Fine Structure Constant and the Minimum Speed of Light

 

 

Fine Structure Constant and the Minimum Speed of Light

 Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

University of Niš, pavleipremovic@yahoo.com, Niš, Serbia

                                                                    

                                                                                                                     When I die my first question to the Devil will be:                                                                                                                     What is the meaning of the fine structure constant?Wolfgang Pauli 

Adopting a priori the Principle of energy conservation, Premović {1} proposed that the speed of light c_t emitted by the nearby or distant galaxy (hereinafter the galaxy) at time t in the cosmological past is lower than the (current) speed of this light c reaching the Earth. He derived the following expression which relates c_t and c

c_t = c/(1 + z_G) ... (1) 

where z_G is the cosmological redshift of the light emitted by the galaxy in the past but measured by an Earth observer at present time. 

The fine structure constant, denoted by α, is a measure of the strength of electromagnetic interaction between light (or photons) and electrons (or charged elementary particles). This constant is considered one of the most fundamental constants in physics. Moreover, cosmologists consider this constant to be one of the fundamental constants that describe the Universe. 

The fine structure constant includes four of nature’s fundamental constants: the permittivity of the free space ε₀, the electric charge of electron e, Planck’s constant h and the speed of light c. This can be formulated by 

a = and²/2e₀hc = 1/137 

where α is dimensionless, ε₀ (= 8.85 × 10^-12 F m^-1), e (= 1.60 × 10^-19 C), h (= 6.63 × 10^-34 J sec^-1) and c (= 2.99792 × 10⁸ m sec^-1).

Multiplying the left and right ends of this equation with c, we get

αc = c/137.

Obviously, αc represents “some” speed of light. In view of the eqn. (1), the product αc represents a minimum speed of light emitted by the galaxy with redshift z_G = 136 in the distant cosmological past. In the equation form,

 αc = c_min = c/137 = c/(1 + z_G) = c/(1 + 136) ... (2). 

Thus, we suggest that the exact formula for the fine structure constant is 

α = c_min/c. 

When the light was emitted from the galaxy with z_G = 136 the speed of this light was 1/137th of its present speed c.^, As the current speed of light c, the minimum speed of light c_min is also independent of the frame of reference. In other words, we define the minimum speed of light c_min as a low-speed limit of the speed of light in the past of our Universe. [1],[2] 

Let us denote by λ_min and λ the wavelengths corresponding to c_min and c. We know that the ratio of the speed of light to the wavelength is equal to the frequency. In this case 

c min/min_min = c/_{l = n}. 

Combining this equation with αc = c_min = c/137 = and after a bit of algebra we get 

λ_min = αλ = λ/137. 

In other words, the Earth’s observer measures the wavelength of the light emitted by the nearby or distant galaxy redshifted for the factor 1/137. 

We know that light (or better a stream of photons) carries the momentum p = h/wavelength. So, the light with c_min carries the maximum momentum 

P_max = h/λ_min = 137h/λ = 137p. 

This momentum is related to the energy by the simple equation  

E = p_maxc_min = hc/λ = hν. 

When the galaxy with z_g = 136 emitted light, the size of the Universe was 1/137th of its current size or its radius was about 0.1 % of its current radius. The age of this ancient Universe of course depends on the dynamics of the space expansion. With the current concordance cosmology of a flat Universe with a matter density Ω_m ≈ 0.25, a cosmological constant Ω_Λ ≈ 0.75 and a Hubble constant H₀ = 72 km sec^-1 Mpc^-1, z_G =136 corresponds to an age of about 11 My or roughly 0.1 % of current cosmological time. At that age, most of the molecular hydrogen (H₂) in the present-day Universe was already formed. This form of hydrogen is the predominant gas constituent in this Universe. Simultaneously, the photons were probably completely decoupled from matter spreading through the Universe to form the cosmic microwave background (CMB) radiation. The Wilkinson Microwave Anisotropy Probe (WMAP) combined with other measurements reveals that the redshift of this decoupling is 1091, implying that the size of the Universe or its radius at the time when the light was emitted was 1/1092th of its present size or radius. [3] 

As we noted above, we interpret the fine structure constant as the ratio between the minimal speed of light c_min to current this speed c. This constant was initially formulated by Sommerfeld in 1916 as the ratio of the speed υ of the electron in the first Bohr’s orbit of a hydrogen atom to the speed of light: α = υ /c. However, these two speeds are completely different in physics and completely independent from each other. Therefore, this ratio has nothing to do with the nature of the fine structure constant.   

Numerous physicists and mathematicians have tried to find the most accurate mathematical formula for the experimental value of fine structure constant α. Excellent reviews of these theories are presented by Sherbon [2, and references therein] and Pellis {3, 2021, and references therein]. The most recent inverse value of this constant α^-1 determined theoretically and experimentally is 137.036 {3}. Einstein was the first to propose in 1909 a mathematical formula for a fine constant structure. His formula is 

a = 7p/10³

with numerical value α^-1 = 136.426. If we multiply by 10⁵ both the numerator and the denominator of the right-hand side of this equation we obtain 

a = 7p × 10⁵/3 × 10⁸. 

At first glance, it can be seen that the denominator represents the approximate current value of the speed of light c ≈ 3 × 10⁸ m sec^-1. Using the mathematical constant π to be approximately equal to 3.142 we estimate that 7π × 10⁵ is approximately equal to the minimum speed of light c_min ≈ 2.2 × 10⁶ m sec^-1. The numerical value of α^-1 is approximately equal to 136.400. 

References

{1} P. I. Premović, The Speed of Light and the Principle of Energy Conservation. The General Science Journal, April 2022.

{2} M. A. Sherbon, Fine structure constant from Sommerfeld to Feynman. J. Adv. Phys. 16, 335-343 (2019).

{3} S. Pellis, Exact expressions of the fine structure constant α.  https://vixra.org/abs/2110.0117, pdf, 2.021. 

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[1] It can be easily calculated that αc = c_min ≈ 2.2 × 10⁶ m sec^-1.

[2] From the frame of reference of galaxy, the speed of light on the Earth is c/137.

[3] The currently highest redshift observed is z_G = 11.01 (the galaxy GNz-11), the Universe was about 1/12th of its present size or its radius was about 1 % of its present radius.