Euclidean Special Relativity

Abstract
A Euclidean interpretation of special relativity is given wherein proper time $\tau$ acts as the fourth Euclidean coordinate, and time $t$ becomes a fifth Euclidean dimension. Velocity components in both space and time are formalized while their vector sum in four dimensions has invariant magnitude $c$. Classical equations are derived from this Euclidean concept. The velocity addition formula shows a deviation from the standard one; an analysis and justification is given for that.

Introduction
Euclidean relativity, both special and general, is steadily gaining attention as a viable alternative to the Minkowski framework, after the works of a number of authors. Amongst others Montanus [1,2], Gersten [3] and Almeida [4] (for references see second attachment), have paved the way. Its history goes further back, as early as 1963 when Robert d'E Atkinson [5] first proposed Euclidean general relativity.

The version in the present paper emphasizes extending the notion of velocity to the time dimension. Next, the consistency of this concept in 4D Euclidean space is shown with the classical Lorentz transformations, after which the major inconsistency with classical special relativity, the velocity addition formula, is addressed. Following paragraphs treat energy and momentum in 4D Euclidean space, partly using methods of relativistic Lagrangian formalism already explored by others after which some Euclidean 4-vectors are established.
With permission of the moderator, I refer to the attached document parts for the remaining sections. Each attachment contains 5 pages of the article.
The article has been accepted for publication in Galilean Electrodynamics and is copied here with permission.

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Related Other Physics Topics News on Phys.org
I thank all the visitors who came to read my post so far but you are of course also welcome to criticize, challenge, discuss etc. the article. More background info about Euclidean relativity, including links to most of the refererenced articles, is available at www.euclideanrelativity.com/links.htm[/URL].
Best regards,
Rob

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member 11137
I did visit your webpage: wouah; a beautiful work. Concerning your theory, I am not able to criticize precisely; I just have some problem with your philosophy (if the solution is complicated this means that you didn't understand the problem) but it is not important and it doesn't matter for the informations that your work is containing. The idea that we are living in a kind of projection of something greater is modern and you can read a lot of articles about this (Pour la Science january 2006: is Gravitation an illusion?...); who knows ? Good luck

Blackforest said:
I did visit your webpage: wouah; a beautiful work. Concerning your theory, I am not able to criticize precisely; I just have some problem with your philosophy (if the solution is complicated this means that you didn't understand the problem)....
Thanks for the compliment, Blackforest!
Never mind my philosophy. It's basically another way of saying "Keep it simple".

I see from some of the links this is being extended into GR as well. As you view the progress t you have an opinion on if Euclidean Relativity (I’ll wait till it’s ‘established’ before shorting that to “ER”) will concluded these “fifth dimensions” will combine in a way that we see our reality based on a dependent background. Or will it be background-independent as Lee Smolin and Loop Quantum Gravity folks expect is true.

Also, do you see Euclidean Relativity developing to a point where it might be able to address issues of entanglement or superposition observations.

RB

RandallB said:
I see from some of the links this is being extended into GR as well. As you view the progress t you have an opinion on if Euclidean Relativity (I’ll wait till it’s ‘established’ before shorting that to “ER”) will concluded these “fifth dimensions” will combine in a way that we see our reality based on a dependent background. Or will it be background-independent as Lee Smolin and Loop Quantum Gravity folks expect is true.
I'll have a look at Smolin's article 'The case for background independence' first before I try to answer this (I'm afraid I'm not at all familiar with the topic).
RandallB said:
Also, do you see Euclidean Relativity developing to a point where it might be able to address issues of entanglement or superposition observations.
There are a couple of speculations on the http://www.euclideanrelativity.com/idea" [Broken] on my website that might indeed be related to entanglement, although these are not directly based on Euclidean relativity pur sang. I'm quite convinced that entanglement can have its basis in closed dimensions. It can however show very differently, depending on the particular dimensions that are taken into account. The page gives three examples: photon/photon, positive/negative charge and schwarzschild/'edge-of-universe' entanglement, respectively based on 3, 4 and 5 dimensional closed manifolds. Again, these are pure speculations.

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RandallB said:
As you view the progress t you have an opinion on if Euclidean Relativity (I’ll wait till it’s ‘established’ before shorting that to “ER”) will concluded these “fifth dimensions” will combine in a way that we see our reality based on a dependent background. Or will it be background-independent as Lee Smolin and Loop Quantum Gravity folks expect is true.
I have read http://arxiv.org/abs/hep-th/0507235" [Broken] with interest (and hope to have grasped correctly some essentials of it). As to your original question whether Euclidean relativity would give rise to background-dependence or rather background-independence, I'm inclined to say: both. The difference seems to have its roots in the absolute versus relative approach of space-time and from the various articles on Euclidean relativity, both standpoints can be defended. The work of Hans Montanus is based on an absolute Euclidean space-time and would thus obviously imply background-dependence. My own work on Euclidean special relativity is based on a relative Euclidean space-time and thus implies background-independence. However, in my second article, "Mass particles as bosons in 5D Euclidean gravity", which in fact deals with general relativity, it becomes clear that whatever is called relative or absolute space-time depends on its turn on the dimensional viewpoint of the observer (as defined in in the article). Any n-dimensional space $X_n$ is absolute to any (n+1)-dimensional space-time $X_{n+1}$ but allows a relative observation of lower-dimensional space-times $X_{n-1}$, $X_{n-2}$ etc. Time $t$ defined as a fifth dimension (with $\tau$ as the fourth) constitutes an absolute background, but only for $X_4$. The overall picture, though, is that the "place on the ladder" could never be determined by any observer, since any $X_n$ would be indiscernible from any other $X_m$ if observational skills of the observer would be limited to the same number of dimensions (see also the considerations in section 6 of the "ideas"-page on my website) and so from this point of view the balance leans over to background-independence (but that point of view would require some sort of dimension-independent "super"-observer).

Smolin also uses the discussion in relation to current stringtheories.
I'm familiar with stringtheories only on a more or less popular level and from this level have a basic understanding of some of its key elements, like dualities, M-theory versus the series of individual theories that lie behind it, Calabi-Yau space and multi-dimensional p-branes. There are a number of parallels that I see between the fractal-like Euclidean model of the universe (described on the "ideas"-page) with its fundamental forces and particles on one hand and stringtheory elements on the other hand. A couple of examples:
- The fractal-universe can be "observed" from different dimensional viewpoints which would each give a different mathematical model as well (each model being associated with a unique number of dimensions). For the "closest" dimensional viewpoints, i.e., the one from our own $X_4$, together with $X_1$, $X_2$, $X_3$ and $X_5$, this would result in rather concrete theories, while the more "distant" viewpoints would be less obvious, but nevertheless mathematically possible. I see here some links with the theoretical possibility of many more stringtheories (in particular in Euclidean space-times) while there must exist a dimension-independent mother-theory that describes the basic principles of each of them. Forgive me my rather non-scientific approach in this description; I'm actually trying to point out a philosophical point of view.
- Dualities in stringtheories could be linked to the dualities that I describe between fermions and bosons. Each fermion in $X_n$ corresponds to a boson in $X_{n+1}$, i.e., they are physically the same entity but described from a different dimensional viewpoint. This may perhaps also be a basis for supersymmetry. In principle, each particle should have a mathematically describable and associated counter-particle from its neighboring dimensional viewpoint. It would however be the same particle in fact, observed from another (higher or lower dimensional) side.
- P-branes may be directly linked to particles in n dimensions as listed in the table of section 6 of the "ideas"-page.

The connection with Euclidean relativity lies in the fact that the Euclidean space-time, extrapolated to the factal-like model of the universe, is far better equipped to support this "visual" interpretation and allows natural interpretations of various elements of stringtheory, the lack of which seems to have been hampering stringtheories from the beginning. The inherently confusing Minkowski geometry is not really helpful in visualizations.

Perhaps the most interesting contribution of the fractal-universe model based on Euclidean relativity is that quantum gravity arises from it completely naturally. The full quantum description of electromagnetism based on a 4D Euclidean space-time can in principle be ported one-to-one to gravity based on a five dimensional Euclidean space-time with mass particles acting as its bosons.

I hope these (admittedly extremely speculative and totally-and-absolutely-not-mathematically-founded) thoughts appeal a bit. I realize that I have been reasoning according to Euclidean space-time models for years already while this all may sound cryptic to anyone who does not have that background. I would not be surprised at all (and not offended either) if anyone with a more thorough mathematical background in stringtheories and QFT wipes the floor with these ideas in an instant.

Rob

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B.t.w., my original post was accepted for this forum based on the first article "Dimensions in special relativity theory" which the moderators have found to be in accordance with the rules for this forum. The subsequent posts have a tendency to divert to other topics (from my other articles) but I'm not sure if this is appreciated by the moderators.
Rob

I understand your point on wanting to stay focused on the "Dimensions in special relativity theory" and that is my intent to stick with your treatment of SR, but that it can ultimately have an effect on future treatment of GR is unavoidable and I’m sure expected.

I must admit I have a little trouble with the idea;
Mortimer said:
As to your original question whether Euclidean relativity would give rise to background-dependence or rather background-independence, I'm inclined to say: both.
I convinced that reality can only be one or the other on this point, and a useful theory should distill from within itself which approach is correct for it. The Smolin’s article is the best background explanation I’ve seen, but fairly new to me as well. For me it seems both GR and QM indicate background-independence. You know your theory best and may want to continue to see if it goes one way or the other – in the long run I think the point would become important. IMO your's also seems to be background-independent, but you would know better about defining a 5th Dimension action of a local time and place, from a coordinate starting from some other location and time action.

If the introduction of the extra dimensions can be shown as retaining background-dependence in your Multi-D SR and that a version of GR could then later be derived. Thus, overturning some the points Smolin has made this would be a very big deal. I think it worth your time to continue looking at his part.

Classical SR is background-dependent mostly because of its simplicity, somehow it doesn’t seem right that it should need to become more complex – but that’s just me. Clearly science cannot survive on classical SR alone anyway.

As to my questions on entanglement and superposition:
I agree this area is speculative at best at this point as it requires advancing this theory into either the GR or Quantum areas first, that should be something for future work.

RB

RandallB said:
I think it worth your time to continue looking at his part.
Your remarks and suggestions have indeed already triggered me to delve somewhat deeper in this topic. It sounds rather interesting and seems like something that I have intuitively missed in my considerations so far.
Rob

Good work

I agree with the proposed Euclidean Relativity.

http://www.euclideanrelativity.com/dim2html/node3.html [Broken]

Mortimer said:
...In that case the photon's velocity vector rotates towards the fourth dimension when nearing an electrical charge, explaining its lower velocity in matter. Mass particles falling into black holes is then equivalent to photons being absorbed by electrical charges.
I think you and I have similar intuitions in physics. This is precisely an idea I have repeated in my mind, and you have the experience to give it scientific support. Thank you so much!

http://www.euclideanrelativity.com/simplified/index.htm [Broken]
Mortimer said:
Like photons are the boson for the electromagnetic field, so are mass particles themselves the boson for the 5D gravity field (so forget about the illustrious graviton). Black holes obviously are its fermions.

Bosons and fermions are one and the same thing. What looks like a fermion from "below" (e.g. a 3-dimensional viewpoint) looks like a boson from "above" (e.g. a 4-dimensional viewpoint).
If this theory is correct, be glad that a 19 year old with an IQ of 131 can understand your theory.

I had the same idea. I have a theory of a fractal universe (so far mostly qualitative) which agrees with these statements, which proposes that our visible universe of galaxies and stars is a boson (specifically a gluon) and that by looking at the "edge of the universe" we may be looking at the surfaces of very large black holes (specificially the surfaces of fermions (quarks)).

My view is that if we could see a "step down", we would have a 4 dimensional-view point of the universes between the quarks in the atoms that makeup everyday objects.

I have posted my idea on a psuedo-journal at: http://academia.wikicities.com/wiki/Cyclic_Multiverse_Theory [Broken]

// sorry I don't know how to speak in "scientific" language yet...

-kmarinas86

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kmarinas86 said:
I think you and I have similar intuitions in physics. This is precisely an idea I have repeated in my mind, and you have the experience to give it scientific support. Thank you so much!
Thank you for your enthusiasm! My website counter works overtime!
I had the same idea. I have a theory....
// sorry I don't know how to speak in "scientific" language yet...
I've read your draft article and I can see the similarities. I like the way you try to visualize your ideas. Perhaps you should try to focus on one thing at a time instead of trying to tell it all at once. You need to work on your formalism indeed, but you seem energetic enough to make that happen (I guess you're a first-year at at Houston university?).

Mortimer said:
(I guess you're a first-year at at Houston university?)
Second year at the University of Houston actually (I graduated high school in 2004).

There is a discussion "Photon's perspective of time" going on in the Relativity forum.
Euclidean relativity offers a solution to the way "time" is perceived by the photon. Euclidean 4-velocities are defined as $dx_\mu /dt$ where $x_4=c\tau$ (as opposed to $dx_\mu /d\tau$ with $x_4=ct$ in Minkowski geometry). The boundary condition is that its magnitude is invariant $c$ (see also this article). From the empirical observation that the speed of the photon is always $c$ it follows that its Euclidean 4-velocity is always pure spatial, i.e. $d\tau$ for the photon is always zero (which poses a problem in Minkowski but not in Euclidean relativity).
In Euclidean space-time this implies that the photon exists in a 3D environment, i.e. the fourth dimension $\tau$ does not exist for the photon. It's Minkowski null-vector is actually a timelike vector in a 3D Euclidean space-"time" where the role of "time" is fulfilled by the third dimension, which is the direction of its travel with speed $c$. The other 2 dimensions form its "space", i.e. the photon is a Flatlander, moving with speed $c$ in its third dimension, like we move with speed $c$ in our fourth dimension.
The only difficulty here is mentally coming to terms with the idea that for a massless particle that travels with speed $c$ there exists one less dimension as compared to mass-carrying particles.

There are some implications though:
- In Euclidean relativity, accelerations in 3D correspond to rotations in 4D. This implies that for an accelerating observer, the photon's velocity vector must rotate along with the observer's frame of reference.
- The electromagnetic field is incontrovertible a 4D thing, which means that photons should consequently be 4D things as well. This may however apply particularly to their wave-nature, i.e. they may behave like waves in 4D and like particles in 3D. The mathematics around such a structure are rather complicated and I admit that I haven't been able to work that out yet.

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Moritmer said:
A spaceship travels relative to Earth....

In proper time the missile hits the asteroid before the spaceship does despite its lower spatial speed. Causality is therefor not violated. The missile runs backwards in proper time.
What happens if the missile is instead sent to a planet? Suppose then, somehow, it returns, in the same fashion it left the first spaceship. Then it would be travelling backwards in proper time with respect to that other planet. Is the proper time of the other planet is synchronous with the proper time at Earth given that they have similar properties?

Does the missile itself become a kind of "antimissile" in reference to the statement you made about antiparticles (running backwards in time)?

Mortimer said:
but from the circle diagram (Fig. 3) it shows that we must now take the negative root...

Note that the cyclic nature of $\gamma$ now also implies that in this situation $\gamma$ has a negative value
I don't know how the circle diagram will have you deal with a negative root.

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kmarinas86 said:
What happens if the missile is instead sent to a planet? Suppose then, somehow, it returns, in the same fashion it left the first spaceship. Then it would be travelling backwards in proper time with respect to that other planet. Is the proper time of the other planet is synchronous with the proper time at Earth given that they have similar properties?
Yes, If the other planet would be stationary with respect to Earth, its proper time would run synchronous with that of Earth (apart from possible gravitational time dilation differences). Actually there is no difference between this planet and the asteroid in the example. The asteroid is also stationary with respect to Earth.
kmarinas86 said:
Does the missile itself become a kind of "antimissile" in reference to the statement you made about antiparticles (running backwards in time)?
I'm very careful with this because the missile's individual elementary particles will obviously not turn each into their corresponding anti-particles. That would potentially violate conservation laws (of e.g. electrical charge). In the article, I'm merely suggesting a possible relation of the negative timespeeds that can occur with the existence of anti-particles that are also sometimes suggested to kind of run backwards in time (in e.g. Feynman diagrams). In the original text this remark was just a footnote but the editor of the journal preferred all footnotes to be placed in the main text.

kmarinas86 said:
I don't know how the circle diagram will have you deal with a negative root.
I have attached two pictures that might clarify why it is necessary to take the negative root.
The first picture is the same as Fig. 3 in the article but I have added the vector $X$, which is the time-velocity vector of the third object (later on the missile). In this picture, $X$ runs along the positive $x_4$ axis.
In the second picture, the actual situation as described with the spaceship and the missile is reflected. The frame $x''$ has now rotated beyond $\pi/2$ relative to frame $x$ and vector $X$ now runs along the negative $x_4$ axis. Vector $W$ is the spatial velocity of the missile (as observed from Earth) with magnitude $v_m$. The magnitude of $X$ is $\chi_m=\sqrt{c^2-v_m^2}$ but when calculating $\tau_m$ from this value one must obviously take the negative value in order to correctly account for the direction of the time-velocity vector $X$ along the negative $x_4$ axis.

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Mortimer said:
Yes, If the other planet would be stationary with respect to Earth, its proper time would run synchronous with that of Earth (apart from possible gravitational time dilation differences). Actually there is no difference between this planet and the asteroid in the example. The asteroid is also stationary with respect to Earth.
I'm very careful with this because the missile's individual elementary particles will obviously not turn each into their corresponding anti-particles. That would potentially violate conservation laws (of e.g. electrical charge). In the article, I'm merely suggesting a possible relation of the negative timespeeds that can occur with the existence of anti-particles that are also sometimes suggested to kind of run backwards in time (in e.g. Feynman diagrams). In the original text this remark was just a footnote but the editor of the journal preferred all footnotes to be placed in the main text.

I have attached two pictures that might clarify why it is necessary to take the negative root.
The first picture is the same as Fig. 3 in the article but I have added the vector $X$, which is the time-velocity vector of the third object (later on the missile). In this picture, $X$ runs along the positive $x_4$ axis.
In the second picture, the actual situation as described with the spaceship and the missile is reflected. The frame $x''$ has now rotated beyond $\pi/2$ relative to frame $x$ and vector $X$ now runs along the negative $x_4$ axis. Vector $W$ is the spatial velocity of the missile (as observed from Earth) with magnitude $v_m$. The magnitude of $X$ is $\chi_m=\sqrt{c^2-v_m^2}$ but when calculating $\tau_m$ from this value one must obviously take the negative value in order to correctly account for the direction of the time-velocity vector $X$ along the negative $x_4$ axis.
Since proper time according to this is negative:

Would it be accurate to say that spaceship is having a clock that runs backwards with respect to earth's clock? Would it also be accurate to say the spacecraft is going back in time? It sounds like a wormhole without the complications of "tunneling".

kmarinas86 said:
Since proper time according to this is negative:

Would it be accurate to say that spaceship is having a clock that runs backwards with respect to earth's clock? Would it also be accurate to say the spacecraft is going back in time? It sounds like a wormhole without the complications of "tunneling".
This would actually apply to the missile, not the spaceship. I assume that is what you mean.
I guess this is indeed what an observer on Earth would measure. For him the clock on the missile would be going backwards. From the missile's perspective, the clocks on Earth would be going backwards of course.
"Going backwards" is relative in this case.

Mortimer said:
This would actually apply to the missile, not the spaceship. I assume that is what you mean.
I guess this is indeed what an observer on Earth would measure. For him the clock on the missile would be going backwards. From the missile's perspective, the clocks on Earth would be going backwards of course.
"Going backwards" is relative in this case.
I see now, thanks!

I found out the old Velocity Addition Derivation which stems from the change in frequency as made by different observers.

Let:

$\alpha=\frac{v_1}{c}$

$\beta=\frac{u}{c}$

$\gamma=\frac{v_2}{c}$

$f_1=f_0 \sqrt{\frac{1-\alpha}{1+\alpha}}$

$f_2=f_1 \sqrt{\frac{1-\beta}{1+\beta}}$

$f_2=f_0 \sqrt{\frac{1-\alpha}{1+\alpha}\frac{1-\beta}{1+\beta}}$

$f_2=f_0 \sqrt{\frac{1-\gamma}{1+\gamma}}$

$\sqrt{\frac{1-\gamma}{1+\gamma}}=\sqrt{\frac{1-\alpha}{1+\alpha}\frac{1-\beta}{1+\beta}}$

$\left(1-\gamma\right)\left(1+\alpha\right)\left(1+\beta\right)=\left(1-\alpha\right)\left(1-\beta}\right)\left(1+\gamma\right)$

$\left(1-\gamma\right)\left(1+\alpha+\beta+\alpha\beta\right)=\left(1-\alpha-\beta+\alpha\beta\right)\left(1+\gamma\right)$

$1+\alpha+\beta+\alpha\beta-\gamma-\gamma\alpha-\gamma\beta-\gamma\alpha\beta=1-\alpha-\beta+\alpha\beta+\gamma-\gamma\alpha-\gamma\beta+\gamma\alpha\beta$

$2\left(\alpha+\beta\right)=2\left(\gamma+\gamma\alpha\beta\right)$

$\alpha+\beta=\gamma+\gamma\alpha\beta$

$\frac{\alpha+\beta}{1+\alpha\beta}=\gamma$

The corresponding page is this:

http://www.euclideanrelativity.com/dimensionshtml/node4.html [Broken]

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Mortimer said:
This would actually apply to the missile, not the spaceship. I assume that is what you mean.
I guess this is indeed what an observer on Earth would measure. For him the clock on the missile would be going backwards. From the missile's perspective, the clocks on Earth would be going backwards of course.
"Going backwards" is relative in this case.
And when they approach each other in a like manner, they would also be running backwards. Is this process symmetrical? Or is it asymmetrical like the twin paradox? My perception is that accelerated frames of reference (that cause asymmetry such as the twin paradox) must be gravitational according to this theory, but if they are based on velocity, then unlike in Einstein's theory, there would be no twin paradox (do I have that right?).

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kmarinas86 said:
I found out the old Velocity Addition Derivation which stems from the change in frequency as made by different observers.
Could you cite your source for this derivation (preferably a link on the web)? In particular
$$f_2=f_1 \sqrt{\frac{1-\beta}{1+\beta}}$$
seems rather odd.

Alternatively, give a definition of $f_0$, $f_1$ and $f_2$ in terms of which frequency from which source is observed by which observer.

kmarinas86 said:
And when they approach each other in a like manner, they would also be running backwards. Is this process symmetrical? Or is it asymmetrical like the twin paradox? My perception is that accelerated frames of reference (that cause asymmetry such as the twin paradox) must be gravitational according to this theory, but if they are based on velocity, then unlike in Einstein's theory, there would be no twin paradox (do I have that right?).
I once gave some comments in a similar discussion in https://www.physicsforums.com/showthread.php?p=591745#post591745". See posts #8 and further.
I must admit that I am not enitirely sure about the correct approach for this particular situation in Euclidean relativity. It seems logical that whenever acceleration plays a role, additional effects should be taken into account, like argumented in http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html" [Broken]. After all, this is not SRT's realm any more.

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Mortimer said:
Could you cite your source for this derivation (preferably a link on the web)? In particular
$$f_2=f_1 \sqrt{\frac{1-\beta}{1+\beta}}$$
seems rather odd.
I used a rather odd substitution (using the wrong letters such as alpha, beta, and gamma). But when this subsitution is undone, it is the same derivation that I found in my Physics textbook by Fishbane, Gasiorowicz, and Thornton.

https://www.amazon.com/gp/product/0136632122/?tag=pfamazon01-20&tag=pfamazon01-20

kmarinas86 said:
I used a rather odd substitution (using the wrong letters such as alpha, beta, and gamma). But when this subsitution is undone, it is the same derivation that I found in my Physics textbook by Fishbane, Gasiorowicz, and Thornton.

https://www.amazon.com/gp/product/0136632122/?tag=pfamazon01-20&tag=pfamazon01-20
I tend to believe some of the reviewer's comments on this textbook seeing this derivation. I have great difficulties in accepting the IMO sloppy assumptions that seem to lie behind the equations in lines 5 and 6 and have never seen this particular derivation before. But perhaps someone else on this forum has another opinion?

Velocity addition formulae also arise outside special relativity. generally, if Shift(velocity) is defined as a frequency ratio f'/f, then we expect a theory to generate a special velocity addition formula when its Doppler relationships have the characteristic:
$$Shift(v_1) \times Shift(v_2) \ne Shift(v_1 + v_2)$$
For emission theory, the Doppler relationship of f' / f = (1 − v) results in a velocity addition formula of
$$V_{TOT} = v_1 + v_2-v_1v_2$$

where $V_{TOT}$ is an equivalent velocity that lets us calculate the same total frequency shift in a single stage.

Although the velocity-addition formula method is general, the appearance of a v.a.f. under special relativity has a very different significance to the use of superficially-similar formulae under other theories.
It seems that your textbook uses a derivation that formally does not apply to relativistic situations but happens to give a similar result.

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