Abstract
Achiral spacetime curvature and chiral spacetime torsion are functionally equivalent. Predicted torsion background angular momentum divergences have not been observed. Rigid polyatomic opposite shoes divergently fitting a trace vacuum left foot are untested. Geometric Eötvös, calorimetry, and molecular rotation temperature experiments are presented. Interactions of torsion trace vacuum chiral anisotropy selective to hadronic matter are measurable, consequential, and more compatible with prior observations than postulated exact vacuum achiral isotropy. (PACS: 04.80.Cc, 11.30.Er)
1. Introduction
Postulating the Equivalence Principle (EP) elevates special relativity to general relativity (GR): All local matter identically vacuum free falls along paralleldisplaced minimum action trajectories. No measurable property of matter causes divergence. Composition and polarized spin Eötvös experiments; Nordtvedt effect and lunar laser ranging; pulsar binary systems with pulsars, white dwarfs, or solar stars all validate GR to limits of measurement in all predictions for all measurable properties and fields ̶ classical, quantum mechanical, relativistic, and gravitational (strong EP).
EinsteinCartanKibbleSciama gravitation (ECKS) contracts to GR if the EP is exactly true (achiral spacetime curvature). ECKS theory contains unobserved angular momentum EP violations given chiral spacetime torsion. Geometric chirality is an absolutely discontinuous symmetry outside Noether's theorems. Geometric chirality cannot be measured. Chiral spacetime torsion is a vacuum trace left foot. Rigid polyatomic opposite shoes geometrically test spacetime geometry to confirm or falsify an EP footnote. Examples for three of five test classes are presented.
2. Boundary Conditions
Massless boson photons detect no vacuum refraction, dispersion, dissipation, dichroism, or gyrotropy. Postulate contingent vacuum symmetries are exactly true for fermionic matter (quarks, hadrons). Parity violations, symmetry breakings, chiral anomalies, baryogenesis, biological homochirality, ChernSimons repair of EinsteinHilbert action are consistent with vacuum trace chiral anisotropy acting only upon hadrons.
Noether's theorems couple exact vacuum isotropy with conservation of angular momentum. Vacuum trace chiral anisotropy selective to hadronic matter leaks as MoND's 1.2×10^{10} m/s^{2} Milgrom acceleration. Dark matter curvefits the TullyFisher relation. Dark matter is empirically undetected. Planck satellite's 5.47:1 (dark matter):(baryonic matter) mass:mass ratio, including sun concentration, appears as less than 1.7×10^{−10} solar masses of dark matter within Saturn's orbital sphere. Without solar dark matter concentration, less than 6.8×10^{−11} solar masses of dark matter (N. P. Pitjev, 2013).
Geometric chirality is easily observed (no S_{n} symmetries) and easily calculated with QCM software (Petitjohn, 1999) but it cannot be measured. αQuartz and γglycine single crystals have calculated CHI = 1 DSI = 0 COR = 1 maximum geometric chirality, but it cannot be measured. Silver thiogallate, AgGaS_{2} in achiral space group (#122), rotates plane polarized sodium Dline light 522°/mm along [100] at 497.4 nm (J. Etxebarria, 2000). P3_{1}21 (#152) quartz and P3_{2}21 (#154) berlinite, mirror image atomic helices, are both levorotatory (A. M. Glazer, 1986). Achiral PhCOCH_{3} impurity changes resolved PhCH(OH)CH_{3} optical rotation (S. Yamaguchi, 1973). CIP notation flips within unchanged geometry, L(2S)alanine but L(2R)cysteine. Solution specific rotations ignore atomic mass distribution, Fig. 1 and Table 1.
Figure 1: Similar absolute configuration mass distribution chiral molecules
Table 1: Uncoupled mass distributions and specific rotations
Formula 
Formal name 
Specific rotation [α]_{D} (°cm^{3}/dmg) 
C_{7}H_{12}O^{a} 
(1R,2S,4S)bicyclo[2.2.1]heptan2ol 
2.3 
C_{7}H_{10}O^{b} 
(1R,4S)bicyclo[2.2.1]heptan2one 
29.8 
C_{9}H_{14}O^{c} 
(1R,4R)7,7dimethylbicyclo[2.2.1]heptan2one 
+73.9 
C_{7}H_{8}O^{d} 
(1S,4S)bicyclo[2.2.1]hept5en2one 
1146. 
C_{6}H_{7}NO^{e} 
(1R,4S)2azabicyclo[2.2.1]hept5en2one 
565. 
^{a}European patent EP0801135 A1 (1997); ^{b,d}J. Phys. Chem. A, 2006: 110(51), 13995  14002, doi:10.1021/jp0655221; ^{c} J. Org. Chem., 1974: 39(12), 1653  1656, doi:10.1021/jo00925a011; Ann. Acad. So. Fenn , Ser A2, 1961: 105(1), 22;^{ e}J. Chem. Soc., Chem. Commun., 1990: 1120  1121, doi:10.1039/C39900001120
3. Dropping Opposite Shoes
Opposite shoes embed within chiral vacuum background (mount a left foot) with different energies. They vacuum free fall along nonidentical minimum action trajectories, exhibiting EP violation. Achiral symmetry plus lattice discretization fails (H.B. Nielsen, 1981). Periodic lattices and quantum spacetime attack vacuum isotropy (Fritz, 2013). Crystallography's opposite shoes are visually and chemically identical, single crystal test masses in enantiomorphic space groups (Souvignier, 2003) (A. F. Palistrant, 1991). Two examples are P3_{1}21 versus P3_{2}21 αquartz or P3_{1} (#144) versus P3_{2}(#145) γglycine. Eötvös experiments are 5×10^{14} difference/average sensitive (T. A. Wagner, 2012). Compositionnulled controls are αquartz versus amorphous fused silica, γglycine versus achiral P2_{1}/n (#14, nonstandard setting) αglycine.
Figure 2: Two geometric Eötvös experiments
The αquartz unit cell is 0.113 nm^{3} volume. 40 grams net as 8 single crystal test masses compare 6.68×10^{22} pairs of opposite shoes (pairs of 9atom enantiomorphic unit cells, the test mass array cube's opposite vertical sides), Fig.2.
Composition and polarized spin Eötvös experiments' net active mass fractions are negligible. Titanium versus beryllium nuclear binding energy is 0.0023974 net active mass fraction (weighted isotopic abundances), the largest relative divergence composition contrast compatible with hard vacuum. Theoretical 100% polarized spin (empirically antiferromagnetic, space group (#225)) octahedral MnO lattice, five parallel unpaired electrons/Mn(II) in the least massive formula unit, is 0.000038669 net active mass fraction, Eq. (1).
5( 5.485799×10^{4} amu/electron)/( 54.93805 amu/Mn + 15.994915 amu/O) 
(1) 
Net active mass fractions in any geometric test of spacetime geometry, nucleus' relative positions in space , are shown in Table 2. They are 400 times the net active mass fraction of the most divergent composition Eötvös experiments.
Table 2: Eötvös experiment net active mass fractions
Net active mass fraction 
Space groups 
Single crystal 
Material 
0.999708 
P3_{1} 
γglycine 


P3_{2} 


0.999713 
P3_{1}21 
benzil 


P3_{2}21 


0.999726 
P3_{1}21 
αquartz 


P3_{2}21 


0.999775 
P3_{1}21 
tellurium 


P3_{2}21 


0.0023974 


Be versus Ti 
0.000038669 


MnO undecatiplet 
4. Melting Opposite Shoes
An Eötvös experiment is geometryconserving. A calorimetry experiment is geometrydestroying. It melts opposite shoes into identical achiral socks, all on a vacuum left foot. Divergent chiral energies of fit transform into a common achiral state. A timevarying EP violation, Earth's inertial rotation versus gravitational orbit, is added to a constant differential chiral vacuum background insertion energy over a 24hour day.
The geometric calorimetric test masses are 94.85 °C melting point, space group P3_{1}21 versus P3_{2}21 single crystal benzil, Fig. 3. QCM software calculates CHI = 1 DSI = 0 COR = 1, maximum mathematical chirality (opposite shoes), for single crystals. Melt, solution, and gas phase host isolated achiral molecules (socks).
Figure 3: Crystalline benzil
Eötvös experiment detection threshold is 5×10^{14} difference/average observing mass/mass divergence. Benzil's enthalpy of fusion, ΔH_{fusion} = 110.6 J/g (G. D. Gatta, 2006) observes mc^{2} massequivalent divergence. The diurnal sinusoidal EP component is Eq. (2.1)  (2.3),
ΔE = (5×10^{17} kg/kg)(299,792,458 m/sec)^{2} sensitivity massequivalent 
(2.1) 
E = 4494 J/kg or 4.494 J/g 
(2.2) 
(4.494 J/g)/(110.6 J/g) = 4.06% ΔΔH_{fusion} 
(2.3) 
Commercial differential scanning calorimeters (DSCs) have 0.1% precision. A geometric calorimetry experiment offers increased sensitivity to vacuum chiral anisotropy toward matter compared to a composition EP Eötvös experiment, Eq. (3), active mass fraction ratio multiplied by signal/instrument resolution,
(0.999713/0.0023974)(0.0406/0.001) = 16,900× sensitivity, 
(3) 
Two horizontally abutted DSCs' sample ports define a geographic northsouth line. Each holds a ~3 mm diameter ~17 mg benzil single crystal sphere with sample carriers crimped against sublimation. One sample port consistently contains one crystal in space group P3_{1}21 and the other sample port one crystal in P3_{2}21. are simultaneously run. New crystals are run at halfhour intervals for 24 hours inclusive. Given sample geographic northsouth alignment and Earth's inertial spin plus gravitational orbit angular momenta, a coordinate system is defined, cycling over 24 hours . For local geographic midnight  achiral, signal node; 0600 hrs  chiral, antinode; 1200 hrs  achiral, node; 1800 hrs  opposite chiral, opposite antinode for EP interaction.
Repeated the next day with eastwest alignment. ΔΔH_{fusion} will have a six hour phase shift. ΔH_{fusion} of finely powdered racemic benzil is baseline, Fig. 4,
Figure 4: Crystalline to molten benzil transition
Case 1: ΔΔH_{fusion} = 0. Zero net signal confirms vacuum achiral isotropy toward matter. P3_{1}21 and P3_{2}21 single crystal ΔH_{fusion} are identical to that of powdered racemic benzil. Values do not change versus time of day and samples' NS or EW geographic orientation.
Case 2: ΔΔH_{fusion} > 0. Net nonzero signal confirms vacuum chiral anisotropy toward matter. Chemically identical enantiomorphic crystals, embedded within a resolved vacuum chiral background (opposite shoes on a left foot), melting into a common achiral state (socks), display different enthalpies of fusion. At least one ΔH_{fusion} will be different from that of powdered racemic benzil.
Case 3: ΔΔH_{fusion} ≠ 0, sinusoidally varying with time of day. The EP has a chiral geometric violation. The angle between Earth's inertial spin and gravitational orbit rotates 360°/24 hours. This sources a composition Eötvös experiment signal. Add P3_{1}21 and P3_{2}21 benzil test masses aligned NS and the coordinate frame cycles chiral, achiral, opposite chiral, achiral every 24 hours.
Case 4: (Case 2)+(Case 3). The vacuum is chiral anisotropic toward chiral atomic mass distribution and the EP has a chiral geometric violation. Dark matter is in fact Milgrom acceleration, SUSY is fundamentally wrong, general relativity is in fact superset ECKS gravitation, quantum gravitation must be rederived to allow for chiral geometric EP violation. No prior observation in any venue at any scale is contradicted.
5. Spinning Opposite Shoes
Observe high resolution rotational spectra of a ~1 kelvin molecular beam racemic rotor mix. Isotropic vacuum elicits no chiral divergence, giving degenerate spectra. Trace chiral vacuum background splits enantiomers' energies (opposite shoes on a left foot) (Moffat, Six lectures on general fluid dynamics and two on hydromagnetic dynamo theory, 1977). Two rotational spectra, perhaps with unequal rotation temperatures, appear. Microwave (μwave) and infrared (IR) spectrometries (dipole transitions) and Raman spectrometry (quadrupole transition) are diagnostic. Centripetal distortion induces a small dipole moment in some zero dipole moment symmetric molecules, e.g., tetrahedral methane spinning about its C_{3} axis. Point group O_{h} sulfur hexafluoride spinning about its C_{4} axis remains unpolarized, by symmetry, as do point group D_{3} molecules spinning about their C_{3} axes.
Required are rigid cage molecules with no pendent spinnable substituents. The EP is compositioninert. Chiral centers should be geometrically homochiral ignoring composition. Rotational symmetries render skeletal positions equivalent. Favor molecules with large CHI values in QCM software. Oblate symmetric tops have simplified rotational spectra. Good yield multigram syntheses afford μwave, IR, and Raman samples. Pentacyclo[6.3.0.0^{2,6}.0^{3,10}.0^{5,9}]undecane, D_{3}trishomocubane, and derivatives (D. I. Sharapa 2012) (I. A. Levandovsky 2010) (W.D. Fessner 1986) (G. J. Kent 1977) qualify, Fig. 5, Fig. 6, and Table 3. D_{3}trishomocubane has 8/11 skeletal atoms being explicit chiral centers. One C_{3} plus three C_{2} rotation axes define three unique skeletal positions: [4,7,11; achiral], [2,9; unnameable chirality], [1,8,6,5,3,10; Rconfiguration displayed]
Figure 5: D_{3}trishomocubane, numbering and CIP chirality
Figure 6: μ wave, IR; and Raman test molecules
Table 3: μ wave, IR; and Raman test molecules
Method 
Molecule 
Moments of Inertia, amuÅ 
~ Dipole Moment μ, D 

I_{x} 
I_{y} 
I_{z} 

μwave, IR 
4one 
282.018 
421.039 
523.570 
3.0 HyperChem 
μwave, IR 
4oxa 
274.348 
278.637 
375.450 
2.4 MARVIN 
Raman 
hydrocarbon 
282.518 
282.518 
384.000 
0.0 symmetry 
Raman 
4,7,11trioxa 
264.653 
264.653 
359.004 
0.0 symmetry 
Raman 
4,7,11trione 
510.318 
510.318 
844.539 
0.0 symmetry 
Raman 
fluorocarbon 
1710.24 
1710.24 
2091.47 
0.0 symmetry 
Table 4 lists some test molecules' QCM outputs. CHI: normalized geometric chirality, 0 to 1. DSI: direct symmetry index, defined in (Petitjohn, 1999) section VI. 0 is best. COR: graph isomorphisms, "correspondences," 1 (identity) to unbounded integer. Smaller is better. Protein amino acid phenylalanine is chiral. Its αcarbon chiral center bears four very different groups: hydrogen, carboxyl, amino, and benzyl. QCM shows its 23 atoms in 3space sum to small geometric chirality. 4oxaD_{3}trishomocubane's 23 atoms sum to a large CHI value, as do its isolated spin0 atom skeleton and its isolated spin½ hydrogens or fluorines. 4,7,11trioxaD_{3}trishomocubane's 19 atoms sum to a nearperfect geometrically chiral object.
Table 4: QCMcalculated geometric chiralities
Molecule 
Moiety 
CHI 
DSI 
COR 
hydrocarbon 
whole 
0.628218 
0.000000 
48 

skeleton only 
0.533027 
0.000000 
6 

hydrogens only 
0.713354 
0.000000 
1 
fluorocarbon 
whole 
0.649502 
0.000000 
48 

skeleton only 
0.506504 
0.000000 
6 

fluorines only 
0.758557 
0.000000 
1 
4oxa 
whole 
0.712524 
0.000000 
8 

skeleton only 
0.542840 
0.000000 
2 

hydrogens only 
0.800458 
0.000000 
1 
4,7,11trioxa 
whole 
0.959321 
0.000000 
6 

skeleton only 
0.484981 
0.281997 
4 

hydrogens only 
0.562643 
0.000000 
6 
phenylalanine 
whole 
0.058600 
0.013103 
8 
5a. Rotational Microwave and Infrared
Heliumentrained vapor of racemic ketone or ether is de Laval vacuum supersonic expanded, reducing rotation temperature to ~1 kelvin. The cryogenic molecular beam passes through a chirpedpulse FT μwave spectrometer (N. M. Kidwell, 2014) (I. A. Finneran, 2013) or FTIR spectrometer in kind to capture its high resolution rotational spectrum. Achiral isotropic vacuum displays enantiomers' degenerate indistinguishable spectra. Trace chiral anisotropic vacuum splits transition energies and/or rotor rotation temperatures, displaying nondegenerate one spectrum for each enantiomer, a left foot bearing opposite shoes.
5b. Rotational Raman Spectra
Molecules lacking both a permanent and a centripetallyinduced dipole moment display rotational spectra with high resolution Raman scattering. SF_{6} centripetal distortion generates no dipole moment, but exquisite rotational spectra (V. Boudon L. M., 2014) (V. Boudon P. A., 2013) are obtained. D_{3}Trishomocubane has small anisotropic polarizability. Its 4,7,11triketone and 4,7,11trioxa derivatives have large anisotropic polarizabilities. Three synthetic approaches to the triketone are past literature (A. P. Marchand, 1987), contemporary methylene to carbonyl oxidations (M. S. Chen, 2010) (H. C. Tung, 1992), and 5methylene1,3cyclopentadiene or 5benzylidene1,3cyclopentadiene (for stability) starting material followed by oxidative cleavage to the carbonyl after photocyclization. The final carbonyl is placed by oxidation. 4,7,11trioxaD_{3}trishomocubane has nearperfect geometric chirality. Synthesis begins with furan plus pyrone rather than cyclopentadiene plus benzoquinone, placing 4,7oxasubstitution . Follow with carbony oxidation l (e.g., Baeyer–Villiger) to the lactone, conversion of the carboxyl to a leaving group with ring opening and epimerization, then endoalkoxide displacement.
Example: Phenyl (4nitrophenyl, 2,4dinitrophenyl, etc.) anion attack upon the lactone obtains the endoalcohol plus basepromoted steric hindrancedriven αcarbon equilibrium inversion to the exophenylketone. 1,2Ethanedithiol obtains the dithioketal, followed by dimethyldioxirane oxidation to the strongly electronwithdrawing disulfone (pK_{a} ~12.5, and lowered by a nitrophenyl). endoAlkoxide displacement at the αcarbon closes the ring ether by displacing the (nitro)phenyl disulfone anion.
5c. Estimated signal amplitude
Boltzmann's constant offers vacuum coupling divergence detection for racemic molecular rotors as energy/mass, as rotational temperature. A one kelvin racemic molecular beam (pairs of shoes) is differentially sensitive to being embedded within vacuum resolved chiral anisotropy (left feet), Eq. (4.1)  (4.3). Differential energy misfits for 4oxaD_{3}trishomocubane (C_{10}H_{12}O; μwave, IR) Eq. (4.4), D_{3}trishomocuban4,7,11trione (C_{11}H_{8}O_{3}; Raman) Eq. (4.5), and perfluoroD_{3}trishomocubane (C_{11}H_{14}; Raman) Eq. (4.6) at Eötvös experiment detection threshold 5×10^{14} difference/average observing mass/mass divergence are
(Δmc^{2})(molar mass)/(molecules/mole)(Boltzmann's constant) = temperature 
(4.1) 
(4.49378 J/g)(molar mass)/(6.02214×10^{23}/mole)(1.38065×10^{23} J/kelvin) 
(4.2) 
(0.540478)(molar mass) = kelvins 
(4.3) 
(0.540478) (148.202 g/mol) = 80 kelvins, 4oxaD_{3}trishomocubane 
(4.4) 
(0.540478) (188.179 g/mol) = 101 kelvins, D_{3}trishomocuban4,7,11trione 
(4.5) 
(0.540478) (398.095 g/mol) = 215 kelvins, perfluoroD_{3}trishomocubane 
(4.6) 
Relative orientation of Earth's spin inertial acceleration and orbital gravitational acceleration varies sinusoidally over 24 hours. Given molecular beam geographic northsouth alignment, for local geographic midnight  achiral, node; 0600 hrs  chiral, antinode; 1200 hrs  achiral, node; 1800 hrs  opposite chiral, antinode for EP interaction. Eötvös experiment minimum detectable divergence corresponds to large net outputs as μwave, IR, and Raman racemate rotational temperature divergences even for modest coupling constants.
6. Historical Precedent
Euclid is incomplete (cartography), then János Bolyai. Newtonian physics is incomplete, then relativity and quantum mechanics. The Dirac equation failed for proton magnetic moment (Otto Stern), then quarks. Particle theory was mirrorsymmetric, then Chen Ning Yang and TsungDao Lee. Examine vacuum symmetry toward extreme enantiomorphic atomic mass distributions. Observe whether vacuum is rigorously achiral isotropic toward fermionic matter as it is toward boson photons. Nothing prohibits empirically falsifying a founding postulate, no matter how "obvious and logically true" it appears to be.
CONCLUSION
Three classes of chemical tests are presented to challenge a physics vacuum symmetry postulate. Existing bench top equipment can heal accumulated elegant but empirically inert quantum gravitation and supersymmetry theories. No prior observation in any venue at any scale would be contradicted by detected vacuum trace chiral anisotropy acting only upon hadrons. Untried geometric Eötvös, calorimetry, and rotational spectrometry are orders of magnitude more sensitive than zero net signal composition and spin Eötvös experiments. Some 45 years of speculation beginning with Leonard Susskind and Murray GellMann in a 1970 Coral Gables, Florida stalled elevator can be reshaped in one day. Perform geometric tests of spacetime geometry, for the worst they can do is succeed.
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