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

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,

(1)

k = 9×10²¹ m kg⁻¹.

(2)

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 (m_s), length (l_s), and time (t_s), 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:

m_s = (ke²/G)^1/2 = 1.859×10^–9 kg,

(3)

l₀ = (ke²G/c⁴)^1/2 = 1.38 ×10^–36 m,

(4)

t₀ = (ke²G/c⁶)^1/2 = 4.6 ×10^–45 s,

(5)

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 m_s 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 R_g = m_sG/c² = 1.38×10^–36 m.

(In educational literature, the gravitational radius of a body R_g = mG/c² is mistakenly confused with its Schwarzschild radius R = 2R_g = 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/m_sc² = 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 x^D (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:

x^D(D > 3) = r = l₀ or x^D(D > 3) = r = ct_{0 .}

(6)

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,

(7)

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,

(8)

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.

(9)

Immediately evident that if the velocity υ = c, this equation becomes invalid because

mc² ≠ (m²c⁴ + m₀²c⁴)^1/2.

(10)

Consequently, equations (8) and (9) are valid if the velocity υ does not exceed a certain value υ_M , which is very close to the speed of light: υ ≤ υ_M , υ_M ≈ c.

Thus, it is clear that the energy of a moving particle cannot increase infinitely, and it has reached the maximum value E_M = lim mc² = Мc² on the certain velocity υ_M :

Mc²= m₀c²/(1 – υ_M²/c²)^1/2.

(11)

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.

(12)

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,

(13)

where a dimensionless factor

n = (e/m₀c)².

(14)

Thus, the factor n is inversely proportional to the rest mass of a charged elementary particle.

For instance, for a proton the factor

n_р = (e/m_рc)² = 1.02×10^–13

(15)

(the rest proton mass m_p = 1.67×10^–27 kg). Further, from equation (12) turns out:

ke² = Rm₀c² = Rn (m₀c²/n) = rMc²,

(16)

where r = Rn,

Mc² = m₀c²/n

(17)

or

M = m₀/n = m₀³с²/e².

(18)

Therefore, considering equation (11), factor

n = m₀c²/Mc² = (1 – υ_M²/c² )^1/2.

(19)

Hence, it is easy to define the limits of velocity (υ_M = lim υ), momentum (p_M = lim p = lim mυ) and energy (E_M = lim E = lim mc²) of a moving charged elementary particle [12]:

υ_M = c(1 – n² )^1/2 (i.e., υ_M ≈ c),

(20)

p_M = Mυ_M = m₀υ_M /n ≈ m₀c/n,

(21)

E_M = Mc² = m₀c²/n.

(22)

For example, the maximum speed of a proton

υ_M = c (1 – n_p²)^1/2 = c [1 – (1.02×10^–13)²]^1/2 = c (1 – 1.0404×10^–26)^1/2,

(23)

and its the maximum energy

М_рс² = m_pc²/n_p = 9.19×10²¹ eV

(24)

(m_pc² = 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 ≤ υ_M , m ≤ M, M ≤ m_s ;

(25)

E = mc²= m₀c²/(1 – υ²/c²)^1/2, υ ≤ υ_M , m ≤ M, M ≤ m_s ;

(26)

p = mυ = m₀υ/(1 – υ²/c²)^1/2, υ ≤ υ_M , m ≤ M, M ≤ m_s ;

(27)

p = m₀υ, if m₀ ≥ m_s .

(28)

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 = m_s. Therefore, it is possible to calculate the limiting rest mass, lim m₀ = m_m , 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 = m_s and m₀ = m_m , we get:

m_s = m_m³c²/e²

(29)

(е = 1.6×10^–25 kg m s⁻¹). Hence,

m_m = (m_se²/c²)^1/3 = 8.0979×10^–26 kg.

(30)

This is the theoretical maximum rest mass that stable charged elementary particles (maximons) can have; rest energy of such particles E_m = m_mc²/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 m_m , then we could easily calculate the exact value of the Stoney mass m_s using formula (29), and then calculate the exact value of the gravitational constant:

G = ke²/m_s².

(31)

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: E_Z = 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 = E_Z/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 (m_m = 8.0979×10^–26 kg, E_m = 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 m_m) of the electron and positron produced during the decay of the Z boson, we can easily calculate the exact value of the Stoney mass m_s using formula (29), and then calculate the exact value of the gravitational constant using formula (31).

Competing Interests: The author declare no competing interests.