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Theoretical Calculation of the Gravitational Constant Using the Elementary Charge, Speed of Light, Z Boson Mass, and Relativistic Mechanics

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23 March 2023

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24 March 2023

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Abstract
The gravitational constant is an unchanging quantity included in the law of universal gravitation. The value of this constant is known much less accurately than other fundamental physical constants, which is due to the experimental difficulties of measuring it. Here we report a simple method for the theoretical calculation of the gravitational constant using the elementary charge, speed of light, Z boson mass, and relativistic mechanics.
Keywords: 
Subject: Physical Sciences  -   Theoretical Physics
There are no infinite quantities in physics at all.
In math, yes, infinities exist, but not in physical reality.

1. Introduction

The gravitational constant is the proportionality constant in Newton’s law of universal gravitation. The value of this constant is known to be much less accurate than that of other fundamental physical constants, and the results of experiments to refine it continue to vary. This is because the gravitational force is extremely weak compared with other fundamental forces. Therefore, the problems are caused by experimental difficulties in measuring small forces taking into account a large number of external factors. The values of the gravitational constant measured by various methods lie in the range (6.67÷6.676)×10–11 m3 kg–1 s–2 [1] and the results of experiments on its refinement continue to differ. The discrepancy in G calculations is such a serious problem that it is considered as a metrological and scientific dead end. But it turns out that there is an unexpected solution to this problem: we can theoretically calculate the gravitational constant.
To solve this problem, in particular, it is necessary to use the value of an elementary electric charge. However, the charge has the different numerical value and dimension in the different systems of measurement.
The most widely used in physics is the International System of Units (SI) composed of other subsystems, in particular, the m-kg-s system of mechanical units (called the MKS system) and the m-kg-s-A system of electromagnetic units (called the MKSA system). Second system differs first system primarily in that, along with the existing three basic units (meter, kilogram, second), it has a fourth basic unit – ampere (A).
In the MKSA system, the elementary charge e = 1.6×10−19 C, and the proportionality coefficient, included in Coulomb’s law, k = 9×109 N m2 C−2.
In 2018, we showed in paper [2] that the electromagnetic units of the MKSA system (A, C, V, Ω, etc.) can be written using the bacic units of the MKS system: m, kg, s. In particular, it is shown that in the MKS system
е = 1.6×10–25 kg m s−1,
k = 9×1021 m kg−1.
Here it is necessary to recall one historical fact.
In 1874 in Belfast, the Irish physicist J. Stoney presented his famous report [3], in which he proposed the first natural system of units of mass (ms), length (ls), and time (ts), built on the fundamental constants c (c = 3×108 m s−1 is the speed of light in vacuum), G, and e. The modern meanings of the Stoney units are as follows:
ms = (ke2/G)1/2 = 1.859×10–9 kg,
l0 = (ke2G/c4)1/2 = 1.38 ×10–36 m,
t0 = (ke2G/c6)1/2 = 4.6 ×10–45 s,
where recommended value G = 6.6743×10–11 m3 kg–1 s–2 [4].
However, physicists failed to understand the significance of Stoney system for science. It is now clear that this was a mistake, because using this system immediately gives important results.
In 2021, we showed in article [5] that the Stony mass ms separates the macrocosm and microcosm; i.e, this mass is the minimum value for the masses of ordinary bodies and the maximum value for the masses of elementary particles. This means the following.
On the one hand, an ordinary body with such a minimum mass would have the smallest gravitational radius Rg = msG/c2 = 1.38×10–36 m.
(In educational literature, the gravitational radius of a body Rg = mG/c2 is mistakenly confused with its Schwarzschild radius R = 2Rg = 2mG/c2; in fact, these are completely different physical quantities [5]).
On the other hand, a charged elementary particle with such a maximum relativistic mass would have the smallest classical radius R0 = ke2/msc2 = 1.38×10–36 m.
Thus, the modern values of the Stoney length and the Stoney time, l0 and t0 = l0/c, are the elementary length and the elementary time, i.e. the smallest values of length and time that exist in nature. Moreover, it turned out that these quantities are associated with the so-called extra dimensions.
The space around us has three dimensions: length, width and height. Scientists consider these dimensions as independent coordinates that are necessary to describe the position of any point in a three-dimensional [3D] space and denote them by x1, x2, x3.
The theorists have suggested that in addition to the usual three dimensions, there are additional 6 dimensions that are curled up in a circle of microscopic radius and, therefore, cannot be detected directly.
In 2022, in the paper [6] the idea was put forward that in three-dimensional space a physical point is, in fact, a minuscule ball. It is shown, that any additional spatial dimension xD (D > 3) is the physical point radius r, which can be expressed in terms of the elementary length l0 or the product of the speed of light in a vacuum c and the elementary time t0:
xD(D > 3) = r = l0 or xD(D > 3) = r = ct0 .
Even more unexpected is that relativistic mechanics is a necessary part of our research.
The relativistic mechanics, as a special branch of physics, arose in 1901, when the German researcher W. Kaufmann discovered an amazing fact in experiments on the study of cathode particles (electrons): the ratio of the electron charge to its mass e/m decreased with increasing speed [7]. Thus, Kaufman’s experiments clearly indicated an increase in the mass of a moving electron. It was a discovery that classical physics could not explain. The mass of the electron turned out to depend on the speed, which in these experiments reached 95% of the speed of light in vacuum. Further research showed that this dependence is well described by the formula
m = m0 /(1 – υ2/c2)1/2,
where m0 is the mass of a rest particle, m is its relativistic mass, υ is speed.
Accordingly, the mass (energy) of other particles also depends on the speed. We see this in cosmic rays, which contain particles of different energies.
In 1966, American scientist K. Greisen [8] and, independently, Soviet physicists G. Zatsepin and V. Kuzmin [9] proposed theoretical upper limit to the energy of charged particles (protons) coming to the Earth from space (GZK limit): 5×1019 eV. Scientists have suggested that the limit is set by slowing-interactions of the protons with the microwave background radiation over long distances. However, this suggestion contradicts to the astrophysical observations: in cosmic rays was detected the extreme-energy protons with energy (1÷3)×1020 eV [10,11]. The observed existence of these particles is called GZK paradox. Thus, these facts indicate that the energy of a moving charged particle (particularly, of proton) is rising only to specific value. The solution of this problem will allow us to better understand the mysterious world of elementary particles.

2. New formulas and the scope of applicability of relativistic mechanics

As is known since the beginning of the 20th century, the energy of a moving particle, Е = mc2, is increased with velocity υ:
mc2= m0c2/(1 – υ2/c2)1/2,
where m0c2 is the rest particle energy.
An assumption arose, which then turned into the assertion that when the particle’s velocity approaches the speed of light (υ→c) the particle’s energy increases without bound (mc2→∞). However, this is simply a historical delusion that is easy to disprove.
Firstly, this postulate contradicts astrophysical observations: in the universe we do not see moving particles with infinite energy.
Secondly, if the speed of the particle is equal to the speed of light, υ = c, then the denominator in equation (6) becomes zero, and this equation itself becomes invalid, because we know from arithmetic that it is impossible to divide by zero.
Thirdly, we can rewrite the equation (8) in the form:
mc2= (m2υ2c2 + m02c4)1/2.
Immediately evident that if the velocity υ = c, this equation becomes invalid because
mc2 ≠ (m2c4 + m02c4)1/2.
Consequently, equations (8) and (9) are valid if the velocity υ does not exceed a certain value υM , which is very close to the speed of light: υ ≤ υM , υM ≈ c.
Thus, it is clear that the energy of a moving particle cannot increase infinitely, and it has reached the maximum value EM = lim mc2 = Мc2 on the certain velocity υM :
Mc2= m0c2/(1 – υM2/c2)1/2.
According to the relativistic mechanics, along the direction of motion of a charged elementary particle decrease its linear dimensions, including classical radius R:
R = ke2/m0c2.
As we mentioned, in the MKS system е = 1.6×10–25 kg m s−1, k = 9×1021 m kg−1, therefore this equation can be written as:
R = km0(e/m0c)2 = km0n,
where a dimensionless factor
n = (e/m0c)2.
Thus, the factor n is inversely proportional to the rest mass of a charged elementary particle.
For instance, for a proton the factor
nр = (e/mрc)2 = 1.02×10–13
(the rest proton mass mp = 1.67×10–27 kg). Further, from equation (12) turns out:
ke2 = Rm0c2 = Rn (m0c2/n) = rMc2,
where r = Rn,
Mc2 = m0c2/n
or
M = m0/n = m03с2/e2.
Therefore, considering equation (11), factor
n = m0c2/Mc2 = (1 – υM2/c2 )1/2.
Hence, it is easy to define the limits of velocity (υM = lim υ), momentum (pM = lim p = lim mυ) and energy (EM = lim E = lim mc2) of a moving charged elementary particle [12]:
υM = c(1 – n2 )1/2 (i.e., υM ≈ c),
pM = MυM = m0υM /n ≈ m0c/n,
EM = Mc2 = m0c2/n.
For example, the maximum speed of a proton
υM = c (1 – np2)1/2 = c [1 – (1.02×10–13)2]1/2 = c (1 – 1.0404×10–26)1/2,
and its the maximum energy
Мрс2 = mpc2/np = 9.19×1021 eV
(mpc2 = 0.938×109 eV is the rest proton energy).
This is the upper limit of the cosmic ray spectrum; it is consistent with observations and at is two orders of magnitude exceeds the GZK limit.
Now we can write down the general formulas:
m = m0/(1 – υ2/c2)1/2, υ ≤ υM , m ≤ M, M ≤ ms ;
E = mc2= m0c2/(1 – υ2/c2)1/2, υ ≤ υM , m ≤ M, M ≤ ms ;
p = mυ = m0υ/(1 – υ2/c2)1/2, υ ≤ υM , m ≤ M, M ≤ ms ;
p = m0υ, if m0 ≥ ms .
These formulas reflect the obtained results, namely:
  • equations (25), (26) and (27) show the existence of limits for the speed, momentum and energy of a stable charged elementary particle, and the limit of the relativistic mass cannot exceed the Stoney mass;
  • equation (28) reflects the invariability (constancy) of the mass for all bodies whose mass exceeds the Stoney mass.
Hence a very important conclusion: the area of applicability of relativistic mechanics is limited to the microcosm and does not extend to the macrocosm, i.e. the ordinary bodies (whose rest mass exceeds the Stoney mass) obey the laws of the classical mechanics.
We can now set out a method for the theoretical calculation of the gravitational constant.

3. Metod

According to the results obtained in paper [5] and equations (25)–(27), the absolute limit of the relativistic mass of a charged elementary particle is equal to the Stoney mass: lim M = ms. Therefore, it is possible to calculate the limiting rest mass, lim m0 = mm , that a stable charged elementary particle can have (it is convenient to call it "maximon"; this name for an elementary particle with a maximum rest mass was proposed by the Soviet physicist M. Markov in 1965 [13]).
Substituting in equation (18) M = ms and m0 = mm , we get:
ms = mm3c2/e2
(е = 1.6×10–25 kg m s−1). Hence,
mm = (mse2/c2)1/3 = 8.0979×10–26 kg.
This is the theoretical maximum rest mass that stable charged elementary particles (maximons) can have; rest energy of such particles Em = mmc2/e = 45.426 GeV (е = 1.6×10–19 C).
Note that we calculated the mass (energy) of the maximon using the Stoney mass, according to the recommended value G = 6.6743×10–11 m3 kg–1 s–2 [4].
Thus, on the contrary, if we knew the exact value of the mass of the maximon mm , then we could easily calculate the exact value of the Stoney mass ms using formula (29), and then calculate the exact value of the gravitational constant:
G = ke2/ms2.
Naturally, the question arises: do the maximons exist?
The affirmative answer is due to the fact that the total energy of the maximon and antimaximon is: 2×45.426 GeV = 90.852 GeV. This needs to be explained.
In 1983, at CERN (European Organization for Nuclear Research) neutral particles (called Z bosons) predicted by the electroweak theory were discovered. Thise particles were born in the collision of colliding beams of protons and antiprotons. In a very short time (~3×10–25 s) Z bosons decay in a fermion and its antiparticle (in particular, in an electron and a positron), flying out in opposite directions.
The theory predicted a Z boson energy of about 90 GeV; the experiment gave a close value.
Based on further experiments, current estimate of the Z boson energy was made: EZ = 91.1876 GeV [14]. Hence it follows that the decay of the Z boson, in particular, produces the electrons and positrons with rest energy E0 = EZ/2 = 45.5938 GeV; this energy corresponds to the rest mass m0 = E0e/с2 = 8.1278×10−26 kg (е = 1.6×10–19 C).
As we can see, the value of the mass m0 (energy E0) of the electron and the positron measured in these experiments is slightly larger than the calculated value of the mass (energy) of the maximon (mm = 8.0979×10–26 kg, Em = 45.426 GeV); this discrepancy leads us to the following conclusions.

4. Conclusions

1. The currently accepted value of the Z boson energy (91.1876 GeV), is slightly overestimated; this means that a system error was made in estimating the energy of the electron and positron produced during its decay (apparently, the increase in energy during their movement is not fully taken into account).
For charged particle with energy E0 = 45.5938 GeV and rest mass m0 = 8.1278×10−26 kg, the factor n = (e/m0c)2 = 4.3165×10−17, and the relativistic mass limit M = m0/n = 1.883×10−9 kg.
In this case, the gravitational constant G = ke2/M2 = 6.506×10–11 m3 kg–1 s–2; this value is completely inconsistent with the experimental values of this constant, which are lying within (6.67÷6.676)×10–11 m3 kg–1 s–2 [1].
Thus, it is necessary to radically analyze the method for estimating the Z boson energy.
2. Having determined the exact value of the energy (mass mm) of the electron and positron produced during the decay of the Z boson, we can easily calculate the exact value of the Stoney mass ms using formula (29), and then calculate the exact value of the gravitational constant using formula (31).
Competing Interests: The author declare no competing interests.

References

  1. Li Q, Xue C, Liu JP et al. Measurements of the gravitational constant using two independent methods. Nature. 2018;560;582–588. [CrossRef]
  2. Abdukadyrov A. Progress of the SI and CGS Systems: Conversion of the MKSA Units to the MKS and CGS Units. American Journal of Electromagnetics and Applications. 2018;6(1);24–27. [CrossRef]
  3. Stoney GJ. On the physical units of nature. Philosophical Magazine. 1881;11(69);381–390. [CrossRef]
  4. "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. https://physics.nist.gov/cgi-bin/cuu/Value?bg.
  5. Abdukadyrov A. Fundamental Values of Length, Time, and Speed. Reports in Advances of Physical Sciences. 2020;4(4);2050008. [CrossRef]
  6. Abdukadyrov A. Fundamental Units of Measurement and Extra Dimensions. Advances in High Energy Physics. 2022;2022;2655733. [CrossRef]
  7. Kaufmann W. Die Entwicklung des Elektronenbegriffs (Development of the concept of electrons). Physikalische Zeitschrift. 1901;3(1);9–15. https://de.wikisource.org/wiki/Die_Entwicklung_des_Elektronenbegriffs.
  8. Greisen K. End to the Cosmic-Ray Spectrum? Physical Review Letters. 1966;16;748–750. [CrossRef]
  9. Zatsepin GT, Kuz'min VA. Upper Limit of the Spectrum of Cosmic Rays. Journal of Experimental and Theoretical Physics Letters. 1966;4;78–80. Bibcode:1966JETPL...4...78Z.
  10. Linsley J. Evidence for a Primary Cosmic-Ray Particle with Energy 1020 eV. Physical Review Letters. 1963;10(4);146–148. [CrossRef]
  11. Bird DJ et al. Evidence for correlated changes in the spectrum and composition of cosmic rays at extremely high energies. Physical Review Letters. 1993;71;3401–3404. [CrossRef]
  12. Abdukadyrov AB. Physics – Unity in Diversity. Bishkek; 2008, pp. 39–40.
  13. Markov MA. Can the Gravitational Field Prove Essential for the Theory of Elementary Particles? Progress of Theoretical Physics Supplement. 1965;E65;85–95. [CrossRef]
  14. Particle Data Group. The Review of Particle Physics (2022). https://pdglive.lbl.gov/Particle.action?node=S044&init=0.
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