NEWTON'S GRAVITATIONAL CONSTANT MEASUREMENTS

The speed of light 'c' has been measured to an accuracy of 8 decimal places and Planck's Constant 'h' is known with an error not exceeding one part per million.

This is not so for Newton's Gravitational Constant 'Big G'.

The National Bureau of Standards (NBS) in Gaithersburg, Maryland, US began measuring 'G' in the 1930's to establish the Luther-Towler-Number LTN=6.67259x10^-11  G-units (m³/kg.s²). 

So it stood until 1994, when the renowned PTB in Braunschweig, Germany's Standards Laboratory measured G much higher, differing in the 3rd decimal place.

Then New Zealand's Measurements Standard Laboratory published a value significantly below the LTN and the University of Wuppertal derived a value in between the NZ one and the LTN.

Notwithstanding the ever improving technological advances and measuring techniques; using torsion pendulums, tungsten cylinders or suspended or accelerating testmasses; 'Big G' has proven to be intractable to conformity. Two of the latest measurements are 6.67327x10^-11  and 6.6742(10)x10^-11 G-units and values by no means definitive.

What is going on?

Shifting heavy objects in the vicinity of the test apparatus seems to influence the atomic structure of the testmasses, irrespective of the isolated environment created for the testing conditions.

The following treatise shall resolve the conundrum and illustrate the unruly behaviour of 'G' as a consequence of the initial boundary conditions for the universe's subsequent evolvement.

It shall indicate that even a 'massless' universe would contain a diminished G-component as the electric permittivity of a massless macroquantised (Hawking) BlackHole and that the present dilemma derives from a finestructure of the nucleonic constituents, which, by definition, must comprise the testmasses.

A precise measurement so would rely on an unambiguous calculation for two neutronic restmasses, a condition which we shall show to be unachievable, because of the nature and interrelationship between the parameters of inertial mass and those of electromagnetic charges.

There are actually TWO G-Constants, one constant as say Go in the quantum gravity models defining the Planck-Scale and another one used for agglomerated masses, say two masses M1,2 being comprised of say N1,2 neutrons.

We shall find a maximised neutron mass mnmax (or nucleon mass via the beta-minus decay of the weak nuclear interaction giving the quark-lepton content of the Standard Model in say protons, electrons and antineutrinos) and a minimum neutron mass mc, and the latter being a direct consequence of the Planck-Mass mP=√(hc/2πGo) from the gravitational finestructure G-alpha=2πGoM²/hc.

So setting G-alpha to unity gives the Planck-Mass.

But setting M=mc gives the G-alpha as the force-interaction ratio between the electromagnetic alpha, defined as:  alpha=2πke²/hc with k=1/4πεo.

One can so immediately calculate the minimum neutron mass as the expression:

mc=√{G-alpha.hc/2πGo}.

Now the ratio between the electromagnetic- and the gravitational interaction strengths is measured and of the order of alpha/G-alpha~10^-39 and one can actually define the G-alpha as a function of alpha and as G-alpha=alpha¹⁸, using the string parameters of Quantum Relativity.

This defines the minimum neutronmass mc explicitely as: mc=√{ke².alpha¹⁷/Go}.

In string parameters, the unification condition for the interactions at the string energy scale demands kGo=1 for a mc=[e/Go].alpha^8.5=9.9247246..x10^-28 kg*. This represents so 58% of the neutron (or nucleon) mass as measured today and is the actual minimum neutron mass.

Now the truly CONSTANT GM² structure in say Newton's Law,  is given by the product Gomc²=1.094446..x10^-64 Nm².

This however is finestructured in introducing a maximum neutron mass given in a unification condition, known as the Euler Identity: X+Y=XY=-1=i²=℮^iπ and applying the absolute value of unitised 1.

We write: Gomc²={GoX^n+k}.{mcYⁿ}.{mcY^k}=Gm.mnmax.mnmin and where Gm is the actual G value as measured and which has proved difficult to do so in the laboratories.

So the applied G value is: Gm(n)=Go.X^n+k and where n is a cycletime n=Hot for a nodal universe with dn/dt=Ho the nodal Hubble Constant Ho=c/Rmax for a Hubble radius Rmax.

The applied Gm so ALWAYS engages a maximised neutron mass (calculated as{mcYⁿ}~ 1.7115x10^-27 kg in string parameters for a present cycletime coordinate np=1.1324..) AND a minimised neutron mass (calculated as {mcY^k}).

The value of k is  so determinative for Gm and differs over the evolution of the universe with respect to cycletime n and as finestructured for an AVERAGE G-value (Gav) obtained in using the geometric mean for the neutron masses in extremum (minmax productation).

One can easily calculate Gav=Go.Xⁿ=6.44317..x10^-11 G-units for a geometric neutron mass product of mnmax.mnmin=mc².Yⁿ =1.69861...x10^-54 kg² for the constancy condition of Gomc²=1.094446..x10^-64 Nm² and omitting the k-factorisation.

But this averaged G value applies for a massless universe under the initial unification condition of the finestructures described in Go.k=1 or Go=4πεo (using Stoney Units for the Planck-Scaling of the chargequantum e).

So BECAUSE an initial mass seedling Mo={mc.mP/me}√E ~ 1.8137..x10⁵¹ kg* became transformed in the de Broglie phase inflation from its preinertial state as gravitational mass into the state of inertia (this is called the Big Bang for a spacetime quanta counter E); this 'Principle of Equivalence' introduced the hitherto massless 'ylemic' 'neutron bosons' as dineutronic states, which under the Higgs mechanism became fermionic and established the mass seedling Mo as the primordial neutron matter, then decaying via beta minus decay into the observed matter in the universe (there was no antimatter).

Subsequently the EMERGENCE of inertial mass under c-invariance also introduced a finestructure for 'G' as described in the above.

One can determine the value of k from finestructuring the critical masses Mo, M∞ and MHawking as boundary Black Hole masses coupled  to the quantum minmax neutron masses.

For curvature radius Rmax and the critical density  ρc=M∞/Vmax=3Ho²/8πGo the Schwarzschild metric gives M∞=Rmax.c²/2Go=c³/2GoHo= ~ 6.47058..x10⁵² kg*.

For the curvature radius RSarkar=2GoMo/c², we have the deceleration parameter qo=½Ωo=Mo/2M∞=2GoHoMo/c³ ~0.014015..     and which so determines the 'missing mass' in the universe to be a consequence of the initial boundary conditions set by the de Broglie inflation and the overall Black Hole evolution of the stringed parameters.

The Mass-Temperature modulus of Stephen Hawking determines MHawking=Constant/THawking for a boundary condition of maximised Black Hole Mass for a minimised Black Hole Temperature in MHawkingTHawking=constant for Constant=hc³/4πGok and k the Stefan-Boltzmann constant.

The relationship is given in superstring (Planck) parameters by

Mmin.Tmax={c/2π}².Mmax.Tmin=hc³/4πGok= ½mP.TP and TP the Planck Temperature TP=mP.c²/k.

This sets the Hawking-Gibbons thermodynamic temperature minima for To=constant/Mo ~ 5.03..x10^-28 K* and T∞=constant/M∞ ~ 1.41..x10^-29 K*.

As the minimum macro Black Hole has Schwarzschild metric λmin/2π=2GoMmin/c² for Tmax=hfmax/k=hc/λmink; and modular duality requires the unification condition for the minimum curvature to relate to a maximum curvature in Rmin=λmin/2π=1/Rmax  or Rmax=2πλmax,  as Rmin.Rmax=1.

In gauge bosonic string parameters, this modular duality then is given in Emax=hc/λmin=mmax.c²=kTmax and Emin=hc/λmax=mmin.c²=kTmin and in the invariance of the lightspeed parameter c as c=fmaxλmin=1/fminλmax or the dimensionless unification conditions:  Emax.Emin=h² and

Emax/Emin=fmax²=1/fmin²={c/λmin}²={c/2πRmin}²=={cRmax/2π}²={cλmax}².

This gives a proportionality: mmax.Tmin=mmin.Tmax for the gauges, which is however modified in the dimensionless factor {c/2π}² for the Black Hole masses for the given temperatures, as bosonic masses describe bosonic Black Holes via E=kT and not the cosmological Black Holes of the Schwarzschild metric.

The c-invariance so uses modular duality in the quantum Black Hole limit c=fmaxλmin=2πfmaxRmin for fmin=c/λmax=c/2πRmax as an unmodulated frequency in Tmin=Emin/k=hc/2πkRmax=hc.λmin/4π²k=3.58856...x10^-26 K* and a temperature above the Hawking-Gibbons limit as required.

This differs in a factor {2π/c}² from the lightspeed inversion in  Tmin=hfmin/k and so 1.574..x10^-41 K*, which violates the Hawking-Gibbons boundaries in NOT using the modular duality and with fmin=1/fmax in frequency units and NOT inverted time units.

And so Mmin.Tmax=hc³/4πGok= ½mP.TP =MHawking.hc.λmin/4π²k and the Hawking Mass is determined as Mmax=MHawking=πc²λmax/Go ~ 2.544690...x10⁴⁹ kg*.

We can see, that this modulation closely approximates the geometric mean of the seedling mass in Mo²/2M∞.MHawking=3.2895..x10¹⁰²/3.2931..x10¹⁰² ~ 0.9989...

This also circumscribes the actual to critical density ratio in the omega of the general relativistic treatment of the cosmologies.

Now recall our applied G value in  Gm(n)=Go.X^n+k and apply our just derived Black Hole Mass modulation coupled to that of the quantum micro-masses.

We had: Gomc²={GoX^n+k}.{mcYⁿ}.{mcY^k}=Gm.mnmax.mnmin and where Gm is the actual G value as measured and which has proved difficult to do so in the laboratories.

Gm(n)=Go.X^n+k=Gomc²/mnmax.mnmin =Gomc²/({mcYⁿ}{mnmin}) and where we have  mnmin={mcY^k} for the unknown value of k.

So Gm(n)=Go.X^n+k=GoXⁿ[mc/mnmin]=Go{mc²/mcYⁿ}.{Mo²/2M∞.MHawking.mav}} and where now {mnmin}={mcY^k}={2M∞.MHawking.mav/Mo²}=1.0011..mav.

mav={Mo²/2M∞.MHawking}{mnmin}={Mo²/2M∞.MHawking}{mcY^k}=0.9989..{mcY^k} and obviously represents a REDUCED minimum mass mnmin=mcY^k.

But the product of maximum and 'new' minimum now allows an actual finetuning to a MEASURED nucleon mass mN by:  mN² = mavYⁿ.mcYⁿ=mav.mnmax.Yⁿ.

So substituting for mav in our Gm expression, will now give the formulation:

Gm(n)=Go.X^n+k=GoXⁿ[mc/mnmin]=Go{mc²/mcYⁿ}.{Mo²/2M∞.MHawking.mav}}

Gm(n)=Go{mc²/mcYⁿ}.{Mo²/2M∞.MHawking}.{mcY²ⁿ/mN²}

Gm(n)=Go{mc²/mN²}{Mo²/2M∞.MHawking}Yⁿ=Go.X^n+k 

The average nucleon mass mN is upper bounded in the neutron mass and lower bounded in the proton mass, their difference being an effect of their nucleonic quark content, differing in the up-down transition and energy level.

For a Neutron Restmass of: mn=1.680717x10^-27 kg* (941.6036 MeV*) the substitution (and using calibrations m=0.9983318783m*; s=0.9990230094s*; kg=0.99626135kg* and C=0.997296076C*) gives: G(np)=6.678764x10^-11 (m³/kgs²).

A perturbation corrected mn=1.681100563x10^-27 kg* (941.818626 MeV*) gives: 
 G(np)=6.675715x10^-11 (m³/kgs²).

The perturbation upper limit is given in the mn=1.681335x10^-27 kg* (941.9506 MeV*) and gives: 
 G(np)=6.6738445x10^-11 (m³/kgs²). The average for the last two values then approximates as a 'best fit' for:

Gm(np)=6.6747798x10^-11 (m³/kgs²).  

This is a best-fit approximation, considering the uncharged nature of the testmasses.

This then gives the value of k from Gm(n)=Go.X^n+k as k=ln(GmYⁿ/Go)/lnX and which calculates as k= -0.073387..

Two protons (mp=1.6789x10^-27 kg* (940.56 MeV*) would give:

G(np)=6.6936x10^-11 (m³/kgs²) and a proton-neutron pair would yield: 

G(np)=6.6791x10^-11 (m³/kgs²); both of the latter  values unsuitable because of the electrocharges increasing the intraquarkian Magnetocharge coupling between the two mesonic rings of the neutron and the single mesonic ring in the proton's down- or KIR-quark.

The best approximation for 'Big G' hence depends on an accurate determination for the neutron's inertial mass, only fixed as the base nucleon minimum mass at the birth of the universe.

A fluctuating Neutron mass would also result in deviations in 'G', independent upon the sensitivity of the measuring equipment. The inducted mass difference in the protonic-and neutronic restmasses, derives from the Higgs-Restmass-Scale and can be stated in a first approximation as the groundstate.

Basic nucleon restmass is mc=√Omega.mP=9.9247245x10^-28 kg*.

(Here Omega is a gauge string factor coupling as:

Cuberoot(Alpha):Alpha:Cuberoot(Omega):Omega for Omega=G-alpha).

KKK-Kernelmass=Up/Down-HiggsLevel=3x319.62 MeV*=958.857 MeV*, using the Kernel-Ring and Family-Coupling Constants.

Subtracting the Ring-VPE (3L) gives the basic nucleonic K-State as 939.642 MeV*. This includes the electronic perturbation.

For the Proton,one adds one (K-IR-Transition energy) and for the Neutron one doubles this to reflect the up-down-quark differential.

Proton (mp=u.d.u=K.KIR.K=(939.6420+1.5013-0.5205)MeV*=940.6228 MeV*. Neutron (mn=d.u.d=KIR.K.KIR=(939.6420+3.0026-1.0410)MeV*=941.6036 MeV*.

This is the groundstate from the Higgs-Restmass-Induction-Mechanism and reflects the quarkian geometry as being responsible for the inertial massdifferential between the two elementary nucleons.  All groundstate elementary particle masses are computed from the Higgs-Scale and then become subject to various finestructures.

Overall, the MEASURED gravitational constant 'G' can be said to be decreasing over time. The ratio given in k is GmYⁿ/Go~0.60073... and so the present G-constant is about 60% of the one at the Planck Scale.

G decreases nonlinearly, but at a present rate of 0.60073/19.11x10^9 year, which calculates as 3.143..x10^-11 G-units per year.

So gravity appears stronger when one 'looks back in time' or analyses cosmological objects at large distances.

The expansion parameter (a) in the Friedmann-Einstein standard cosmology can be rewritten as a curvature ratio R(n)/Rmax={n/(n+1)} and describes the asymptotic universe in say 10 dimensions evolving under the inertial parameters of the c-invariance.

This 'lower dimensional universe' is open and expands under hyperbolic curvature under the deceleration parameter qo=½Ωo=Mo/2M∞=2GoHoMo/c³ ~0.014015... This open universe is bounded in the 'standing wave' of the Hubble Oscillation of the 11D and 'higher dimensional universe'.

The boundary is given in the omega of the 'missing mass' of the volumes, which differ in a factor of V11/V10=nRmax³/(n/(n+1))³Rmax³=(n+1)³/n²=DIM-Factor (and which assumes its minimum for one complete oscillation for n=2 as DIM=27/4=6.75 so 14.7  Billion years from the present).

Presently, for n=1.132419.. DIM=7.561.. and so the 'missing mass' will be measured as a 'dark matter' distribution of 'dark haloes' etc. around the luminous matter given in the ylemic mass seedling Mo of the baryonic matter.

As Mo is just 2.8% of M∞, but is subject to a 'growth' in the maximising factor Yⁿ=1.724.. for the present epoch, one can take the factor Mav==Mo.√Yⁿ=1.313.. for a 'dark matter' percentage upper bounded in 2.8%(1.724)~4.83% and lower bounded in 2.8%(1.313)~3.68%.

But so 7.56 open universes are contained within the closed and spherical universe given in the Hubble bound. And the 'dark matter' will be 7.56 times the luminous baryonic matter in the interval {27.82%, 36.51%} as percentage of the total energy of closure for Ωo=1 and the critical density ρc=M∞/Vmax=3Ho²/8πGo.

Our Big Bang happened at the modular time 1/fmax=tmin=fmin=3.33..x10^-31 seconds*, coinciding with the end of the stringed inflation epoch of the standard cosmology.

The 'de Broglie' inflation established the crucial boundary parameters as say given in the Mo and M∞ Black Hole masses described.

As the baryonic mass seedling Mo sets the Sarkar Scale for the cosmic architecture in the size of galactic superclusters as the limit for the gravitationally interacting systems before cosmic homogenuity; there must be a Black Hole evolution superposed onto the expansion of the 10D universe and the oscillation of the 11D universe which 'adds' a 'electromagnetic' volume of 2π²Rmax³ at the Hubble nodes every 16.9 Billion years.

In terms of the dimensional 'intersection' this can be described as a 'Strominger Brane' evolution with the Sarkar Scale set at the instanton, decreasing as a 'shrinking' Black Hole until it becomes massless at the wormhole scale defined in the minimum macro Black Hole λmin/2π=2GoMmin/c²=1.591549..x10^-23 metres*. 

This then resets the bosonic micro Black Holes with their macro counterparts under the modular duality.

This Black Hole evolution is higher dimensional and purely electromagnetic, not being observable due to its noninertial nature, except the so called 'dark matter' and 'dark energy' scenarios of the boundary- and initial conditions. This can lead to a feasible model for the phenomenon of consciousness.

The process will take place in a DIM factor of about 457 as: Mmin.√Y^N=Mo   and for
N=2ln(Mo/Mmin)/lnY~454 and so in 16.9x454 Billion years, which are about 7.673 Trillion years. The gravitational constancy of Gom