When I got this result that the rate of atomic clock (frequency of emission standard) will increase near a big mass, my first thought was about a possible error in calculation steps. There exists a lot of science books on general relativity which explain about time slowdown as a solid experimental fact. I scrutinized again my deduction in search of mistakes. There was no mistake. Actually, the phenomena of time acceleration near a big mass does not requires sophisticated mathematics. We can consider at qualitative level how the atom mass changes in gravitational field.
The electron/proton charge does not change in gravitational field. However, the change in particle mass is described by formula for mass (3.2). The decline in electron mass makes the electron to take a lower orbit, so the atom size decreases as well. But the electron velocity within the atom tends to increase. All that goes against the teaching of general relativity that all physical processes slow down in the vicinity of a big mass.
As soon as I recognized that the atomic clock rate has to speed up near a big mass, I decided to learn more about time-gravity experiments. One amazing thing about these experiments that they have been conduced long ago, in 1970s. However, technologies of those years did not allow to make conclusive experiments, since the high-accuracy atomic clocks were not invented yet. Today the best accuracy of high-end atomic clock is about 10–15. Experiments with this type of clock will answer the question where the clock rate is faster: at the basement or atop a high building (the relative contribution of gravitational effects is gH/c2, that is 10–16 per every meter of elevation). However, technologies of 1970s would failed to make this accuracy. This is a short info about this subject [45,p.73]:
The primary time standard for the U.S., a cesium fountain clock installed in 1999 by the National Institute of Standards and Technology (NIST) at its Boulder, Colo., laboratory, is good to one part in 1015 (usually written simply as 10-15). That is 500 times the accuracy of NIST’s best clock in 1975.
We see that experimental technique was unable to provide the needed accuracy for conclusive time-gravity tests. When what was kind of experiments they wrote about in the books on general relativity theory?
Assume we have a couple of high-accuracy atomic clocks. A first device is placed on the Earth surface, and another clock is placed atop of a high building. How can we compare their rates? There exist two options.
The first approach. We compare periodically the clock readings and make conclusion about their rates. This approach is quiet difficult, since we have to track down the billionth parts of second (this was impossible forty year ago).
The second approach. The readings of atomic clock are determined by the frequency of a basic element, i.e., the quantum generator. Instead of comparing absolute readings, let us compare the frequencies of quantum generators.
How can we compare the frequencies of two identical generators (one at the basement and one atop a high building)?
Numerous experiments (conducted in 1970s) demonstrated that the frequency of quantum generator at the height Н exceeds the frequency of analogous device at zero level by relative value gH/c2. This was a start of a wrong conclusion that the pace of atomic clock increases with height. This conclusion was unjustified. The key moment here is how exactly these two frequencies were compared.
If we want to compare the frequencies of two electromagnetic waves, we have to arrange their meeting in one place. For example, the generator at the top produces a wave which is being sent downward; here this frequency is compared with the frequency of the first (basement) generator. Experiment revealed that the upper frequency is slightly higher (the relative difference is gH/c2). But what is the meaning of this fact?
According to the teaching of general relativity, the atomic clock in the upper position (elevated by height Н) runs faster by factor of gH/c2. And the quantum generator frequency is also higher by factor gH/c2. Then the emitted electromagnetic wave travels downward and this frequency is compared against another standard at the basement level (the frequency of a similar generator at the zero level). This logic assumes implicitly that while the wave has been traveling to a lower position, its energy and frequency did not change at all. This is an obvious contradiction to the Universal law of gravity.
Anyway, if we accept the idea that the frequency of the upper quantum generator is higher, we have to consider the fact that a photon (wave) traveling downward has a changing energy and frequency!
This problem was discussed in journal “Doklady Physics” [116,117] and was debated in journal “Physical Sciences – Uspekhi” .
Therefore, these famous experiments only testified about existence of so-called gravitational shift of spectral lines, but not about rate of time. That’s all.
The frequency of the upper generator is different from the frequency of the lower one. However, the frequency of the upper electromagnetic wave will change during the downward trip. The sum of these two effects gives the relative shift of frequency by gH/c2 obtained in experiments. This subject was discussed also in [194,195].
We see that the second method fails to give a conclusive result about clock rates at different heights. As for direct comparison of reading of two identical clocks places at different levels, this type of experiments has not been conducted yet. At least the multi-volume “The Encyclopedia of Physics”  tells nothing about this option, and some thought experiments were discussed in “Modern Physics Letters” .