Gravitational phase shift
The gravitational phase shift is a phenomenon, in which the components of the gravitational four-potential and the gravitational tensor independently change the phase and frequency of periodic processes, as well as the time flow rate. This phenomenon can be detected by comparing the results of two experiments, conducted in the gravitational field with different potentials or mismatching field strengths.
Historically, the first predicted effects were the gravitational time dilation and gravitational redshift.  In the first effect, deceleration of the clock rate is detected, when placed in the gravitational field, which can be explained by the influence of the scalar gravitational potential on the clock. In the second effect, the difference of the received radiation wavelength from the standard value arises in the case, when the radiation source and the radiation receiver are placed in regions with different gravitational potentials. In both the general relativity and the covariant theory of gravitation, these effects are caused by the influence of the field on its proper time at the observation point and are calculated with the help of the metric tensor.
In case when the gravitational field strength and the gravitational torsion field in the covariant theory of gravitation (the gravitomagnetic field in the general theory of relativity) are equal to zero, the phase shift due to the action of the gravitational field potentials can be considered as the gravitational analogue of the Ehrenberg–Siday–Aharonov–Bohm effect.
The idea that the action function has the physical meaning of the function, describing the change of such intrinsic properties of bodies and reference frames, as the rate of proper time flow and the rate of rise of the phase angle of periodic processes, appeared in works of Sergey Fedosin in 2012. 
Covariant theory of gravitation
For comparison, the formulas for calculation of the gravitational phase shift presented below are supplemented by similar formulas for the phase shift due to the electromagnetic field.
Influence of the four-potentials of fields
Respectively, for the gravitational and electromagnetic fields difference of the clock in the weak field approximation is described by the formulas:
Here is gravitational 4-potential , where is the scalar potential and is the vector potential of gravitational field; the electromagnetic 4-potential , where is the scalar potential and is the vector potential of electromagnetic field; means 4-displacement, is the speed of light, and are the mass and the charge of the clock.
The clock 2, which is out of the field and measures the time , is check one and the clock 1 measures the time and is under the influence of 4-field potentials or . Time points 1 and 2 within the integrals indicate the beginning and the end of the field action.
From the time difference we can move to the phase shift for the same type of processes in the field and outside it, or occurring in different states of motion. To do this, in the denominators it is necessary to replace by the value of the characteristic angular momentum. For the level of atoms it will be the Dirac constant :
The phase shift, obtained due to the electromagnetic 4-potential , acting on a particle with the charge , is proved by the Aharonov-Bohm effect in quantum physics. The phase shift in the gravitational 4-potential is also confirmed in the papers,   where it was found that the phase shift is proportional to the integral of the gravitational vector potential :
From the integral equations given above, we can go to differential equations. It is convenient to denote as the coordinate reference clock time of external observer located beyond the field of the system. If is a displacement of the clock 1 in the field and is the speed of the clock, for gravitational and electromagnetic fields respectively we can write:
Here is the rate of time change of the clock 1 due to the scalar field potential and motion in vector potential of corresponding field, is the rate of time change of the clock 2 with the same motion without the field, and is the angular frequency of a process associated with the control object 2 located outside the field.
In the static experiments in gravitational or electric field it is convenient to consider the difference between the rate of time of the clocks or the frequency difference of periodic processes in the two neighbouring points in space where all the clocks and objects are stationary and their velocities are zero. The last four equations can be written for the clock 3 and the object 3, located in the neighbouring point 3 and then subtracted from the equations for the clock 1 and object 1. If , we have:
This shows that the rates of the clocks at the points with different scalar potentials of the field do not match. In case of the gravitational field it gives the gravitational time dilation, which results in the gravitational redshift. The similar effects are also expected, if the gravitational field is replaced by the electromagnetic field. These effects in the electromagnetic field have not been measured yet because of their smallness.
The gravitational potential on the Earth’s surface is defined by the formula:
where and are the mass and radius of the Earth, is the gravitational constant.
At the point, which is located at the distance meter above the Earth's surface, the potential will be equal to:
Therefore, for the difference in the clock rate at points 1 and 3, which differ in height by 1 meter, we can write:
Here is the gravitational field strength, which is equal in the absolute value to the free fall acceleration 9.8 m/s2. As we can see, if a period of time second passes, the lower clock will lag behind the upper clock by about 10-16 seconds.
Angular frequencies in the last two equation are meaningful local reduced Compton angular frequencies in the given points of the field and related to the rates of the fixed clocks, and it can be written as:
here is the reduced Compton angular frequency in the absence of gravitational or electromagnetic field.
The work in the gravitational field of moving masses between points with different scalar potentials is , and the work on charge transfer in an electric field is equal to . In carrying out this work there is a change of location of mass or charge in the field, as well as change in the local reduced Compton angular frequency. It may be noted, that the work is equal to the product of the Planck constant to the change of reduced Compton angular frequency: 
Influence of the field tensors
The energy of the fields associated with the matter unit with mass depends not only on the absolute value of the four-potentials, but also on their rates of change in the spacetime, that is on the field strengths. Each additional energy must influence the inner properties of the matter, including the proper time flow rate. The field strengths are included in the action function through the field tensors, so that for the corresponding time shifts in the gravitational and electromagnetic fields we can expect the following:
From these formulas it follows that the gravitational field strength inside the volume of the clock must accelerate their rate and the electromagnetic field strength must on the contrary slow down the clock rate, as opposed to the case, when there is no field.
To estimate the effect in the gravitational field we will use the weak field approximation, in which we can assume that , , and the volume element . For the two clocks, located at adjacent points 1 and 3, in the absence of the torsion field , which usually makes small contribution, we can write the following:
Suppose points 1 and 3 are located near the Earth’s surface and are separated in height by the distance meter. We will choose the mass and the volume of the clock in such a way, that the relation would hold, where is the average density of the Earth. Under these conditions, we find:
which is comparable in magnitude to the effect of gravitational time dilation from the action of the gravitational scalar potential, but has the opposite sign.
On the line, connecting the two bodies, we can find a point, where the total gravitational field strength vanishes and the total scalar potential becomes equal to the sum of the potentials of these bodies. At this point, the gravitational time dilation does not depend on the field strengths of these bodies.
- A. Einstein, "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen", Jahrbuch der Radioaktivität und Elektronik 4, 411–462 (1907); English translation, in "On the relativity principle and the conclusions drawn from it", in "The Collected Papers", v.2, 433-484 (1989); also in H M Schwartz, "Einstein's comprehensive 1907 essay on relativity, part I", American Journal of Physics vol.45,no.6 (1977) pp.512-517; Part II in American Journal of Physics vol.45 no.9 (1977), pp.811–817; Part III in American Journal of Physics vol.45 no.10 (1977), pp.899-902, see parts I, II and III.
- Fedosin S.G. The Hamiltonian in Covariant Theory of Gravitation. Advances in Natural Science, 2012, Vol. 5, No. 4, P. 55-75. http://dx.doi.org/10.3968%2Fj.ans.1715787020120504.2023.
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- John A. Macken. The Universe is Only Spacetime. Chapter 8. Analysis of Gravitational Attraction. Gravitational Energy Storage. p. 8-25.