice spacings - are fulfilled. 1 fm = 0.00001 Å.

Table XX. CHI FLUCTUATIONS AT SMALL RADIUS INCREMENTS

Radius
Interval, Å
Radius
Increment, fm
Plot
  100 - 101.50000
  200 - 201.50000
100.0
100.0
Graph 1
  100 - 100.45000
  300 - 300.04500
100.0
  10.0
Graph 2
  900 - 900.00450
2700-2700.00045
    1.0
    0.1
Graph3

Graph 1 shows a visibly different frequency of CHI fluctuations for 100 Å and 200 Å starting radii given the same radius increments. Graph 2 and Graph3 show that multiplying the starting radius by three and dividing the radius increment by ten gives an apparently constant CHI fluctuation frequency. Multiplying the radius by 3.16228 is stable over a broad observed range. If RÅ is the starting radius in angstroms and rfm is the corresponding radius increment in femtometers that always adds atoms, then

rfm = 106/(RÅ)2
A 0.5 cm radius test mass always adds atoms, has CHI fluctuations, with a radius increment of 4·10-10 fm. Oxygen and silicon nuclear diameters are 6.05 and 7.29 femtometers respectively. Thermal fluctuations are ~5% of bond length.

Chirality is not a point phenomenon. It is demonstrated that enantiomorphic centimeter-diameter quartz balls are deeply asymptotic to theoretical maximum parity divergence. As the diameter decreases the parity divergence decreases. Chirality and parity divergence abruptly vanish at scales smaller than a body-centered SiO4 tetrahedron, or a radius of 01.609 Å.

Table XXI. QUARTZ CHI AT SMALL RADII

ContentsAtomsCHI DSICOR
SiO4   5 0.000238 0.658392 2
SiO4Si4   9 0.606391 0 1
unit cell 11 0.408110 0 1

Green's function monotonically increases with the number of massed points included in a spherical envelope of increasing diameter. Parity divergence oscillates about its trend line with significant amplitude. Green's function analyses do not constrain parts-per-trillion gravitation parity anomalies.

Gravitational Scaling

Gravity may weakly couple to mass because it leaks across tightly compactified extra dimensions[41,113] only perceptible at or below their scale lengths[114] of 10[(30/n)-19] meters, or more explicitly[115]
R = [h-barc]/(Msc2)](Mp/Ms)(2/n)
R = compactified dimension radius
h-bar = Planck's constant/2pi
c = lightspeed
Ms = energy unification scale, ~1 TeV in M-theory
Mp = Planck mass, 1.221·1016 TeV
n = number of compactified dimensions
r = centers of mass separation
alpha = anomaly magnitude vs. Newtonian gravity; (n+1) or 2n (n-sphere or n-torus winding)
lambda = total anomaly radius, compactified dimension plus fringing

The "extra" six dimensions loosely comprise one charge dimension, two isospin dimensions, and three color dimensions. Their physical meaning is debatable. Below compactified dimensions' radius gravity non-classically varies as 1/r(2+n). At larger spans the anomaly exponentially decays as a Yukawa potential, [1 + alphae-R/lambda]. If compactified dimensions exist, spacetime can be homogeneous but not isotropic at small scales. The parity Eötvös experiment is a powerful test of spacetime isotropy at small scales.

Table XXI. COMPACTIFIED DIMENSIONS' EFFECTS

Compactified
Dimensions
Anomaly
Radius, Å
Empirical Observation,
Lower Limit
One 1021-1023 planetary orbits affected; wrong
Two 106-107 1/r2 deviates; not at 0.01 mm[113]
Three 10 crystal unit cells; (parity Eötvös)
Four 10-1-3·10-2 200x uranium nuclear diameter;
1/18 Bohr hydrogen diameter
Five 10-2-10-3 7x uranium nuclear diameter
Six 10-4 2/3 uranium nuclear diameter
Seven  2·10-5 proton Compton wavelength

Gravitation empirically varies as 1/r2 in 59 - 1150 Å effective gap separations (9600 - 10,690 Å center of mass separation) in atomic force microscope Casimir Effect experiments[116]. An active radius (with decay at its edges) implies a characteristic anomalous volume[117] whose contents' geometry interacts with its n-space container. Parity pair Eötvös experiments directly probe three to four compactified dimensions.

The emergent scale of alpha-quartz parity is ~0.484 nm. Given a gravitational parity interaction at much smaller scale, would it show? Consider a deep bed of close-packed bowling balls and a deep bed of monodisperse 1-micron radius silica balls. They have identical void space, 25.952 vol-% (1-[pi/3sqrt2]) and identically shaped voids. A methane molecule has an effective diameter of 0.000409 microns [118]. Methane roars through the bowling balls but exceedingly slowly diffuses through the silica balls.

For cubic or hexagonally close-packed identical balls with radius 1, tetrahedral holes will contain a ball with radius [(sqrt6)/2]-1 or 0.2247. Thus 1-micron radius silica spheres have voids holding 0.4495 micron diameter spheres. This is 1100X larger than methane's effective diameter. The four trigonal windows to each tetrahedral hole will contain a sphere with radius 0.1547 radius or 0.3094 micron diameter in our case. Methane can arithmetically freely pass through the silica ball bed by a generous factor of 756, though in the real world it certainly doesn't - not by a long shot. A gravitational non-interaction cannot be arithmetically discounted by "common sense."

For three mutually tangent spheres with radii r1 r2 r3 there is the Soddy circle solution:

r4 = (r1r2r3)/{r1r2 + r1r3 + r2r3 (±)2sqrt[r1r2r3(r1 + r2 + r3)]}
The positive solution is the inscribed fourth circle tangent to all three. (The negative solution is the circumscribed fourth circle tangent to all three.) Octahedral and cubic lattice holes are larger in radius and have larger radiused windows.

If gravitation did have an antisymmetric interaction with maximal parity-pair divergent crystal lattices, a very small scale for gravitational granularity might have difficulty "oozing through," though the spacetime interpretation (with Heisenberg uncertainty?) is certainly more recondite than 3-D diffusion.

Other Chiral Test Masses

Polycrystalline masses have large ratios of damaged area to volume (air interfaces, grain boundaries, dislocations, twinning). The Sagnac effect in ring laser gyroscopes[119] is a spacetime topological probe proportional to the scalar product of area and angular velocity vectors, and inversely to perimeter length (proportional to subtended area and compactness). Self-similar single crystals are superior to disordered materials as test masses.

Carbon, nitrogen, and oxygen atoms have similar small masses, and hydrogen is negligible. Light atoms have perceptible de Broglie wavelengths and participate in tunneling reactions[120]. Nucleic acids, proteins, sugars... CHNO organics in general are uninteresting for being much more like flat vacuum than tellurium:

Table XXII. MEAN ATOMIC WEIGHTS

SubstanceMean atomic
weight
Formula (space group)Comments
tellurium 127.60     Te               (P31,221) 99.98% chiral rest mass
PdSbTe 118.59     PdSbTe       (P213) 99.98% chiral rest mass
alpha-cinnabar 116.33     HgS             (P31,221) 99.97% chiral rest mass
alpha-quartz 20.03   SiO2            (P31,221) 99.97% chiral rest mass
benzil 8.09 C14H10O2   (P31,221) 99.97% chiral rest mass
tartaric acid 9.38 C4H6O6     2 chiral centers
cyclooctaamylose 7.72 C48H80O40   40 chiral centers
palytoxin 6.55 C129H223N3O54   64 chiral centers
insulin 7.37 C257H383N65O77S6   51 amino acids
somatotropin 7.16 C990H1529N263O299S7 191 amino acids

Chiral polymers[121] have loosely packed crystal lattices. Aggregated helicenes[122] can have [alpha]D=170,000° (1400°/mm gyrotropy neat) as can neat cholesteric (nematic-C) liquid crystals, but optical chirality is irrelevant. Binaphthyls are [5]helicenes less a ring junction. Jacobsen's ligand chelates uranyl[123], tin, or lead[124] with small geometric chiralities. Helicates[125] including iron[126] and silver helicates[127] ((R,R)-ligand gives (S)-helices) have large unit cell volumes. An alternating platinum-silver P61 helix[128] has a 7898.1 Å3 unit cell. A claimed intensely chiral octahedral complex[129], tris[1,2-dithiolatophenylenetungsten(IV)][130], has a 5649.0 Å3 unit cell.

Conclusion

All gravitation theories without exception are either symmetric or anti-symmetric to parity transformation. There are no observations constraining gravitation to be either. External symmetry parity is tied to physical geometric parity by the usual commutation relations. Geometric parity divergence can be ab initio quantitatively calculated. Test mass geometry is a natural test of spacetime geometry. Composition Eötvös experiments in crossed gravitational and inertial accelerations validate pursuit of parity Eötvös experiments. Parity pair tellurium, alpha-cinnabar, or alpha-quartz test masses offer at least 520 times composition test masses' differential input versus overall rest mass. The deep connection between explicitly calculated CHI and geometrically modeled CHI is reassuring.

Equivalence Principle violation falsifies metric theories of gravitation (leaving affine, teleparallel, and noncommutative theories unscathed). Contrasted test mass compositions afforded 400+ years of null results; calculated parity pair test masses have never been examined. A non-null parity Eötvös experiment supercedes parity-violating energy difference[131] explanations of biological homochirality.

Gravitation applies to free elementary particles (single neutrons fall[132]). Classical physics, relativity, and quantum mechanics are all point phenomena. A "point" spans the Planck length[133], (h-barG/c3)1/2 or 1.616·10-26 nm and thus implies a spherical volume approximating 2.21·10-78 nm3. Three atoms define an achiral plane. Geometric parity is an emergent phenomenon vanishing at smaller than unit cell scales. A tellurium unit cell contains three atoms as six half-atoms in the walls of a 0.1018 nm3 irreducible configuration. A non-null parity Eötvös experiment breaches "point phenomenon" by a volume factor of 1077, confronting Planck-regime physics with a bench top experiment.

"If we want to solve a problem that we have never solved before, we must leave the door to the unknown ajar," Richard Feynman[134]. The parity Eötvös experiment is a powerful new test of spacetime theories. Somebody should look.

Acknowledgements

FastCHI software (Matthew Francey), BigCHI upgrade and statistics package (Tony Lapen), BigCHI runtime optimization (John Hooper), parallelized CHIpir (John Edward Scott), Athlon/Linux calculation (Siddhartha Mukherjee to 100K Å radius, Steve Thompson and Robert Newson to 204K Å) Opteron-244/Linux calculation to 505K Å radius (John Edward Scott); Opteron-848 16-cluster calculation to 1.1 million Å radius (AMD Developer Center, John Edward Scott); space group geometry (JC Osborn), lattice manipulation software (John W. Hooper, Jon Wright), donated crystal data (Steve Simpson, Pieter Kuiper, Bruce Ravel; Armel Le Bail and Yvon Laligant; Armin de Meijere and Sergej Kozhushkov; Saicheong Chung, Christine Flaschenriem, Cambridge Crystallographic Data Centre; Inorganic Crystal Structure Database); locality violations (Steven Carlip); compactified dimensions (Mike Varney); electrodynamics (Florian Dufey); geocenter (Matthew Francey) and barycenter (Myles Standish) ephemerides; link updates (Shannon). The Adelberger and Luo physics groups for queuing the parity Eötvös experiment in P3121 versus P3221 alpha-quartz for empirical testing. We shall see.

Dr. Petitjean generously donated thousands of Alpha 2100 workstation and RS6000/Power3 CPU hours exploring CHI for molecules and crystal lattices. Advanced Micro Devices donated a cluster of 16 Opteron-848s for 900 hours. This author acknowledges a great debt of gratitude.

References


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