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# Newton's Gravitational Constant Calculated by a Pure Quantum Electrodynamics Equation

Emmanouil Markoulakis

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Emmanouil Markoulakis. Newton's Gravitational Constant Calculated by a Pure Quantum Electrodynamics Equation. 2025. hal-05074171

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Distributed under a Creative Commons Attribution 4.0 International License
Newton's Gravitational Constant
Calculated by a Pure Quantum
Electrodynamics Equation

Emmanouil Markoulakis, Department of Electronic Engineering, Computer Technology Informatics & Electronic Devices Laboratory, Hellenic Mediterranean University, Romanou 3, Chania 73133, Crete, Greece. Author E-mail:markoul@hmu.gr

Abstract

We take the Brandenburg, J., GEM theory equation first published at 1988, for “Big” G and combine with our previous research derived equation for the Fine Structure Constant α, expressed as a pure ratio of magnetic flux. We then derive we believe a novel and accurate theoretical equation which is within less than one part per thousand of the measured CODATA value for G in SI units, expressed purely in quantum electrodynamics QED physical constants without the use of any free parameters. By observing and interpreting the physical meaning of this newly derived pure QED equation for gravity G constant we have been shown for a real possibility that gravity is quantized and its quanta also theorized being the massless spin-2 graviton, is somehow responsible for the mass of the electron and the proton and consequently their mass ratio $m_p/m_e$ but also their electromagnetic characteristics like the FSC α, charge radius and Compton wavelength $\lambda_p$ and $\lambda_e$ of the particles and electromagnetic characteristics of the free space vacuum like vacuum permeability $\mu_0$ physical constant and the speed of light c of the vacuum. The author herein concludes from this new novel derivation of big G constant equation that the possible quanta of gravity or graviton yet to be found is probably the most fundamental universal and omnipresent particle in our universe that is also the fundamental particle quanta of the vacuum and the fabric of spacetime and is the particle that links and possible also sources all known forces in our universe macroscopic and quantum like gravity and electromagnetism but also possible the two nuclear, strong and weak forces. The original Brandenburg, J., equation and this new derivation presented herein, both support the findings and conclusions of our previous related sere arch and our massless cold graviton superluminal model.

Keywords: Gravitational constant, graviton, electromagnetism, quantum electrodynamics, quantum gravity, vacuum.
# 1. Derivation of a pure QED equation for G gravity constant

Using novel Brandenburg's equation (1) [1] [2] [3] [4] [5] [6] for G and modified for SI units and our novel electromagnetic flux ratio FSC α equation in SI units (2) [7] [8] solved for e²:

$$G \approx \frac{9e^2a \times 10^9}{m_p m_e} \times e^{-\left[ \frac{4m_p}{m_e} \right]} \quad (1)$$

Where $m_p$ and $m_e$ the proton and electron mass, $a \approx 1/137$ is the fine structure constant FSC, $e$ is the elementary electron charge and $e = 2.71828$ is Euler's number, the base of the natural logarithms.

$$e^2 = \frac{2ah}{c\mu_0} \quad (2)$$

Where $h$ is the Planck constant, $c$ the speed of light if the vacuum and $\mu_0$ is the magnetic permeability of the vacuum.

Substituting eq. (2) inside eq. (1) above and also expressing the mass of the proton and electron in terms of Compton wavelengths using the general equation $\lambda = \frac{h}{mc}$, thus $\lambda_e$ and $\lambda_p$ values for the electron and proton correspondingly, we get the **final equation (3) in SI units** below for the Newton's gravitational physical constant G, expressed purely in QED physical constants with no free parameters and $e$ being Euler's number:

$$G \approx \frac{18a^2c\lambda_p\lambda_e \times 10^9}{h\mu_0} \times e^{-\left[ \frac{4\lambda_e}{\lambda_p} \right]} \quad (3)$$

Notice, eq. (3) above is accurate within less than one part per thousand of the measured CODATA value for $G = 6.67430(15) \times 10^{-11} \text{m}^3 \cdot \text{kg}^{-1} \cdot \text{s}^{-2}$ in SI units.
## 2. Discussion and Conclusions

According to this analysis and physical interpretation, these result complies and supports our previous research findings [9] [10] [11] and our novel superluminal massless spin-2 cold graviton model as the quanta of spacetime and the vacuum and also quanta of gravity being the most fundamental particle of the universe from which all other known forces emerge thus electromagnetism and nuclear strong and weak forces.

As we can see eq. (3) above can be written with the general mathematical form of equation (4) and its resulting xy graph plot of illustrated in fig. 1 below:

$$y=G=b \cdot e^{-x} \qquad (4)$$

$$\text{With, } b = \frac{18a^2c\lambda_p\lambda_e \times 10^9}{h\mu_0} \approx 1.107 \times 10^{27} \qquad (5)$$

<figure><img src="image_4.png" /><figcaption>Fig. 1 Plot function of G values from a varying $x = \sqrt{4\lambda_e/\lambda_p}$ Compton wavelength square root ratio of the electron to the proton. The approximate calculated value of the G gravitational constant at the graph point (x=85.7, G≈6.685 x 10⁻¹¹) is shown above. The slope of the function is very step and almost vertical to the x-axis indicating how crucial for the universe the G constant value is controlling the mass ratio $m = h/\lambda c$ of proton to the electron $m_p/m_e$. See online version of the graph of fig. 1 here, https://www.desmos.com/calculator/c3se5munju.</figcaption></figure>
The plot function in Fig. 1 shows the varying G values from a hypothetical varying $x = \sqrt{4\lambda_e/\lambda_p} \approx 85.7$ Compton wavelength square root ratio of the electron to the proton from is nominal value of $\approx 85.7$. The approximate calculated fixed value of the G gravitational constant at the graph point $\{x=85.7, G \approx 6.685 \times 10^{-11}\}$ is shown above in fig. 1. The slope of the function is very step and almost vertical to the x-axis indicating how crucial for the universe the G constant value is, controlling also the mass ratio $m = h/\lambda_{(Compton)}$ of proton to the electron $\sqrt{4mp_e/m_e} \approx 85.7$. See online version of the graph of Fig. 1 here, https://www.desmos.com/calculator/c3se5munju.

**We observe in Fig.1 that although the gravitational constant G has a very small positive numerical value of the order of magnitude of $10^{-11}$ and although not sensitive to the increase of its value, however, any small reduction from its currently fixed value even of one order of magnitude will result in dramatic explosive increase in the proton to electron mass ratio destroying possible any coherency of baryonic matter in the universe with catastrophic results. This result comes as a surprise since it is general accepted that at the quantum level, gravity has little to no effect to matter and the quantum properties of particles and the other three forces electromagnetism and the strong nuclear and weak forces.**

**Nevertheless,** our research here clearly demonstrates that something much more elaborate and subtle is going on and that gravity must be quantized and having a crucial role at the formation of matter and mass at the quantum level despite its tiny amplitude at the quantum scale. This is explained in our previous research [9] [10] [11] of our novel graviton-vacuum model and how really the massless spin-2 graviton is the source and origin of all other particles of the standard model, but with a twist.

Also another novel observation from eq. (3) & (5) is that apart of the speed of light c value which is an inherent property of light from the free space vacuum's $c = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$ permeability $\mu_0$ and permittivity $\varepsilon_0$ characteristic as shown by Maxwell theory, we see here now also maybe for the first time a clear connection of G with the vacuum and its magnetic permeability $\mu_0$ physical constant. This is also supported in our previous research [9] [10] [11] which regards the graviton being the quanta of gravity but also the quantum of the vacuum itself and dark energy. The author suggest herein from our findings in this paper also that permeability $\mu_0$ and permittivity $\varepsilon_0$ physical constants effectively represent for the free space accordingly to a classical analogue of the elasticity to torsion and elasticity compression of a medium and big G is actually an analogue to the general stiffness of a medium.

Leaving with a futuristic note here at this point, we see from equation eq. (3) and also supported by our previous research [9] [10] [11] that the only way to actually achieve ever superluminal speed FTL space travel and “antigravity” is to get rid of the vacuum by artificially somehow displacing the vacuum around the spacecraft and creating a bubble of finite wall thickness of “true void” or lower energy density than the nominal free space vacuum zero-point energy (ZPE) currently about $10^{-9} \text{ J/m}^3$. This will allow the ship to effectively have no mass inertia and even lower mass than the tiny photon upper limit is predicted to have $m < 1 \times 10^{-18} \text{ eV/c}^2$ (experimental limit)<fnref n="1" />. Therefore is predicted to reach FTL c speed and will not be restricted by special and general relativity theory and no subject of time dilation and length contraction because essentially the spacecraft will travel in a type of hyperspace free of gravity, vacuum and Einstein spacetime and not ruled by SR and GR.

<footnote n="1">¹ https://en.wikipedia.org/wiki/Photon</footnote>
# Appendix

The original novel Brandenburg G equation [6] in CGS units:

$$G = (e^2 / m_p m_e) \alpha \exp \left( -2 \left( \frac{m_p}{m_e} \right)^{1/2} \right) = 6.6684 \times 10^{-8} \text{ dyne} - \text{cm}^2 \text{gm}^{-2}$$

Also, in this link here<fnref n="2" /> (see footnote) is a very comprehensive explanation to why the gravitational constant G computes to a different numerical value in CGS units than its value in the SI units.

# References

[1] Brandenburg J.E. (1988). GEM Theory III, APS Bull.,33, 1, p.32.

[2] Brandenburg J. E. (1992). Unification of Gravity and Electromagnetism in the Plasma Universe, IEEE Transactions on Plasma Science, Plasma Cosmology Issue Vol 20, 6, p944.

[3] Brandenburg, J.E. (1995) “A Model Cosmology Based on Gravity Electro-Magnetism Unification”, Astrophysics and Space Science, Vol 277, p133-144. ( also in “Plasma Astrophysics and Cosmology” (1995) A. L. Peratt Editor, Kluwer Academic Publishers)

[4] Brandenburg, J.E. (2007). The Value of the Gravitation Constant and its Relation to Cosmic Electrodynamics, IEEE Transactions On Plasma Science, Plasma Cosmology Issue Vol. 35, No. 4., p845.

[5] Brandenburg J.E. (2011). The Value of the Newton Gravitation Constant Derived From a Combined Sakharov and Kaluza-Klein Model of Baryo-Genesis and Gravity EM-Unification, Meeting of the Division of Particles and Fields of the APS August 2011.

[6] J.E. Brandenburg, Experimental and Theoretical Progress on the GEM Theory, Phys. Procedia. 20 (2011) 197-205. doi:10.1016/J.PHPRO.2011.08.018.
https://www.researchgate.net/publication/251709895_Experimental_and_Theoretical_Progress_on_the_GE_M_Theory

[7] E. Markoulakis, E. Antonidakis, A ½ spin fiber model for the electron, Int. J. Phys. Res. 10 (2022) 1-17. doi:10.14419/ijpr.v10i1.31874. (hal-03544897) (https://tinyurl.com/2s44f7h7)

<footnote n="2">² https://testbook.com/question-answer/the-value-of-the-gravitational-constant-in-the-cgs--607085bb6c5e001788eb1806</footnote>
[8] E. Markoulakis, E. Antonidakis, Fine Structure Constant Model Demonstrates the Electron Elementary Charge of Having an Intrinsic Manifold, J. Appl. Math. Phys. 10 (2022) 2923–2939. doi:10.4236/jamp.2022.1010196. [hal-03814472]

[9] E.N. Markoulakis, The superluminous vacuum, Int. J. Phys. Res. 11 (2023) 18–25. doi:10.14419/IJPR.V11I1.32269. https://doi.org/10.14419/ijpr.v11i1.32269. Also found here, https://www.researchgate.net/publication/370773578_TheSuperluminous_vacuum and [hal-04098550]

[10] E.N. Markoulakis, Superluminal graviton condensate vacuum, Int. J. Phys. Res. 12 (2024) 45–61. doi:10.14419/BMR9G725. [hal-04663412v2]

[11] Markoulakis, Emmanouil, Higgs Boson LHC detection by two γ-photon decay emission described as a superluminal Relativistic Image Doubling effect (April 10, 2024). Available at SSRN: https://ssrn.com/abstract=4789892 or http://dx.doi.org/10.2139/ssrn.4789892