The Tired-Light Hypothesis: Derivations of Basic Relations

 

The Tired-Light Hypothesis: Derivations of Basic Relations

 

Pavle I. Premović

Laboratory for Geochemistry, Cosmochemistry&Astrochemistry,

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

The Big Bang model is the generally accepted theory for the origin and evolution of the Universe although it faces many challenges. There are other alternative theories about the origin of the Universe and its evolution. One of them is the "tired-light” theory.

This theory was originally proposed by Zwicky {1} in 1929 immediately after Hubble {2} discovered the linear distance-redshift relationship of some galaxies (outside local group) through observations. This theory explains the cosmological redshift z as a result of energy loss of photons of the light emitted by a nearby or distant galaxy.[1] This loss is due to the interactions of photons with material particles as they travel through intergalactic space. According to the tired-light theory, space is Euclidean, static, slowly evolving {4} and probably infinite.

In this communication, we will derive the basic equations of the tired-light theory using a derivation similar to that for radioactive decay.

We denote with E₀ the initial energy of a photon emitted by a nearby or distant galaxy and with E_d its energy at a distance d away which remains after n light-years. Of course, n is an integer. Let us assume that after each unit of distance, in this case, a light year (ly), the same fraction of the photon energy is transferred to the intergalactic material particles; we will call it the energy fraction f. It can be easily shown that after n light-years the energy of a photon would be

E_d/E₀ = fⁿ    … (1).

Applying the natural logarithm to both sides of this equation, we get

ln (E_d/E₀) = n × lnf    … (2)

The photon distance d away from the galaxy is related to the number of light-years as follows

d = n × (ly)    … (3).

Combining eqns. (2) and (3) and rearranging we get

ln (E_d/E₀) = [lnf/(ly)] × d    … (4).

The term in square brackets on the right side of this equation is constant and LaViolette {5} defined it as “the energy attenuation coefficient” (or “the rate of energy attenuation”) and denoted with β. The term lnf is negative because f is a fraction then

β = - [lnf/(ly)]    … (5).

This coefficient is a positive number and has the unit of (ly)^-1. Introducing β instead the term of lnf/(ly) of eqn. (4) we obtain

ln (E_d/E₀) = - βd

or

E_d = E₀ e^{- βd}.

This formula, at first sight, is similar to the corresponding formula for more familiar radioactive decay: t_1/2 = 0.692λ where t_1/2 is the half-time of a radioactive atom and λ is its decay constant. There is, however, a fundamental distinction between them. In radioactive decay, the radioactive atom decays. In contrast, in the tired-light case, the photon emitted by the galaxy does not decay.

If D represents the distance between the galaxy and the Earth then E_d = E_D where E_D is the energy of the photon when it reaches the Earth then eqn. (1)

E₀/E_D = 1/fⁿ.

We can easily find that

λ_D/λ₀ = fⁿ 

where λ₀ and λ_D are the corresponding photon’s wavelengths. We know that the greater distance between the galaxy and the Earth, the larger the wavelength of light of this photon reaching the Earth

Subtracting 1 from both sides of the last equation and after a bit of algebra we have

(λ_D − λ₀)/λ₀ = 1/fⁿ − 1.

We know that the left side of this equation is a redshift z of a photon reaching the Earth. So

1 + z = 1/fⁿ

or

f = nth root of 1/(1 + z).

The cosmological redshift z is a universal physical constant and is independent of the wavelength of a photon emitted by a galaxy. This independence is regularly found in all galaxy spectra with a precision of about 10^-4 [6]. So, according to the last equation, the energy fraction f is independent of a photon wavelength. Hence the energy attenuation coefficient β must be also independent of this wavelength [see eqn. (5)], as well as the attenuation time constant τ (= βc) and the attenuation half-time θ_1/2 [see eqn. (6)].

References

{1} F. Zwicky, On the red shift of spectral lines through interstellar space. Proc. Nat. Acad. Sci, 15, 773-779 (1929).

{2} E. Hubble, A relation between distance and radial velocity among extra-galactic nebulae. Proc. Nat. Acad. Sci, 15, 768-773 (1929).

{3} P. I. Premović, Distant galaxies in the non-expanding (Euclidean) Universe: the light speed redshift. The General Science Journal, December 2021.

{4} P. A. LaViolette, Genesis of the Cosmos. The ancient science of continuous creation. Bear & Company (2004).

{5} P. A. LaViolete, Is the Universe really expanding? ApJ. 301, 544-553 (1986).

{6} I. Ferreras and I. Trujillo, Testing the wavelength dependence of cosmological redshift to Δz ∼ 10^-6. ApJ, 825 – 842 (2016).

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[1] We will define nearby galaxies as those whose redshift z is from 0.001 to 0.1 (or 0.001 ≤ z ≤ 0.1) and distant galaxies with z > 0.1. Of course, there is no sharp boundary between nearby and distant galaxies {3}.