What are the implications of a Euclidean interpretation of special relativity?

[Mortimer]

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.

 

[Mortimer]

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].

Rob

 

[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

 

[Mortimer]

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".

 

[RandallB]

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

 

[Mortimer]

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).

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" 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.

 

[Mortimer]

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" 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

 

[Mortimer]

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

 

[RandallB]

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;

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

 

[Mortimer]

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

 

[kmarinas86]

Good work

I agree with the proposed Euclidean Relativity.

http://www.euclideanrelativity.com/dim2html/node3.html

...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

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

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

-kmarinas86

 

[Mortimer]

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?).

 

[kmarinas86]

(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).

 

[Mortimer]

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 http://www.euclideanrelativity.com/dim2html/node2.html). 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.

 

[kmarinas86]

http://www.euclideanrelativity.com/dimensionshtml/node4.html

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 traveling 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)?

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.

 

[Mortimer]

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 traveling 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.

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.

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.

 

[kmarinas86]

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".

 

[Mortimer]

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.

 

[kmarinas86]

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!

 

[kmarinas86]

Velocity Addition Deriviation

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

Start Velocity Addition Derivation

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$

End Velocity Addition Deriviation

The corresponding page is this:

http://www.euclideanrelativity.com/dimensionshtml/node4.html

 

[kmarinas86]

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?).

 

[Mortimer]

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.

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" . After all, this is not SRT's realm any more.

 

[kmarinas86]

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

 

[Mortimer]

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

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?

 

[Mortimer]

From http://en.wikipedia.org/wiki/Velocity-addition_formula" :

Velocity-addition in other theories
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 let's 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.

 

[robphy]

It looks like the derivation transcribed by kmarinas86 is velocity-composition derived from the Doppler Effect. The Doppler factors (the square-root quantities) are the Bondi k-factors.

$\sqrt{\frac{1-\gamma}{1+\gamma}}=\sqrt{\frac{1-\alpha}{1+\alpha}\frac{1-\beta}{1+\beta}}$
is
$k_{20}=k_{10}k_{21}$

When rewritten in terms of the associated relative-velocities, one gets the velocity-composition formula.

 

[Chronos]

The right equation must work in both time directions. If you get a result at T=1 that cannot be reversed to T=0, you have either made a mistake, or disproven causality.

 

[Mortimer]

The right equation must work in both time directions. If you get a result at T=1 that cannot be reversed to T=0, you have either made a mistake, or disproven causality.

Please explain the context of your remark? What equation are you referring to? What reversal?

 

[Chronos]

Please explain the context of your remark? What equation are you referring to? What reversal?

It was a pretty obscure comment now that you mention it! I was thinking about the negative timespeed thing. Can you define this in terms of a path integral?

 

[Mortimer]

I do not have a full explanation for all the implications of the negative timespeed. I more or less accept it as an odd consequence of the Euclidean interpretation that doesn't seem to lead to inconsistencies as long as proper time $\tau$ is taken as the basis for causality. The negative timespeed is however Achilles' heel of the article anyway.
I can't define this in terms of a path integral but there may be something that comes close to the concept. On the http://www.euclideanrelativity.com/idea" ).
Despite this divergence of proper time coordinates, all objects remain observable in 3D space which implicitly demands that every point in 3D must contain all proper time coordinates, so the proper time dimension is fully contracted or curled up in space.
This means that any event with coordinates $(x_1, x_2, x_3, c\tau)$ includes the proper time coordinates of all other events, i.e., if the asteroid of the example coincides with the spaceship, the event also contains the proper time coordinate that correlates to the event in which it was earlier (in proper time) hit by the missile. So essentially, the event is the sum of all events of earlier and later proper times, hence the similarity to Feynman's path integral. I'm sure this is not the total answer to it though. I'm still pondering this myself.

 

[RandallB]

Mortimer; Been a while since I peeked in.
Have you decided if the (x₁, x₂, x₃) part the (x₁, x₂, x₃, c'tau') event coordinates for your theory, expects to correlate directly with a x,y,z location in our locally observed reality.

I.E. Have you decided if your theory is background-independent as I suspected. Or do you think you can preserve background-dependence.

 

[Mortimer]

Have you decided if the (x₁, x₂, x₃) part the (x₁, x₂, x₃, c'tau') event coordinates for your theory, expects to correlate directly with a x,y,z location in our locally observed reality.
I.E. Have you decided if your theory is background-independent as I suspected. Or do you think you can preserve background-dependence.

I'm quite convinced now that it is definitely background-independent, including $c\tau$.

 

[Hurkyl]

I think I'm beginning to figure out why I have a problem with this theory. Don't get me wrong -- I think it's a very neat idea, but the development seems lacking.


In SR, it was very clear what everything "meant". Space-time is a 4-D Minowski space. We can put coordinates $\langle ct, x, y, z\rangle$ onto Minowski space, and that specifies a place and a time. $\tau$ is simply the variable we use to denote a good parameter for a worldline. The expression $(d(ct))^2 - (dx)^2 - (dy)^2 - (dz)^2$ is the expression for a pseudometric on Minowski space, as represented in orthonormal coordinates. (i.e. an inertial reference frame)


When I read the presentation for Euclidean relativity, I feel like it's doing nothing more than playing around with the equations of SR to put them into a neat form. It talks about a space parametrized by 5 coordinates: $\langle ct, x, y, z, c\tau \rangle$. The equation $(d(ct))^2 = (dx)^2 + (dy)^2 + (dz)^2 + (d(c\tau))^2$ is merely an equation of motion -- it is not a metric. In fact, there seems to be no geometry done at all!

Presumably the central equation of motion $(d(ct))^2 = (dx)^2 + (dy)^2 + (dz)^2 + (d(c\tau))^2$ should be some sort of invariant: for any two "good" choice of coordinates, it should hold in one if and only if it holds in the other. But what does that say about the geometry? Should the expression $(d(ct))^2 - (dx)^2 - (dy)^2 - (dz)^2 - (d(c\tau))^2$ be considered an invariant? If so, then we've simply moved into a (4+1)-dimensional Minowski space!

Do we break up 5-d space into one-dimensional time and 4-dimensional $\langle x,y,z,c\tau \rangle$ space, ala Newtonian mechanics? If so, then it makes sense to treat the 4-dimensional space as Euclidean. (But then again, if we foliate Minowski space, we can treat each slice as Euclidean as well) Then we have the neat fact that the equations of motion say that all particles travel with the same, fixed speed. But, we've reverted to a theory that has an absolute time parameter, and all of the philosophical problems with that.

Stepping back to special relativity, proper time has only translational symmetry -- a proper duration is an invariant quantity. In other words, $\tau$ doesn't "mix" with space-time coordinates.

In the passage to Euclidean Relativity, we've lost that -- it attempts to treat $\tau$ as just another spatial coordinate. Now that $\tau$ is no longer an invariant, what do clocks measure?


Another crucial feature of special relativity is that photons travel along null geodesics. In other words, worldlines satisfying $d(c\tau) = 0$ were very special.

When we pass to Euclidean relativity, it seems to me that this crucial feature of special relativity has been entirely lost: it is destroyed by just about any Euclidean transformation of $\langle x, y, z, c\tau\rangle$-space.

In other words, I think the laws of physics are not invariant under Euclidean transformations! And that's a big problem.

Maybe as I read more, I will be satisfied on these issues, but skimming through the article doesn't make me very optimistic.

 

[Mortimer]

Thanks for your elaborate comments, Hurkyl. They once more set me thinking. It keeps being extremely difficult to switch between the Minkowski and the Euclidean interpretation.

Stepping back to special relativity, proper time has only tanslational symmetry -- a proper duration is an invariant quantity. In other words, $\tau$ doesn't "mix" with space-time coordinates. In the passage to Euclidean Relativity, we've lost that -- it attempts to treat $\tau$ as just another spatial coordinate. Now that $\tau$ is no longer an invariant, what do clocks measure?

This is consistent in all Euclidean 4-vectors. The invariant in Minkowski space-time is no longer the invariant in Euclidean space-time. The original Minkowski time-component becomes the invariant. Also in the Euclidean interpretation, $\tau$ is defined as what a clock reads that moves along with the object. For me it makes more sense that this value is not invariant. After all two observers that move relative to each other will both say that the other clock is moving slow, so how can proper time than be invariant during a transformation between frames?
At one time I came to the conclusion that Minkowski geometry is a pure mathematical thing that merely gives the correct mathematical results when given the correct input. It has no link to physical reality. I've tried to express this in a separate article at http://www.euclideanrelativity.com/4vectors . Here you will find more on the geometrical background of the Euclidean interpretation. It is perhaps easier to initially treat time $t$ as a parameter for tracking worldlines, similar to the way $\tau$ is used in Minkowski geometry. Other endorsers of Euclidean relativity, like e.g. Hans Montanus do that too. The sole reason why I don't is that its designation as a real dimension is used explicitly in a follow-up article that can be found at http://www.euclideanrelativity.com/dim2html .

Another crucial feature of special relativity is that photons travel along null geodesics. In other words, worldlines satisfying $d(c\tau) = 0$ were very special. When we pass to Euclidean relativity, it seems to me that this crucial feature of special relativity has been entirely lost: it is destroyed by just about any Euclidean transformation of $\langle x, y, z, c\tau\rangle$-space.

I've realized this too. In section 4 of the article is mentioned:

Equation (18) is in fact based on the universality of light speed and the basis for reasoning is that an object, e.g. a photon, having speed $c$ for an observer in frame $x$ will still have that same speed for an observer in frame $x'$. This is one of Einstein's original postulates and also in this Euclidean approach it will still be maintained as a valid postulate, which essentially means that the photons velocity vector, as measured from the moving frame, must have rotated along with that frame.

I just accept this as being a mandatory consequence of the Euclidean interpretation. I do not try to explain it in the article. So far, I have not run into anything that would be inconsistent with it.

In other words, I think the laws of physics are not invariant under Euclidean transformations! And that's a big problem.

The point is that, although there are some strange consequences associated with the Euclidean interpretation, I did not find any law that would be violated by them after careful consideration. But being not even graduate level, I cannot rule out to have missed something crucial.

The biggest problem in getting to appreciate the Euclidean interpretation is to get rid of the Minkowski way of thinking that has been taught in all relativity courses for nearly 100 years now. In particular when you have grown accustomed to it over the years, it's hard to avoid it in the Euclidean interpretation. For me it's often the other way around. It's hard to avoid the Euclidean way of thinking in Minkowski discussions and I am aware that this is in fact a "handicap".

 

[Hurkyl]

In other words, I think the laws of physics are not invariant under Euclidean transformations! And that's a big problem.

I wanted to elaborate more upon this one. I will use the case of a photon again.


Suppose we had a worldline (along with its proper time) defined by the equations:

x = ct
y = 0
z = 0
$\tau$ = 0

One orientation-preserving Euclidean motion is the one that cyclically permutes the y-, x-, and c$\tau$-axes. So, in Euclidean relativity, we have the equally valid coordinatization:

x' = 0
y' = 0
z' = 0
$\tau$' = t

which is, of course, a perfectly good worldline according to special relativity, just not that of a photon.

Also in the Euclidean interpretation, $\tau$ is defined as what a clock reads that moves along with the object.

which essentially means that the photons velocity vector, as measured from the moving frame, must have rotated along with that frame.

These weird problems suggest that they are coordinate-dependent ideas, and not true geometric entities.

But, it struck me how you can maintain Euclidean symmetries without any of these weird problems.

Instead of insisting that $\tau$ be what clocks measure, which breaks Euclidean symmetry, you could introduce a preferred direction into 4-d space.

In other words, you postulate the existence of a 4-dimensional vector q.

Then, instead of defining $\tau$ to be what clocks measure, you could say that clocks measure $\mathbf{v} \cdot d\mathbf{q}$. (Where v is the velocity 4-vector). In other words, clocks measure displacement along the q direction.

Similarly, photons would be moving in a direction perpendicular to q.

This restores Euclidean symmetry, because q would rotate along with the coordinates.

Going back to my original example:

x = ct
y = 0
z = 0
$\tau$ = 0
In this <x,y,z,c$\tau$> coordinate chart, we have

q=<0,0,0,1>

Now, if I applied the Euclidean rotation, I result in

x' = 0
y' = 0
z' = 0
$\tau$' = t

with q=<0,1,0,0>

And we see that the velocity vector <0,0,0,c> is perpendicular to the preferred direction <0,1,0,0>.


Maybe this is what your article said, but I feel better having written it my way.



By the way, allow me to suggest that telling people they have their understanding of SR wrong is not the right way to get people interested! When I see statements like:

Figure 3 shows the background of this multiplication and it is clear that the components of the Minkowski 4-vector can have no physical meaning.

I get quite put-off. When I read this paper, I'm strongly inclined to think of reasons why the author has no clue about special relativity than attempt to appreciate the new approach.