I Two twins moving towards each other in a circular orbit

[Bartolomeo]

The photon is a green photon in Joe's frame all the time, and the photon is a blue photon in the mirror frame all the time..

At least we agree, that if we say that „the aliens move“ or „the mirror move“ a photon of „green“ proper frequency appears to be „blue“ at reception. It can be only blue, not red. Blue means time acceleration.
We also agree, that if we say, that „the source move“,
SR claims, however, that photon of proper frequency „green“ can only be observed as „red“ at reception, i.e. clock dilates.

In the mirror frame the blue photon is produced by a device that emits wave crests at slow pace, the wave crests move away from the device at slow pace, because the device is moving, chasing the crests. The crests are placed close to each other, in the mirror frame. And that's why it's a blue photon in the mirror frame: Wave crest distance is a blue photon wave crest distance, in the mirror frame.
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Your jump into another frame. In that frame Joe moves but not Aliens. It is a different story. This effect is not transverse, but longtitudal in this frame since photon does not move at normal between patches of motion of source and receiver. It takes long path. This interpretation says nothing about clock rate. But yes, there maybe this interpretation of observations. Either Transverse, or longtitudal, this is a matter of taste.

 

[Bartolomeo]

I think this is true, but it's worth pointing out that these are two separate experiments. They are done at different times, one when the aliens see Joe moving purely transversely and one when Joe sees the aliens moving purely transversely.

It is exactly what Special Relativity does – conducts two separate experiments and they lead to the same outcome. I have already noted that in previous post. Reciprocity of observations takes place when we change frame and conduct another experiment. The task about Aliens means, that it is not that easy and each observer has to take into account „opinion“ of other one. They have decide who at which relative velocity moves relatively to each other. They cannot admit state of proper rest simultaneously.

Again, say that in Joe's frame the aliens travel along the x-axis and pass through the origin at t=0, and Joe lies on the y-axis at y=Y. Let Joe continually illuminate the alien ship with his laser. The light Joe emitted at t=-Y/c is the light Joe considers to have been emitted parallel to the y-axis (he has to lead a moving target) while the light he emitted at t=t'=0 is what the aliens consider to have been emitted parallel to the y-axis (he's a moving source aiming at the spatial origin). It's this difference in times that drives the difference in received frequency in your two cases.

Light received when they are at the closest approach. That means the Aliens are in motion and Joe is at rest (Aliens see blueshift, since their own clock dilates).

Light emitted when they are at the closest approach. That means the Aliens are at rest and Joe is in motion (Aliens see redshift, since their own clock tick as usual and moving clock dilates)

„ simple way of expressing this is to point out that the null frequency shift occurs for the pulse that travels the shortest distance from emitter to receiver, and this pulse is obviously neither emitted nor received at the point when the emitter and receiver are at their point of closest approach” http://mathpages.com/home/kmath587/kmath587.htm
That means, Joe and Aliens admit motion at „equal“ velocities (simplified 0,45 c and 0,45 c) and their clocks dilate at the same magnitude, i.e. they see the same clock rate. They tilt their tubes at equal angles backward and into front respectively.

What if the Aliens admit that they move with velocity 0,1 c and Joe with 0,8 c?

But which observation is the "correct" one? First? Second? Third? Fourth? Only the second one?

Do you still think that it doesn‘t matter what the Aliens think about their own state of motion?

this is all irrelevant to the original case of circular motion compared to an inertial observer on the axis, because there's a time symmetry there that isn't present in the linear motion being discussed since #15.

This animation is on the first page of google: Please look at the last episode.
For rotating observer accumulated amount of time dilation solely depends on linear velocity. If two observers rotate with the same LINEAR velocity, accumulated time dilation amount will always be the same and their clock will always show the same time. Direction of rotation does not affect it.