Time dilation problem question

[stevendaryl]

"Seeing" in special relativity means comparison of your clock with another clock in immediate vicinity, just in front of you, nothing else. Not turning you head up and down, left and right and observing distant material bodies.
Well, excuse me, we can also make pictures and measure frequency. But there are very important details in this case,

You cannot determine whether Herb's clock is running faster or slower than John's clock by only looking at nearby clocks.

 

[Battlemage!]

"Seeing" in special relativity means comparison of your clock with another clock in immediate vicinity, just in front of you, nothing else. Not turning you head up and down, left and right and observing distant material bodies.
Well, excuse me, we can also make pictures and measure frequency. But there are very important details in this case,

Two clocks separated by large distances cannot be compared in the way you describe, but you can certainly look at a clock far away. What will your eyes tell you? Is this the same as what the equations tell you?

If you don't calculate for the finite speed of light then what you see with your eyes will be incorrect, or at least ambiguous.

 

[Bartolomeo]

Two clocks separated by large distances cannot be compared in the way you describe, but you can certainly look at a clock far away. What will your eyes tell you? Is this the same as what the equations tell you?

If you don't calculate for the finite speed of light then what you see with your eyes will be incorrect, or at least ambiguous.

Yes! That's why observer takes readings in immediate vicinity. Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

https://en.wikipedia.org/wiki/Observer_(special_relativity)
https://arxiv.org/abs/physics/0512013

 

[Bartolomeo]

You cannot determine whether Herb's clock is running faster or slower than John's clock by only looking at nearby clocks.

It is the core of special relativity. Einstein's work of 1905. Observer introduces a reference frame and allocates AT LEAST two clocks in different spatial positions. AT LEAST two clocks.
Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

https://en.wikipedia.org/wiki/Observer_(special_relativity)
https://arxiv.org/abs/physics/0512013

 

[Battlemage!]

Yes! That's why observer takes readings in immediate vicinity. Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

https://en.wikipedia.org/wiki/Observer_(special_relativity)
https://arxiv.org/abs/physics/0512013

This doesn't really address the point that what you see with your eyes isn't necessarily the same thing as what is going on locally at what you are looking at, due to the finite speed of light. The Doppler effect needs to be taken into account if you're going by what you see.

 

[Battlemage!]

It is the core of special relativity. Einstein's work of 1905. Observer introduces a reference frame and allocates AT LEAST two clocks in different spatial positions. AT LEAST two clocks.
Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

https://en.wikipedia.org/wiki/Observer_(special_relativity)
https://arxiv.org/abs/physics/0512013

What does Jim conclude as he passes both? (or rather, as they pass him, given that he is justified in claiming he is at rest)

 

[Bartolomeo]

This doesn't really address the point that what you see with your eyes isn't necessarily the same thing as what is going on locally at what you are looking at, due to the finite speed of light. The Doppler effect needs to be taken into account if you're going by what you see.

"To see by eyes" can be interpreted in various ways. A scientist has tools. Rods, clocks, spectrometers, photo cameras. You can do no measurements without introducing a reference frame. Reference frame is synchronized clocks in different spatial positions.

 

[Bartolomeo]

What does Jim conclude as he passes both? (or rather, as they pass him, given that he is justified in claiming he is at rest)

That depends on how Jim will measure. He can introduce his own reference frame. He hires Jack and gives him a clock. Jack goes into another spatial position. He sends a beam of light to Jack. They adjust clocks , taking into account that speed of light was c. Now their clocks show the same time.
Tony and Bill fly towards them from the deep space. They chase each other. They have clocks too. Tony and Bill synchronize their clocks too with the same assumption.
Tony passes by Jack first and compare clocks. Let's their clocks show 12 o'clock each. Then Tony passes by Jim. Tony's clock show 3 PM and Jack's 7 PM o' clock.
Jim and Jack think, that Tony's clock runs slower. Tony thinks, that Jim's and Jack's clocks run faster.

Now upside down.

Jim passes by Tony. They compare clocks. For example, their clocks shows 12 each. Then Jim passes by Bill. Jim's clock shows 3 PM and Bill's 7 PM.
Tony and Bill together think, that Jim's clock runs slower. Jim thinks, that Tony's and Bill's clocks run faster.
https://en.wikipedia.org/wiki/Time_dilation#/media/File:Time_dilation02.gif

 

[Ibix]

Jim thinks, that Tony's and Bill's clocks run faster.

Only if Jim assumes Bill and Tony's clocks are synchronised. He has no reason to believe they are, though, since you seem to regard looking at anything not right in front of you as forbidden.

If he doesn't look, he's guessing. If he does look he's assuming. But what I do not understand is why you would choose to make anything other than an assumption of isotropy. You seem to pick one frame and let them assume isotropy, then force every other frame (edit: or rather, everyone not at rest in your chosen frame) to make a more complicated assumption. Why? Just to make things more complex?

 

[Bartolomeo]

Only if Jim assumes Bill and Tony's clocks are synchronised. He has no reason to believe they are, though, since you seem to regard looking at anything not right in front of you as forbidden.

If he doesn't look, he's guessing. If he does look he's assuming. But what I do not understand is why you would choose to make anything other than an assumption of isotropy. You seem to pick one frame and let them assume isotropy, then force every other frame to make a more complicated assumption. Why? Just to make things more complex?

Well, Jim is too stupid to assume something. He is very plain guy. He compares clocks and makes his own conclusions. I do not like or dislike certain synchronization procedures. I am a bit suspicious. Just in case. But I know, that another measuring technique of time dilation (transverse Doppler Shift) does not make any assumptions. And observations are not reciprocal. And that makes me even more suspicious.
Either Doppler effect and photo camera is rubbish or Einstein clock synchronization for every observer.

 

[Ibix]

Well, Jim is too stupid to assume something. He is very plain guy. He compares clocks and makes his own conclusions.

...thereby assuming that Bill and Tony's clocks are synchronised.

I do not like or dislike certain synchronization procedures. I am a bit suspicious. Just in case. But I know, that another measuring technique of time dilation (transverse Doppler Shift) does not make any assumptions. And observations are not reciprocal. And that makes me even more suspicious.

They aren't expected to be reciprocal in the Joe/alien example we discussed in the other thread because the situation you are examining is not symmetric betwen Joe and the aliens. As you yourself pointed out. If you reverse the experiment completely the results are reciprocal, as noted in the Wikipedia page you linked to in that thread.

 

[Bartolomeo]

...thereby assuming that Bill and Tony's clocks are synchronised.

They aren't expected to be reciprocal in the Joe/alien example we discussed in the other thread because the situation you are examining is not symmetric betwen Joe and the aliens.

Exactly. In no way they can be symmetrical. In no way. Green photon turns blue at the moving mirror. Not red. In no way red, whatever you do. Mirror dilates itself and concludes that source oscillates faster. Joe sees redshift (dilation), Aliens see blueshift (acceleration of time). Aliens see redshift (dilation), Joe sees redshift (acceleration of time). Joe makes picture of moving Aliens and see contraction. Aliens make picture of Joe and see stretching, since their photo - film contracts.

 

[Ibix]

Aliens see redshift (dilation), Joe sees redshift (acceleration of time). Joe makes picture of moving Aliens and see contraction. Aliens make picture of Joe and see stretching, since their photo - film contracts.

Not if they use the same experimental procedure. If they use different procedures it's hardly surprising that the results are different.

 

[Bartolomeo]

Not if they use the same experimental procedure. If they use different procedures it's hardly surprising that the results are different.

There are no procedures. What kind of procedures? Joe releases green photon straight up. The aliens have a mirror. Photon reflects and goes back to Joe "from the top". It comes back green again. What color was at mirror? Red? Blue! Only blue, newer red. For Joe the mirror is moving source, and photon which comes back to Joe from moving source redshifts, since moving source dilates. Blue and only blue! It is so simple. Never red!

 

[Ibix]

There are no procedures. What kind of procedures? Joe releases green photon straight up. The aliens have a mirror. Photon reflects and goes back to Joe "from the top". It comes back green again. What color was at mirror? Red? Blue! Only blue, newer red. For Joe the mirror is moving source, and photon which comes back to Joe from moving source redshifts, since moving source dilates. Blue and only blue! It is so simple. Never red!

If the aliens fire a green photon straight down so that it strikes Joe then bounces back you will find the exact same result with the roles reversed. They can't do it at the same time as Joe is doing his experiment because Joe is not in position to do so in their frame at that time - so they would be following a different experimental procedure if they try.

 

[Bartolomeo]

If the aliens fire a green photon straight down so that it strikes Joe then bounces back you will find the exact same result with the roles reversed. They can't do it at the same time as Joe is doing his experiment because Joe is not in position to do so in their frame at that time - so they would be following a different experimental procedure if they try.

Yes, but they have to take physical action and change direction of their tube. They can't look with one eye in two different direction at the same moment. But, so what? Yes, the Aliens will release green photon and it will be blue for Joe, Joe reflects it back and the Aliens see it red again. Now the photon is blue at Joe. Now Joe is in motion and his clock dilates. That what I told all this time. Who ascribes himself state of rest, sees dilation. Who ascribes himself state of motion, sees acceleration since he dilates himself.
But they can't ascribe themselves equal states of proper rest or motion.

 

[jbriggs444]

But they can't ascribe themselves equal states of proper rest or motion

This is unclear. What cannot be equal to what? And what is "proper rest".

Certainly two observers can each maintain that they are at rest and that their peer is not.

 

[Bartolomeo]

This is unclear. What cannot be equal to what? And what is "proper rest".

Certainly two observers can each maintain that they are at rest and that their peer is not.

Proper rest is a state, when you consider yourself as being at rest and interpret observations from the point of view observer at rest. Imagine that you stay at certain point at axis y, at point Y. You posses a long thin tube. You ascribe yourself a state of proper rest. You want to measure frequency of moving source - green monochromatic lamp. You turn the tube into the origin. You stay and wait. The lamp moves along x in negative direction. At certain moment you see a short flash, when the lamp passes the origin and a photon goes through you tube. The flash will be red in color due to Transverse Doppler Effect.
Then you ascribe yourself small velocity. You repeat the experiment but do nothing special in this case - just turn the tube a bit into front. Why? Because if photon flies along y-axis up, you approach the point of "intersection from the left" and you have to take aberration into account. You tilt tube so as the photon would be able to pass through it. You see a flash again. The photon will be a bit less red, than it was in the first experiment.
Then you repeat the experiment again, but ascribe yourself even larger proper velocity. You think that since your velocity increased, you tilt you tube even further an see that color of photon moved into blue range of spectrum.
In fact you do nothing, just turn you tube further an further into front and interpret every observation as transverse doppler effect, since photon according to you comes to you along y-axis and due aberration becomes more and more blue with each experiment.
So, ascribing yourself state of proper motion means that you simply turn your tube or gaze further and further into front.

If lamp emits dispersed light, you will always see it. But if the source emits narrow beam, the source will always have to adjust angle of emission at corresponding angle, so as the photon will go through your tube,
These angles are always tied with relativistic aberration formula. The more you tilt you tube into front, the less the source has to tilt his laser pointer back. At the moment, when these angles are the same, you will see no shift, neither dilation nor acceleration, because the source and you ascribe themselves equal velocities.
So, state of proper motion means, that you look into front but not straight down

 

[Ibix]

Yes, but they have to take physical action and change direction of their tube. They can't look with one eye in two different direction at the same moment. But, so what? Yes, the Aliens will release green photon and it will be blue for Joe, Joe reflects it back and the Aliens see it red again. Now the photon is blue at Joe. Now Joe is in motion and his clock dilates. That what I told all this time. Who ascribes himself state of rest, sees dilation. Who ascribes himself state of motion, sees acceleration since he dilates himself.
But they can't ascribe themselves equal states of proper rest or motion.

No. The result of Joe and the aliens isn't symmetrical because the circumstances aren't symmetrical. Joe fires the laser; the aliens do not. This is not the same as two people watching each other's clocks, which is a symmetrical situation and the results must be symmetrical, barring an absolute rest frame of some kind.

 

[jbriggs444]

Proper rest is a state, when you consider yourself as being at rest

So it's not a state at all. It's a decision about what reference frame to use.

 

[Bartolomeo]

So it's not a state at all. It's a decision about what reference frame to use.

Well, this decision looks like that. If you choose a frame, in which you move with equal velocities, the source and you have to turn tubes at the same angle. If you choose a frame, in which you at rest, you look at right angle to direction of motion of the source. Angle depends on chosen reference frame.

 

[Ibix]

Proper rest is a state, when you consider yourself as being at rest and interpret observations from the point of view observer at rest. Imagine that you stay at certain point at axis y, at point Y. You posses a long thin tube. You ascribe yourself a state of proper rest. You want to measure frequency of moving source - green monochromatic lamp. You turn the tube into the origin. You stay and wait. The lamp moves along x in negative direction. At certain moment you see a short flash, when the lamp passes the origin and a photon goes through you tube. The flash will be red in color due to Transverse Doppler Effect.
Then you ascribe yourself small velocity. You repeat the experiment but do nothing special in this case - just turn the tube a bit into front. Why? Because if photon flies along y-axis up, you approach the point of "intersection from the left" and you have to take aberration into account. You tilt tube so as the photon would be able to pass through it. You see a flash again. The photon will be a bit less red, than it was in the first experiment.
Then you repeat the experiment again, but ascribe yourself even larger proper velocity. You think that since your velocity increased, you tilt you tube even further an see that color of photon moved into blue range of spectrum.
In fact you do nothing, just turn you tube further an further into front and interpret every observation as transverse doppler effect, since photon according to you comes to you along y-axis and due aberration becomes more and more blue with each experiment.

So you are just saying that if you do different experiments you get different results. This is true (and frame invariant) but trivial.

 

[Bartolomeo]

So you are just saying that if you do different experiments you get different results. This is true (and frame invariant) but trivial.

Yes, outcome of experiment depends on how I make it. and what I think about my own state of motion. If I am at rest, I do not consider aberration. If I am in motion, I consder aberration. You told me in your very first post, that outcome of experiment does not depend on how it is done, don't you? And what I think about my motion. No, it does. The faster I move, the blueish source will be.

 

[jbriggs444]

Well, this decision looks like that. If you choose a frame, in which you move with equal velocities, the source and you have to turn tubes at the same angle. If you choose a frame, in which you at rest, you look at right angle to direction of motion of the source. Angle depends on chosen reference frame.

When asked what "proper rest" means, you give a longwinded and incorrect response. When this is objected to, you try to clarify by invoking aliens and laser pointers.

I conclude that "proper rest' is not a meaningful term.

 

[Bartolomeo]

When asked what "proper rest" means, you give a longwinded and incorrect response. When this is objected to, you try to clarify by invoking aliens and laser pointers.

I conclude that "proper rest' is not a meaningful term.

Especially for you I even posted a video.

 

[Ibix]

Yes, outcome of experiment depends on how I make it. You told me in your very first post, that outcome of experiment does not depend on how it is done, don't you?

No. I told you that the outcome of one experiment did not depend on which frame you use to describe it. See the second quote and my response in this post.

 

[Bartolomeo]

No. I told you that the outcome of one experiment did not depend on which frame you use to describe it. See the second quote and my response in this post.

I have nothing to say.

 

[Ibix]

I have nothing to say.

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

 

[Bartolomeo]

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

It is very difficult graphically demonstrate coincidence of commoving and rotating observer, we need spatial imagination. But I think you can understand that. Just try to move either observer relatively to rotating circle and central lamp or the whole rotating establishment relatively to resting observer.
Fig. 1 – Commoving observer moves in the reference frame of the lamp (blue shift)
Fig. 2 – Commoving observer and the lamp are at equal velocities to each other (no shift)
Fig. 3 – Lamp is in motion in the reference frame of commoving observer (reds shift)
Fig. 4 – Rotating observers in the same spatial positions as commoving ones.

Length of arrow V reflect "proper velocities"
http://[url=https://postimg.org/image/8mvqlxk0z/][PLAIN]https://s29.postimg.org/8mvqlxk0z/Fig_1.jpg

 

[Bartolomeo]

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

Relative velocity is the same in every give experiment. It doesn’t change.

Experiment is to show the independence of observations from the arbitrary choice of the reference system. Of course, if you do not change anything, nothing changes.

The purpose of any given experiment is to confirm or deny certain suggestions. Suggestion is the following: observation of clock rate of relatively moving clock does not depend of state of rest or motion of any relativistic observer. If Ibix and Bartolomeo conduct certain experiment, Ibix must measure dilation of Bartolomeo’s clock (to register redshift) and Bartolomeo has to measure dilation of Ibix’s clock (to register redshift) in any given experiment despite of what they think about their state of motion or rest.

Amount of redshift must depend on their relative velocity solely. Let’s say 0.9 c.

They conduct experiment at 10 AM and write a report. Ibix releases a photon along y axis, Bartolomeo reflects it and Ibix gets it back. Report claims, that Ibix measured redshift and Bartolomeo blueshift. Relative velocity was 0.9 c.

They conduct another experiment after lunch at 2 PM. Relative velocity was 0.9 c again. velocity solely. Let’s say 0.9 c. Bartolomeo releases a photon along y axis, Ibix reflects it and Bartolomeo gets it back. Report claims, that Bartolomeo measured redshift and Ibix blueshift.

They can conduct 1000 experiments or even 1000000 experiments. Relative velocity is still 0.9 c. They change angle of rotation of their tubes in every experiment at 0.01 degree. In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Amount of time dilation, which corresponds to relative velocity 0.9 c is the same in every experiment. It is mutual property. The more Ibix takes for himself, the less remains to Bartolomeo.

Time dilation and length contraction is like a short blanket. You cover your head but it is too short to cover legs and vice versa.

Well, Ibix had finally found THE brilliant solution! Ibix saw time dilation at 10 AM, Bartolomeo at 2 PM. They publish a report, that they both saw time dilation fully in accordance with SR.

 

[Battlemage!]

Amount of redshift must depend on their relative velocity solely. Let’s say 0.9 c.

The Doppler shift formula recognizes the notion of the source of the emitted frequency, so how can the source-emitted frequency be arbitrarily interpreted as the received frequency simply by changing reference frame?

 

[Battlemage!]

To expand a little - in the post I linked in my last, you wrote "And what will a co-moving inertial observer (an observer of a co-moving inertial frame of reference) see, who at some point in time ends up near one of the observers hurtling around the circle? This will depend upon what velocity it ascribes to itself." There is only one point in space where it makes any sense to compare the viewpoint of an instantaneously co-moving inertial observer and a circular orbiting observer: where the two are co-located and co-moving. At that point there is only one setting of the tube angle that will receive light, and the frequency that you measure is obviously frame invariant. It does not matter if the co-moving observer decides to use some other frame - they will conclude that the tube is set in the correct position to reveive light at the moment they co-move with the orbiter. If they move the tube they will be doing a different experiment (which may obviously have different results), an experiment which is irrelevant to the question at hand because it won't work when they are co-moving with the orbiter.

I now think you were referring to running multiple experiments which can, of course, have different results. But they are irrelevant to the circular motion thread as I just noted, and you are misapplying them in this one. When the two parties do the same experiment (i.e., set their receiving tube to the same angle to the other's direction of travel by their own local measurements) they receive the same frequency that the other received. This is the reciprocity that is expected, and is consistent with symmetric time dilation. The fact that, in any given realisation of the experiment, the two parties do not see the same frequency is not a counter-argument because there is no symmetry in that realisation. One of the parties is emitting a light pulse at a time of their choosing and the other is passively receiving. The symmetry is broken and no reciprocity is expected.

They can conduct 1000 experiments or even 1000000 experiments. Relative velocity is still 0.9 c. They change angle of rotation of their tubes in every experiment at 0.01 degree. In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Hmmm... Let me take a gander at making them see the same Doppler effect, in my obviously unqualified noobish ways... What about this scenario?Ibix shoots the light wave. Bartolomeo sees Ibix moving away at 0.9c. Since Ibix is moving away according to Bartolomeo's frame, Bartolomeo sees the following redshift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Now, according to Ibix, when the light wave comes back, to him it is emitted from the mirror. As the mirror is moving away from him, he'll see the following Doppler shift:

$$f_O = f_E \sqrt{\frac{1-0.9}{1+0.9}}$$

Looks like it's exactly the same for Ibix if the reflected light can be treated as if it emitted from the mirror.

Unless... that assumption is not valid?

 

[Bartolomeo]

The Doppler shift formula recognizes the notion of the source of the emitted frequency, so how can the source-emitted frequency be arbitrarily interpreted as the received frequency simply by changing reference frame?

Very good question! I have exact answer in my files, just copy - paste:

In this case you can say that source is in motion and dilates, or observer is in motion and dilates. Doppler Shift will be the same as measured by observer. Graphically:

If an observer and a source approach each other head – on, the observer will see Doppler blueshift. In this case the observer may explain his observations in different ways depending on chosen reference frame. He may consider himself at rest, and wave fronts gather „on the nose“ of the source then, but because of dilation of the source‘s clock it will be „less blue“ than it were in classical case. In classical case wave fronts would hit him almost instantly. Either he may ascribe to himself a state of proper motion and source is at rest then. In this case maximum frequency he may register (in classical case) has to be 2f. But, since his own clock dilates, as he approaches the speed of light the frequency he measures will tend to an infinitely large value.

Sure, he can use a frame in which they have equal proper speeds or any other. Measured frequency will be the same, but relative contributions of time dilation will be different and frame dependent.

Amount of time dilation every observer possesses is in accordance with his "proper velocity". Total amount of time dilation is in accordance with relative velocity 0.9 c

This is purely longitudinal Doppler Shift, not Transverse.

All the ideas prior to these considered Transverse effect, when beam of light travels up and down at normal to direction of travel of observers.

 

[stevendaryl]

It is the core of special relativity. Einstein's work of 1905. Observer introduces a reference frame and allocates AT LEAST two clocks in different spatial positions. AT LEAST two clocks.
Observer in SR is not a "physical person". It is the whole reference frame, which is filled with many people and each of them has a clock. They synchronize clocks by light signals and make judgement then. John is certain point of reference frame K. John has a clock. Bill is far away. Bill has a clock too. John and Bill are the relativistic observer together. They synchronize clocks by light. Jim passes by John first and compares his clock with Johns at this moment in immediate vicinity. Then Jim passes by Bill. They again compare clocks in immediate vicinity. John and Bill together make conclusion, that Jim's clock runs slower.

Right, but that conclusion requires synchronization of different clocks. For clocks at rest relative to Jim and synchronized according to Jim, the opposite conclusion follows: that John's and Bill's clocks are running slow. There is no situation in which Jim "sees" the other clocks to be running faster than his. What he sees is a sequence of different clocks, not a single clock running fast or slow.

 

[Ibix]

Fig. 1 – Commoving observer moves in the reference frame of the lamp (blue shift)

Blue shift due to time dilation. No shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer.

Fig. 2 – Commoving observer and the lamp are at equal velocities to each other (no shift)

No shift due to time dilation. Blue shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer, the same as in figure 1.

Fig. 3 – Lamp is in motion in the reference frame of commoving observer (reds shift)

Red shift due to time dilation. Blue shift due to relative motion of source and receiver. Net result is that the receiver measures a blue shift. I assume here that the co-moving observer is co-located and co-moving with the orbiting observer, the same as in figure 1 and 2.

Fig. 4 – Rotating observers in the same spatial positions as commoving ones.

...but not actually co-moving here. The rotating observer's velocity is tangent to the circle and a single inertial frame cannot have a velocity that is tangent to the circle in more than one place. So at least two of these measurements are irrelevant - they are comparing different experiments, as I have been saying for some time now.

Length of arrow V reflect "proper velocities"

This is a bizarre terminology. My proper time is the time measured with a clock attached to me. My proper length is length measured with a ruler attached to me. Proper velocity, by extension, is the distance I travel measured with a ruler attached to me (i.e. zero) divided by the proper time. So proper velocity is always zero, which is why you never see the term. All of these are invariants since I've specified what is being used to make the measurements.

You seem to arbitrarily choose a set of rulers and clocks that aren't attached to me, change sets when you feel like it and call whatever result you get my proper velocity. It's not a proper velocity. It's just my velocity in some frame you've picked. Please stop using the term, or provide a valid reference for its use.

Amount of redshift must depend on their relative velocity solely.

This is incorrect in general. The redshift measured in a given frame also depends on the angle between the 3-velocity vector of the emitter and the spatial component of the line joining the emission and reception events. Hence the necessity for radial and transverse expressions for red shift.

They conduct experiment at 10 AM and write a report. Ibix releases a photon along y axis, Bartolomeo reflects it and Ibix gets it back.

"Along the y-axis" according to who? Bartolomeo and Ibix do not, in general, agree that the photon was emitted along the y-axis and hence are unsurprised by differing readings.

They conduct another experiment after lunch at 2 PM. Relative velocity was 0.9 c again. velocity solely. Let’s say 0.9 c. Bartolomeo releases a photon along y-axis

"Along the y-axis" according to who? Bartolomeo and Ibix do not, in general, agree that the photon was emitted along the y-axis and hence are unsurprised by differing readings.

In any experiment, they cannot find, that Bartolomeo fixes redshift and Ibix redshift. It is ABSOLUTELY IMPOSSIBLE.

Indeed. Because there is no symmetry in a given experiment - as I said in my last post.

Well, Ibix had finally found THE brilliant solution! Ibix saw time dilation at 10 AM, Bartolomeo at 2 PM.

I wouldn't put it like that, but you can if you like. Your experimental descriptions are incomplete, but if I infer correctly what you mean, in the 10am experiment Ibix saw pure transverse motion but Bartolomeo did not and vice versa in the 2pm experiment. The experiments are anti-symmetric setups, so anti-symmetric results are to be expected.