Speed of Light and the Principle of Energy Conservation

 

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The Speed of Light and the Principle of Energy Conservation

 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

                                                                                                  

Abstract

Adopting a priori the Principle of energy conservation, we found that the speed of light emitted (in the cosmological past) by nearby or distant galaxies with the cosmological redshift z_G is lower by the factor (1 + z_G) than the (current) speed of this light c (= 299792 km sec^-1) reaching the Earth. We have also suggested that the radius of the Hubble sphere in the same past was smaller by the same factor. We find that the light speed increased by nearly 2 cm sec^-1 y^-1 for about the last 12 Gy.

 

Keywords:  Principle of energy conservation, speed of light, nearby galaxy, distant galaxy, cosmological past, Hubble sphere 

Introduction 

Until recently, numerous scientists proposed their varying speed of light theories. Excellent reviews of these theories are presented by Magueijo {1} and Farrell and Dunning-Davies {2}. All of these scientists suggest that the speed of light continuously decreased to its current speed c over the lifetime of the Universe. In this communication, we will consider the possibility that the speed of light of nearby or distant galaxies increased to its present speed c through the cosmological history of the Universe. This possibility is entertained in communication by Premović {3} as an explanation of the redshift of the Lyα emissions sourced by the distant galaxy EGSY8p7. 

According to the standard cosmology, nearby and distant galaxies recede from the Earth because the Universe is expanding at a constant rate or better because of the space expansion. The distance between a nearby or distant galaxy and the Earth is related to their "cosmological redshift" z_G (> 0). The larger this distance, the higher the measured cosmological redshift. This shift at the present time is defined by the following equation 

zG = (λ – λ G)/λ _G ... (1) 

where λ_G is the wavelength of monochromatic light (or the stream of monoenergetic photons) (hereinafter light) emitted by a nearby or distant galaxy in the cosmological past and λ is the wavelength of this light measured by an Earth’s observer at present time. 

The energy loss of the wavelength of redshifted light is a major concern for cosmologists. Indeed, when this light moves from a nearby or distant galaxy to the Earth its energy appears to decrease. This violates the Principle of energy conservation. This issue has been of interest to many cosmologists but consideration of their approaches and solutions is beyond the scope of this work. 

Discussion and Derivations 

Modern cosmology claims that the universe of over 12.8 billion years ago (or 1 billion years after the Big Bang) was similar to today's Universe. In addition, after billions of years in the future, the Universe will be similar to the primordial (modern) Universe. The law of energy conservation is one of the most important laws of physics and has not been violated in any known physical process in our Universe. Hence, it is reasonable to assume that this law was not violated in the Universe from 12.8 billion years ago to the present day. Nor will it be violated in some distant future of the Universe. This communication adopts, a priori, the Principle of energy conservation. 

There are E – the energy of light emitted by the nearby or distant galaxy towards the Earth and measured by the Earth’s observer at present and E_G – the energy of this light when was emitted by the galaxy at time t (or in the cosmological past) and estimated by this observer knowing the value of E. These energies are given by 

E = hν and E_G = hν_G 

where h (= 6.63×10^-34 J sec) is Planck’s constant and ν_t and ν are the corresponding frequencies. 

At last, we know that the frequency equals the speed of light divided by the wavelength or ν = c/λ.  By the Principle of energy conservation E_G = E or ν_t = ν. If we denote with λ_t and c_t the wavelength and the speed of light emitted by a nearby or distant galaxy (in its cosmological past) then we have ν = c_t/λ_t = c/λ or c_t/c = λ_t/λ. Combining c_t/c = λ_t/λ with z_G = (λ – λ_t)/λ_t [(derived from eqn (1)] we arrive at

c _t = c/(1 + z_G) ... (2)

 

Hence, from the perspective of the Earth’s observer, the speed c_t of the light emitted by the nearby or distant galaxy at time t in the cosmological past is lower than the (current) speed of this light c reaching the Earth. If z_G → 0, c_t → c and if z_G → ∞, c_t → 0. Thus, for the Earth’s observer the speed of light is not constant throughout the Universe, but variable in time.  

In the near or distant cosmological future, the speed of light emitted by the nearby or distant galaxy reaching the Earth will be greater than c.[1] For the initial speed of this light to be the same, the speed at which it reaches the Earth must be proportional to (1 + its redshift). This and related issues will be considered in our separate communication.

 

By symmetry, from the perspective of the galaxy’s observer, the speed of light, c_t, emitted by our Milky Way is lower for (1 + z_G) than the speed of this light reaching the galaxy c in its present time. There is no reason why these two speeds would be the same for these observers in their reference frames. This is by Newtonian as well as Einstein physics. 

Let D_G stand for the distance (hereinafter distance) at time t in the cosmological past between the nearby or distant galaxy and the Earth. The galaxy’s light will cover the distance D_G with an average speed c_av = [c_t + c)]/2 < c. Since this speed is less than the current speed of light c, light from the galaxy will travel D_G longer time. So, the light will take longer to cover the distance D_G than if its speed is c. In other words, we deal with cosmological (or cosmic) time dilation. This dilation can be expressed by the following relation 

τ_C = D_G(1/c_av – 1/c)    … (3) 

(or τ = [c – c_t][c + c_t))](D_G/c)].[3]  

As the cosmological redshift of nearby and distant galaxies, z_G increases with their distance to the Earth, the speed of their emitted light will simultaneously increase with this distance. 

Hubble’s constant H₀ (= 7 × 10^-11 y^-1) represents the constant rate of the space expansion. The subscript “0” indicates the value of the Hubble constant today. The terms 1/H₀ and c/H₀ are the current age of the Universe and the current radius of the Hubble sphere (or the current Hubble’s length), respectively. For the above value of H₀ and the current speed of light c, their values are about 13.8 Gy and 13.8 Gly.

When the galaxy with z_G emitted light, the size of the Universe was 1/(1 + z_G)th of its current size or the radius of the Hubble sphere was about 1/(1 + z_G)th of its current radius. In other words, the ratio of the current radius of the Hubble sphere, R_H(0), and the radius of this sphere in the cosmological past, R_H(t) is equal 1 + z_G. In expression, 

                                                              R_H(0)/R_H(t) = 1 + z_G

or 

R_H(t)/R_H(0) = c_t/c    … (4).

We know that c_t/c = λ_t/λ then 

λ/λ_t = R_H(0)/R_H(t) = 1 + z_G. 

This equation corresponds to these relations 

λ/λ_t = S_o/S_t = 1 + z_G 

where S_t and S₀ are the scale factor values at the time t when the light was emitted by a nearby or distant galaxy and the time t₀ when it was detected by the Earth’s observer (t > t₀).

 

According to the Friedmann–Lemaître–Robertson–Walker metric, which is used to model the expanding Universe, if at present we receive light with a redshift of z_G, then the scale factor at the time the light was originally emitted is given by{\displaystyle a(t)={\frac {1}{1+z}}}

 

S_t = 1/(1 + z_G).

 

Multiplying this equation with the speed of light c we get

 

cS_t = c/(1 + z_G) = c_t.

 

The number of periods would be identical for R_H(t) and R_H(0). As we noted above, the speed of light divided by the frequency equals the wavelength of light (or ν = c/λ) then the number of their wavelengths would be also the same. In expression,

R_H(0) = Nλ and R_H(t) = Nλ_t

 

where N can be approximated as a very large natural number if it is larger or equal to say 10⁴ [6].

 

Examples: 

1.     As one of the direct distance measurement methods, the “megamaser” method has demonstrated its capability for precise distance measurement of nearby galaxies. However, it appears this method is suitable for very few of these galaxies - “megamaser” galaxies. For the present case, we select two “megamaser” galaxies with a negligible peculiar speed: the closest NGC 1052 and the farthest NGC 6264 {4}.

      NGC 1052 has z_G = 0.00493 and D_G = 65 Mly. Applying eqn. (2), we calculated that the speed of light emitted by this galaxy is about 298000 km sec^-1 and the average speed of this light to the Earth is (299792 km sec^-1 + 298000 km sec^-1)/2 ≈ 299000 km sec^-1. The time dilation of light from NGC 1052, calculated using eqn. (3), is about 0.43 ky. Using eqn. (4) we found that the radius of its Hubble sphere: 13.8 Gly/1.0043 ≈ 13.7 Gly.

 

2.    NGC 6264 has z_G = 0.0338 and D_G = 447 Mly. Using the same equations as above we find that the speed of its light is about 290000 km sec^-1 and the average speed of this light to the Earth is about (299792 km sec^-1 + 299000 km sec^-1)/2 ≈ 299000 km sec^-1. The time dilation of light from this galaxy, calculated by eqn. (3), is about 3 ky. Using eqn. (4) we found the radius of its Hubble sphere: 13.8 Gly/1.0338 ≈ 13.4 Gly.

 

3.      The quasar APM 08279+5255 has zG = 3.91. Its age of about 12 Gy was determined by measuring the Fe/O ratio [5, and references therein]. Applying eqn. (2), we find that the  speed of light emitted by APM 08279+5255, cAPM ≈ 61000 km sec-1 and the average speed of this light to the Earth is about (299792 km sec-1 + 61000 km sec-1)/2 ≈ 180000 km sec-1. The light-travel distance is about 12 Gly ≈ 137 × 1023 km. However, we suggest that this distance is smaller and can be calculated by the following formula 12cAV ≈ 732 × 1021 km. The cosmological redshift of APM 08279+5255 zAPM = 3.91 implies that the Universe expanded about 4.9 times between the emission of the light by this quasar and the observation of the same light by the Earth’s observer. Using eqn. (4) we found that the radius of its Hubble sphere: 13.8 Gly/4.9 ≈ 2.8 Gly. In all current cosmological models, the recessional speed of distant galaxies with zG > 1.5 exceeds the speed of light {6}. (Although there are still many debatable questions on this issue). Hence, this speed of APM 08279+5255 with zG = 3.91 exceeds the speed of light at present time and in the cosmological past. Of course, the motion of this galaxy at present time does not affect the light that it emitted 12 Gy ago. Using the above data for NGC 1052 and APM 09279+5255 we estimate that the speed of light increased by about 2 cm sec-1 y-1 for about the last 12 Gy. Of note, that Sanejouand {7}. estimated that this speed decreased by nearly 2 cm sec-1 y-1 for about the last 12.8 Gy.   Conclusions   If we apply the Principle of energy conservation, then we show that the speed of light ct emitted by nearby or distant galaxies at time t in the cosmological past is lower than the (current) speed of this light c reaching the Earth. The extent of this lowering depends on the factor (1 + zG). The Hubble sphere was smaller in the past than today for the same factor. We estimate that the speed of light increased by nearly 2 cm sec-1 y-1 for about the last 12 Gy.   References   {1} J. Magueijo, New varying speed of light theories. Rept. Prog. Phys., 66, 2025-2068 (2003). {2} D. J. Farrell and J. Dunning-Davies, The constancy, or otherwise, of the speed of light.

arXiv:physics/0406104[physics.gen-ph] (2004).

{3} P. I. Premović, The Big Bang Universe and the Principle of energy conservation. The General Science Journal, December 2021.

{4} P. I. Premović, The age of the “megamaser” galaxies in the Big Bang Universe. The General Science Journal, December 2021.

{5} J. van Royen, The Speed of Light Is Not Constant in Basic Big Bang Theory. Journal of Modern Physics, 14, 287-310 (2023).

{6} T.M. Davis, C.H. Lineweaver, Expanding confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe, Publ. Astronom. Soc. Australia 21, 97–109 (2004).

{7} Y.-H. Sanejouand, A simple varying-speed-of-light hypothesis is enough for explaining high-redshift supernovae data. arXiv:astro-ph/0509582v1-20 Sep 2005.

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[1] For the nearby galaxies is valid the Hubble law. By this law, the Hubble constant H₀ is the constant of proportionality between the cosmological redshift z_G of a nearby galaxy and its distance D_G to the Earth or z_G = [(H₀D_G/c]. Combining this equation with eqn.

[2], we find that c_t = c²/(c + H₀D_G). For the nearby galaxies with z_g ≤ 0.01 c_t ≈ c. [3] The redshift of some nearby galaxies z_G ≤ 0.01 then in their cases (1 + z_G) ≈ z_G.