Is there tested proof of time dilation in the stationary frame, from the perspective of the moving frame?

Indeed, an inertial observer can ascribe himself a state of "proper rest" or "proper motion". However, in SR an observer rarely founds himself in a "moving frame", an observer is usually "at rest" in his own frame. Two spatially separated and Einstein - synchronized clocks of his "rest frame" measure longer time interval than a single clock, which is changing spatial position (is moving) in his frame (time dilation).

SR recognizes only one synchronization of spatially separated clocks - Einstein's .

However, there were Mossbauer rotor experiments (time dilation tests) in centrifuge;

• if an observer (absorber) is at rest in the center of the centrifuge and a source of radiation is attached to the rim of centrifuge, this observer would measure $gamma$ times lower frequency of radiation, or "moving clock is running slower than his own"

• If an observer (absorber) is attached a rim of an centrifuge and a source of radiation is located in the center, this observer would measure $gamma$ times higher frequency of radiation, or that the a "clock at rest is running faster than his own"

• If two observers are located on the opposite sides of a rim of rotating ring, they would measure absence of dilation of each other clocks (Champeney and Moon time dilation test)

The circumference can be of arbitrarily large diameter; i.e. this rotating observer can be quasi - inertial; that doesn't change things much. Rotating observer simply cannot ascribe himself state of rest.

Good to note, that A. Einstein in his celebrated 1905 paper teaches, that from the point of view of "moving observer" a clock "at rest" is ticking $gamma$ times faster than his own.

One clock is slower that the other and vice versa - is nonsense - even in Special Relativity

@Mohammad Javanshiry, I have provided the quote already. The quote was taken straight from Einstein‘s paper, just please read it carefully. Einstein clearly indicated that the source was „at rest“ and the observer was „moving“. Note that Einstein attached time dilation to the observer. Indeed, relativistic Doppler blueshift doesn‘t mean that the source‘s clock is running faster. It doesn‘t also mean, that it is running slower. It means, that source‘s clock is running either slower, or faster, or at the same rate – at any rate you wish, that purely depends on what an observer thinks about his own motion. One can attach time dilation either to the observer, or to a source or even to the both.

If it is still not clear, please read 34-6 The Doppler effect, Relativistic effects in radiation, Feynman lectures. Note that Feynman considers the effect in the frame of the stationary observer AND in the frame of the stationary source. In the first case he attaches time dilation to the source (34.12), in the second to the moving observer. In the second case(34.14) Feynman divides source‘s frequency by $sqrt {1-v^2/c^2}$.

If an observer is moving towards a stationary source of radiation, frequency of the source increases (blueshift) due to dilation of observer‘s clock. Since his clock is running slower, the „outside world“ appears to him as if in fast-forward mode.

@foolishmuse I don’t know why they debate twin paradox. Twin paradox has trivial resolution in the framework of Lorentz Ether Theory.

I don't also know why they have decided that the speed of light is isotropic in all frames of reference.

If an observer in inertial laboratory wants to measure rate of “moving” clock, he must set up laboratory equipment first, or synchronize two spatially separated clocks within his laboratory, say A and B. So as to synchronize these clocks, he must know, how long a light pulse travels from clock A to clock B, i.e. he must to know the one – way speed of light. But, so as to measure the one - way speed of light he must synchronize clocks. Hence, there is a circular reasoning. It is not possible to measure one – way speed of light prior to certain synchronization scheme. However, it is possible to measure speed of light back – and – forth by means of single clock.

Since one – way speed of light depends on synchronization convention, one - way dilation also depends on synchronization convention. All that is clear for 100+ years, Einstein understood that perfectly well.

SR assumes, that one – way speed of light is isotropic in all relatively moving reference frames, but it is not experimentally confirmed fact. It is a convention, a.k.a. Einstein synchronization, or standard synchrony convention. If every observer synchronizes clock according to Einstein, every “moving” relatively to him clock would appear running slower than his own, spatially separated ones.

Einstein’s synchrony convention is only a special case of Reichenbach’s synchrony convention, or non – standard synchronization. This synchronization allows anisotropic one – way speeds of light, but keeps two – way speed of light isotropic.

For example, this observer can assume, that his laboratory is moving relatively to a “stationary” clock. In this case he must take into account his own velocity in the stationary clock’s frame and re-synchronize clocks in his laboratory according to anisotropic (Reichenbach’s) synchrony convention. In this case “stationary” clock would measure longer interval of time, or would appear running faster.

The same is about the relativistic Doppler effect. As soon as you re-adjust laboratory equipment or change interpretation, you can make “moving clock” running at any rate you wish - slower (if you think, that clock is moving within your frame) or faster (if you think, that you are moving himself relatively to a stationary clock).

However, raving fans of SR do not want to admit, that an observer can be “moving” himself, even though they admit, that motion is relative. This is the paradox.

I wrote this note so as to avoid synchronization issue and work it out only by means of relativistic Doppler effect; this demonstrates that these wonders like “one clock is slower than the other and vice versa” is simply a piece of nonsense.

By the way, there is a good article which addresses twin paradox.