In the past, the stars did not exactly follow the supposedly perfectly circular trajectory, it was necessary to imagine the existence of epicycles … until new elliptical trajectories were accepted. Today, the distant galaxies not having to exceed the speed of light in spite of the law of Hubble, it is imagined a space which would create itself its own space, and the distant supernovae not following exactly the speed away calculated there, it is imagined the existence of a repulsive dark energy … unless we come to accept a new formula for the speeds away as a function of the redshift. This is what we will try to demonstrate in this article.

After having recalled the definitions and uses of redshift and wavelength, we will analyze the research status on the relativistic redshift formula. Then we will propose the neo-Newtonian derivation to get to the expression of a slightly modified formula. Finally, we will discuss the consequences or not of this change of formula.

The redshift, noted z, is the fringe shift measured on light and electromagnetic waves from moving objects. From this measurement, first the speed away for stars and galaxies is calculated and then their distance is deduced using Hubble’s “law”.

Yet, at the horizons of the Universe, by correlating with the distances derived from their luminosity, supernovae appear to be at distances greater than those calculated by the redshift method (see Figure 1). To explain this additional distance speed and therefore this additional distance, it is invoked the existence of a dark energy with a repulsive effect, dark energy which has so far not been able to be directly apprehended. And more simply, why not question the formulas used to calculate the very fast speeds and the far distances?

Light and electromagnetic waves in general are characterized by their wavelength λ, their propagation speed c and their frequency f.

We can characterize them only by two out of three parameters because they are related to each other by the relation:

λ = c / f (1)

Since the speed of light in vacuum is assumed to be constant, it has become customary to characterize the electromagnetic waves by only one parameter, the wavelength λ(i.e. a distance). Yet this is the frequency (i.e. a time, or more exactly the inverse of time) that we perceive or directly measure. For example, an optical detector does not measure a length, but is excited at certain frequencies. This is why in this article we will favor the explanation by frequencies f (rather than by wavelengths λ).

The relativistic expression allowing to deduce the distance speed V from the redshift z is as follows:

V = − ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 c (2)

This formula comes up against two difficulties.

We have derivated within the framework of neo-Newtonian mechanics the value V of the speed from the source in function of the redshift to get to a new formula:

V = z 1 + z c . This formula has the double characteristic of being equivalent to

relativistic predictions for low redshifts, but of deviating from them by up to 17% for high redshifts.

What can be the consequences of future research?

For any observed physical phenomenon, we seek an explanation. For example:

1) As the speed of expansion of the Universe cannot exceed the speed of light, it is assumed that space itself is expanding.

2) As the light interference remains identical despite the movement of the source, it is assumed that the speed of light would be constant in any frame of reference.

3) As our Galaxy being in motion with respect to the Cosmic Microwave Background (in contradiction with the isotropy of the theory of Relativity), it is assumed that it would be subjected to the Dipole Repeller.

4) As the Pioneer spacecrafts seem to be slowing down more than expected, it is assumed that they would be slowed down by their own flow of photons.

5) As distant supernovae appear less bright than expected, it is assumed that there is a dark energy that makes them accelerate.

Faced with these difficulties of interpretation, we propose here a change of paradigm, namely that resulting from neo-Newtonian mechanics:

1) We have expressed and demonstrated a slightly different formula relating the redshift to the speed of movement of the source. Thus, it is no longer necessary to imagine an expanding space by itself, which does not prevent galaxies from moving away from each other as a result of an explosive-type phenomenon.

2) We have shown that a variable frequency and speed of propagation depending on the movement of the source can explain a constant wavelength and therefore stable light interferences.

3) The observed anisotropy of the Universe can be explained very well by considering the Cosmic Microwave Background as the expression of a fixed or barycentric frame R₀.

4) The slight blueshift effect of Pioneer spacecrafts is proof of the validity of this new redshift formula:

5) The Universe retains many mysteries, and not just about distant supernovae and quasars.

This new analysis grid proposed here (within Neo-Newtonian Mechanics) would allow any researcher interested in it to shed new light on his research, to test it and would surely open up the field of possibilities.

I would like to thank BGD for the reading and for the support.

The author declares no conflicts of interest regarding the publication of this paper.

Serret, O. (2021) A New Formula of Redshift vs. Space Expansion and Dark Energy. Journal of Modern Physics, 12, 229-253. https://doi.org/10.4236/jmp.2021.123018

Let us have

· An observer without motion (he is in his own frame)

· A source moving at the velocity v s o u r c e (in the observer’s frame). So

β = v s o u r c e c (1.1)

Relativistic equation hereafter

1 + z = 1 + β 1 − β (1.2)

is equivalent to

β = ( 1 + z ) 2 − 1 ( 1 + z ) 2 + 1 (1.3)

The links between the frequencies and the periods are

1 + z = f s o u r c e f o b s (1.4)

1 + z = T o b s T s o u r c e (1.5)

We’ll check two cases, the redshift and the blueshift one.

1) Edshift Case

See specific configuration of Figure 6.

· The distance of emission source of the photon (B) at the time t 1 = T s o u r c e :

d s o u r c e = v s o u r c e ( t 1 − t 0 ) (1.6)

with v s o u r c e and so β < 0 .

· And the length traveled by the photon (B) to reach the observer is

L = v p h o t o n ( t 2 − t 1 ) (1.8)

L = c ⋅ ( T o b s − T s o u r c e ) (1.9)

And with the Equation (1.5)

L = c ⋅ ( ( 1 + z ) − 1 ) ⋅ T s o u r c e (1.10)

2) Blueshift Case

See specific configuration of Figure 7.

· The distance of emission source of the photon (B) at the time t 1 = T s o u r c e :

d s o u r c e = λ (1.12)

λ = v p h o t o n T s o u r c e (1.13)

λ = c T s o u r c e (1.14)

Then

· And the length traveled by the photon (B) to reach the observer is 16

L = v p h o t o n ( t 2 − t 1 ) (1.16)

L = c ⋅ T o b s (1.17)

Numerical application (with for example T_source = 1 second):

Let us apply in Table 2 previous formulas with some example of z and with the difference of length:

δ = L − | d | (1.19)

So to explain the redshift, you need to suppose a Universe expansion (for example a Universe expansion of 0.4*c*T for z= 1).

And to explain the blueshift, you need to suppose a Universe contraction (of −0.1*c*T for z= −0.1)!

 Without a Universe which would expand by itself, the relativistic formula of redshift is uncoherent!

Let us have a coordinate system oriented according to the propagation of the photons. According to the motion of the Source and the motion of the Observer, there are 4 cases. We illustrate 2 cases (see Figure 8 and Figure 9).

1) About the source:

Let us have a source emitting photon (A) at time t₀ and photon (B) at time t₁

By definition, the source period T s o u r c e is

T s o u r c e = t 1 − t 0 (2.1)

d s o u r c e is the distance traveled by the source at the velocity v s o u r c e during a period T s o u r c e

d s o u r c e = v s o u r c e ⋅ T s o u r c e (2.2)

D_A is the distance traveled by the photon (A) at the velocity v_A during a period T s o u r c e

D A = v A ⋅ T s o u r c e (2.3)

λ is the distance between photon (A) and photon (B)

λ = D A − d s o u r c e (2.4)

Please note these are algebraic value. If d s o u r c e is positive, λ will be lower than D_A. If d s o u r c e is negative, λ will be greater than D_A (please check Figure 8 with t₁ for d s o u r c e positive, and Figure 9 with t ′ 1 with d s o u r c e negative).

2) About the observer:

Let us have an observer receiving photon (A) at time t₂ and photon (B) at time t₃

By definition, the observer period T o b s is

T o b s = t 3 − t 2 (2.5)

l o b s is the distance traveled by the observer at the velocity v o b s during a period T o b s

l o b s = v o b s ⋅ T o b s (2.6)

L B is the distance traveled by the photon (B) at the velocity v B during a period T o b s

L B = v B ⋅ T o b s (2.7)

λ ′ is the distance between photon (A) and photon (B)

λ ′ = L B − l o b s (2.8)

Please note these are algebraic value. If l o b s is positive, λ ′ will be lower than L B . If l o b s is negative, λ ′ will be greater than L B (please check Figure 8 at t 3 for l o b s positive, and Figure 9 at t ′ 3 for l o b s negative).

Because the motion of the source is continuous, the velocities of the photons are all the same

v B = v A (2.9)

Because the velocities of the photons are the same, the distance between photons remain the same

λ ′ = λ (2.10)

What is perceived is the frequency, not the wave length. So the redshift z is, with f for frequency

1 + z = f s o u r c e f o b s (2.11)

Remark 1: In Newtonian mechanics and so in Neo Newtonian Mechanics, there is no difference of length whatever it is measured from the Observer frame or from the Referential frame.

Remark 2: when c is constant please note Equation (2.11) is equivalent to

1 + z = λ o b s λ s o u r c e (2.12)

By definition of the frequency

f = 1 T (2.13)

And let us define V the distance speed of the source from the observer by

V ≅ v s o u r c e − v o b s (2.14)

In Neo-Newtonian Mechanics, the velocities cannot be directly added for very high velocities. In the present case, we consider that the source and the observer are in usual velocity, so we can use the approximatively equal symbol.

Using Equations (2.4), (2.3), (2.2)

T s o u r c e = λ v A − v s o u r c e (2.15)

Using equations (2.6), (2.7), (2.8), (2.9), (2.10)

T o b s = λ ′ v B − v o b s (2.16)

Using equations (2.9) and (2.10)

T o b s = λ v A − v o b s (2.17)

Using Equations (2.11) and (2.13)

1 + z = T o b s T s o u r c e (2.18)

Then using Equations (2.15) and (2.17)

1 + z = v A − v s o u r c e v A − v o b s (2.19)

v A − v o b s + z ( v A − v o b s ) = v A − v s o u r c e (2.20)

z ( v A − v o b s ) = v o b s − v s o u r c e (2.21)

So using Equation (2.14)

V ≅ − z ( v A − v o b s ) (2.22)

· When V is positive, the source gets closer to the observer, it is the blueshift.

· When V is negative, the source gets away from the observer, it is the redshift.

Lest us now consider 2 cases, either the photon is moving in the same direction than the source ( v s o u r c e > 0 ), either the photon is moving in the opposite direction from the source ( v s o u r c e < 0 ).

In Neo-Newtonian Mechanics, a photon cannot go faster than the asymptotic limit s, which means that the faster velocity of the photon is c.

{ if   [ v s o u r c e > 0 ]     then   [ v A = c ] if   [ v s o u r c e < 0 ]     then   [ v A ≅ c + v s o u r c e ] (2.23a-b)

Please note that in the second case, v A is not greater but lower than c, because v s o u r c e < 0 .

1) v s o u r c e > 0 (see Figure 10)

Using Equations (2.22) and (2.23 a)

Please note that usually v o b s ≪ c , then we get

V ≅ − z ⋅ c (2.25)

That means that when the source is coming toward us relatively to the absolute frame, we will get a blueshift except if the Earth is moving away faster the the source.

Physically, this is not the usual case.

2) v s o u r c e < 0 (see Figure 11)

Using Equations (2.22) and (2.23b)

V ≅ − z ( c + v s o u r c e − v o b s ) (2.26)

Then with Equation (2.14)

V ≅ − z ( c + V ) (2.27)

( 1 + z ) V ≅ − z ⋅ c (2.28)

So we get the general formula of the redshift with the algebraic value V:

· When z is positive, V is negative, the source moves away from the observer, it is the redshift.

· When z is negative, V is positive, the source moves toward the observer, it is the blueshift.

Please note that when z ≪ 1 , then we get

V ≅ − z ⋅ c (2.30)

Let us have

· An observer without motion (he is in his own frame)

· A source moving at the velocity v s o u r c e (in the observer’s frame). So

β = v s o u r c e c (3.1)

NNM equation hereafter for the redshift, when the axis is oriented with the photon

1 + z = 1 1 + β (3.2)

Is equivalent to

β = − z 1 + z (3.3)

Period from redshift

1 + z = f s o u r c e f o b s (3.4)

1 + z = T o b s T s o u r c e (3.5)

We’ll check two cases, the redshift and the blueshift one.

1) Redshift case

See specific configuration of Figure 12.

· The distance of emission source of the photon (B) at the time t 1 = T s o u r c e :

d s o u r c e = v s o u r c e ( t 1 − t 0 ) (3.6)

with v s o u r c e < 0 and so β < 0 .

· And the length traveled by the photon (B) to reach the observer is

L = v p h o t o n ( t 2 − t 1 ) (3.8)

for the redshift ( v s o u r c e < 0 ) using the Equation (2.23b) already seen

v p h o t o n ≅ c + v s o u r c e (2.23b bis)

with v s o u r c e < 0

L = ( c + v s o u r c e ) ⋅ ( T o b s − T s o u r c e ) (3.9)

Using Equations (3.1) and (3.2):

c + v s o u r c e = c + β ⋅ c (3.10)

1 + β = 1 1 + z (3.11)

So

c + v s o u r c e = c 1 + z (3.12)

And using Equation (3.5)

z ⋅ T s o u r c e = T o b s − T s o u r c e (3.13)

Then from Equation (3.9)

L = c 1 + z ⋅ ( z ⋅ T s o u r c e ) (3.14)

So in the case of the redshift, δ is the difference

δ = L − | d | (3.16)

δ = c ⋅ z 1 + z ⋅ T s o u r c e − ( | β | ⋅ c ) T s o u r c e (3.17)

According to NNM, for the redshift when the axe is oriented with the photon:

| β | = z 1 + z (3.18)

Then

2) Blueshift case

See specific configuration of Figure 13.

· The distance of emission source of the photon (B) at the time t 1 = T s o u r c e :

λ = D − d (3.20)

λ = v p h o t o n ⋅ T s o u r c e − v s o u r c e ⋅ T s o u r c e (3.21)

for the redshift ( v s o u r c e > 0 ) using the Equation (2.23a) already seen

v p h o t o n = c (2.23a bis)

λ = c ⋅ T s o u r c e + z ⋅ c ⋅ T s o u r c e (3.22)

with z< 0

· And the length traveled by the photon (B) to reach the observer is

L = v p h o t o n ( t 2 − t 1 ) (3.24)

L = v p h o t o n T o b s (3.25)

Using Equation (3.3)

T o b s = ( 1 + z ) T s o u r c e (3.26)

So in the case of the blueshift:

δ = L − λ (3.28)

δ = c ⋅ ( 1 + z ) T s o u r c e − c ( 1 + z ) ⋅ T s o u r c e (3.29)

δ = 0 (3.30)

Numerical applications (with T_source = 1 second).

Let us apply in Chart 3 previous formulas with some example of z and with the difference of length δ with:

for the blueshfit, δ = L − | λ | (3.31)

for the blueshfit, δ = L − | d | (3.32)

Chart 3. Neo-Newtonian numerical application.

1 + z = f s o u r c e f o b s (4.1)

1) According to NNM for the redshift

if   [ v s o u r c e < 0 ]   then   [ v A ≅ c − | v s o u r c e | ] (4.2)

v s o u r c e = z 1 + z c (4.3)

So

v A ≅ c ( 1 − z 1 + z ) (4.4)

v A ≅ c 1 1 + z (4.5)

λ o b s = v A f o b s (4.6)

λ o b s ≅ ( c 1 1 + z ) ( 1 + z f s o u r c e ) (4.7)

λ o b s ≅ c f s o u r c e (4.8)

λ o b s ≅ λ s o u r c e (4.9)

2) And according to NNM for the blueshift

if   [ v s o u r c e > 0 ]   then   [ v A = c ] (4.10)

λ o b s = v A f o b s (4.11)

λ o b s = c ( 1 + z f s o u r c e ) (4.12)

λ o b s ≠ λ s o u r c e (4.13)