In general relativity gravitational time dilation and gravitational redshift are treated as two separate phenomenon because time in this theory is linear flowing in one direction and imaginary because it has no connection with physical reality. Over and above this, gravity is a geometrical effect in this theory.

In an alternative theory called periodic relativity, time is nonlinear periodic phenomenon associated with particle waves at the fundamental level. T = 1/f is only fundamental relation that connects time with physical reality. In this theory there is no difference between gravitational time dilation and gravitational redshift. In both cases frequency of light change due to change in gravitational force. When you talk about atomic clock most of the time people do not bother to consider how time is measured by atomic clock. In atomic clock the change in the frequency of hyperfine splitting of 6s electron in the outer shell is measured. This is the measurement on the light spectra just like the gravitational redshift. What you are doing in both cases is the measurement of frequency shift. So there is no difference between two. But when you relate the atomic clock frequency change with some imaginary time scale then all the problems arise. This not permitted in periodic relativity.

The concept of time that is directly connected to physical phenomenon is the Period of the wave or the oscillator. This is why all time measuring devices are oscillators without any exception. So you can only compare one period of time between two events with another period of time between other two events. More details in following articles.

http://dx.doi.org/10.1007/s10714-009-0908-5

https://www.researchgate.net/publication/225712819_Periodic_relativity_basic_framework_of_the_theory?ev=prf_pub

https://arxiv.org/abs/0707.4539

The difference between redshift experiments and atomic clock delay is, in the redshift experiments the emitter and the detector are at different altitudes, while in atomic clock the clock readings are obtained locally. However from the following perspective you are right. The reading difference is created because of the difference of the dilation rates between separation and reunion of clocks. The clocks don't delay at all while being at rest in the gravitation field. This fact can be understood when the accelerated frame is observed from an inertial Frame.

To prevent my answer to be musunderstood I want to add following remarks. Bu above statement I mean that clocks are delayed during separation and during reunion . In an accelerated reference Frame this causes a time difference after reunion to compare the clocks that were synchronized before separation. This is because of ordinary clock dilation of special relativity.

You said:" But if both effects exist namely if the clock at lower altitude is actually delayed and the frequency of the beam is shifted on the way because of Doppler equivalent effect just because equivalence principle, these effects would add up to each other".

The conclusion that the the clock at lower altitude actually is delayed is based on the approximate calculation based on the classical Doppler effect. In this approximation the effects of SR,.i.e. Lorentz transformation are dismissed.(are not necessary in this order of aproximation).

There is subtlelty however. Though the Doppler effect is based on Newtonian physics(no Lorentez transformation used) the conclusion that the lower clock is delayed is conclusion based on SR principles.

You said: "The conclusion that the the clock at lower altitude actually is delayed is based on the approximate calculation based on the classical Doppler effect."

(Leaving the discussion of the difference between newtonian and SR calculation of doppler effect which is of second importance) the doppler effect does not lead to clock dilation at all. It leads to the frequency shift of the emitted radiation "on the way" from emitter to detector but this does not imply that clock rates at the location of emitter and at the location of detector are changed. Application of equivalence principle and taking SR into the account leads to gravitational clock dilation. Thus gravitational redshift and gravitational time dilation are different phenomena. They have different origins.

I agree with your interpretation except for the conclusion that graviational redshift and gravitational time dilation are different phenoma.

That's why I said there is a subtlelty in using Newtonian mechanics to calculate the Doppler shift. From a Newtonian point of view the frequency change it's not owed to time dilation, it is owed to change in relative light velocity. To correctly interpret the frequeny change according to SR we must use time dilation/contraction since there is no change in the relative velocity of light.

As you initially correctly indicated SR time contraction owed to relative velocity of receiver also takes place and rigorously must be added however it is a second order effect in this approximation so it suffice to interpret frequency change as time dilation/contraction.

Clocks do slow down in gravitatinal fields and the observed red shifts are experimental proofs of that.

May it help?

C. H. Wörner & R. Rojas: “An elementary approach to the gravitational Doppler shift”, Eur. J. Phys., 38, 015604 (5pp). doi: 10.1088/0143-0807/38/1/015604 (2016).

Who can explain why in this paper https://www.researchgate.net/publication/331048388_Coordinate_velocity_and_desynchronization_of_clocks

the authors argued in contrary with W. Kundig paper [W. Kundig, Phys. Rev. 129, 2371 (1963).] that in Mossbauer rotor experiment the blue-shift has been obtained?

Independent of the MM experiment: The Fourier time series proves all Doppler shifts change observed time and adds dimensions to time.  It does this by breaking any signal into sine and cosines if the frequency of each are multiplied a Doppler shift frequency ratio,  each value now happens at t/K instead of t. Therefore by superposition it is true for the original function.   That adds dimensions to time because the axial shift is directional. That impacts a lot of relativity. Also, we sense all things by forces traveling at the speed of light and have Doppler shifts. For the full proof and another different one and some impacts open this link:

https://drive.google.com/file/d/16oJTvHAaV6TYPXMtGdz-NBqc1RW3yY-n/view?usp=sharing

The importance of that is for most angles the axial shift is larger than the transverse; the axial shift is directional giving direction to all things it affect and we measure all things by forces whose transmitted energy have Doppler shifts.The source of Einstein and Lorentz NOT giving the axial shift that power is Lorentz wrote about aether winds which cancel each other out in the axial direction. Which enabled isolation of the transverse shift. But after the light in the Michelson experiment hit the first stationary mirror, the Doppler shift puts it in the stationary reference plane. No more shifts to cancel the first. Also in most situations the axial shift exists.

Link to a paper describing an omission in the application of relativity to beams in most books: https://drive.google.com/file/d/17wia8LJNxjRAnDqK_XwAOABK5yqs8qM/view?usp=sharing

This link is to a paper on: One important constant is NOT invariant during change of reference frame: https://drive.google.com/file/d/1ljTtjvx5HiFi3vguznj8p-8XRtXa85NV/view?usp=sharing

Thank you for your time and I hope the above gave you some new ideas. Samuel Lewis Reich [email protected]