Discrete Expressions for the Gravitational Constant Offer Improved Precision

 

G=6.6740779428(56) 10-11m3/kg s2

2018 CODATA Measurement: 6.67408 10-11 m kg-1 s-2
MQ Calculation: 6.67407 10-11 m kg-1 s-2

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Inputs

  • θsi can be measured as the polarization angle of quantum entangled X-rays at the degenerate frequency of a maximal Bell state. As an angle θsi=3.26239 rad ± 2 μrad; as a momentum θsi=3.26239030392(48) kg m s-1 and with respect to the Target Frame, θsi has no units. The relation of angle and mass is mathematically demonstrated, as well, by No-Ping Chen, et. al.
  • c is the speed of light which may also be written as c=nLlf/nTtf=299,792,458 m/s such that nL=nT=1 is physically significant.

Terms

  • QL is the fractional portion of a count of lf when engaging in a more precise calculation.
  • nLr describes the count of lf representative of the position of an observable with respect to the frame of a center of mass.
  • S is the unknown constant when resolving a description of gravity, latter replaced with θsi.
  • r is the distance between an observer and a target.
  • G is the gravitational constant, 6.6740779428(56) 10-11m3kg-1s-2 such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure not at the limit (e.g., G=6.6738448362(53) 10-11m3kg-1s-2 at the blackbody demarcation).

Calculations


Experimental Support

P. Mohr, B. Taylor, D. Newell, CODATA Recommended Values of the Fundamental Physical Constants: 2014, p. 3, (2015), arXiv: 1507.07956v1, doi:10.1063/1.4954402.


Discussion

When classical expressions are written in terms of physically significant units of measure and counts of those measures, an expression and value for the gravitational constant can be resolved with quantum precision. There are two prerequisite inputs; they are c and θsi.

A Measurement Quantization (MQ) description of the gravitational constant is resolved as a difference between two frames, the discrete Measurement Frame of the observer and the non-discrete Target Frame of the universe. We describe this difference as a function of the non-discrete count QL of a more precise calculation.

To this we add a new constant. Describing momentum with respect to the Measurement Frame of the observer, θsi=3.26239030392(48) is also the angular measure of the polarization field with respect to the plain of X-rays when at their degenerate frequency. Measurements were first carried about by Shwartz and Harris and published in their 2011 paper entitled, Polarization Entangled Photons at X-Ray Energies.

Application of MQ offers us the ability to resolve descriptions of phenomena not just quantum in measure, but also cosmological. For instance, one may replace a description of fundamental mass mf with terms defined with respect to the universe(1,Eq. 88). As demonstrated below, a description of fundamental mass then becomes a description of the diameter and age of the universe.

FundamentalMass.png

Importantly, scaling between frames is physically supported both by paradox and cross-epoch measurements. Paradoxes include descriptions of a speed of light that differ as a function of measurement scope (i.e., quantum, macroscopic and cosmological). As MQ recognizes that the speed of light is constant for all frames this paradox is avoided, but should scaling not be physically valid, this issue would have to be addressed.

A second support regards descriptions of properties of early universe phenomena compared to their measure in the present epoch. For this we offer several descriptions, a majority describing the Cosmic Microwave Background (CMB). We can resolve a no parameter calculation of its age, quantity and present-day density and temperature. Correspondence between MQ calculations and measurement all align digit-for-digit and within the margin of uncertainty associated with each measure.

In short, we find support for scaling of the fundamental expression across all measurement domains.

[24] D.J. Fixsen, The Temperature of the Cosmic Microwave Background, (2009), arXiv: 0911.1955, doi:10.1088/0004-637X/707/2/916.

[25] Mohamed Abdullah, Anatoly Klypin, Gillian Wilson, Cosmological Constraints on Ωm AND σ8 from abundances using the GALWCAT19 optical-spectroscopic SDSS catalog (2020), arXiv:2002.11907, doi:10.3847/1538-4357/aba619.

[26] Planck Collab. 2018 Results VI (2018), arXiv:1807.06209, doi:10.1051/0004-6361/201833910.

Let us now turn our attention to the properties of gravity. While we have used the equality symbol in the initial expressions to describe gravitational curvature(1,Eq. 8), said descriptions carry a prerequisite understanding of the conditions associated with their measure. The reason for this relates to the length contraction associated with discrete measure.

We find the effects of length contraction physically relevant in all descriptions incorporating the measures of G or ħ(4,Sec. III.E). Specifically, the value of G is a function of distance associated with a phenomenon. In that G is typically measured macroscopically, for instance, where the measure of G is resolved at a distance of one meter, then the effects described by the Informativity differential are far less than the precision afforded. The same would be true if we measured the reduced Planck constant as a function of a quantum distance (i.e., blackbody radiation). Combining these two measures combines two frames of reference and therein we find disagreement in solutions to component terms. For instance, using Planck's unit expressions to solve for the value of G will produce different results.

We can solve the problem by resolving a value for G and ħ at a common distance, both as a function of an electromagnetic phenomenon (a quantum distance) or both with respect to a macroscopic distance. Typically we solve measures at their upper count limit. For the latter case, we do not italicize the term.

A second observation is realized when G is described in terms of the fundamental measures(4,Eqs. 67-79). We discover the expression for G has two components. First, there is the length to time ratio (lf/tf), which we refer to as the length frequency. Second, there is the mass to time frequency, which we refer to as the mass frequency (mf/tf). While such descriptions are sometimes presented in terms of fundamental measures, it is understood that we are referring to the corresponding count terms (i.e., the count frequency). As such, the gravitational constant is length frequency cubed - once for each spatial dimension - divided by the mass frequency.

The expression describes length, mass and their relation to time. The correlation between mass and time is one dimensional. But, the correlation between length and time is three dimensional, representative of each of the three physical dimensions that describe space(2,Eq. 102).

We will lastly note, there are mathematical arguments that support greater dimensions in a description of space. MQ offers new tools to physically validate such claims. Where the Informativity differential is a function of the Pythagorean theorem - dimensions two or greater - it follows that any mathematical model employing dimensions 2 or greater with respect to any dimension, must also account for the contraction of measure in that same dimension.

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