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. ^{[1]}
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. ^{[2]}

**Contents**

- 1 Theoretical description
- 1.1 Covariant theory of gravitation
- 1.1.1 Influence of the
four-potentials of fields
- 1.1.2 Influence of the
field tensors
- 2 References
- 3 See also
- 4 External links

**Theoretical**** description**

**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: ^{[2]}

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, ^{[3]}
^{[4]} 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/s^{2}.

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: ^{[5]}

**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:

Here is the vacuum permeability, is the __gravitational tensor__;
is the electromagnetic tensor; is the determinant of the metric tensor.

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.

**References**

- 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.
^{ }^{2,0}^{2,1}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.- B. S. DeWitt,
Superconductors and gravitational drag. Phys. Rev. Lett., Vol. 24, 1092–3
(1966).
- G. Papini,
Particle wave functions in weak gravitational fields. Nuovo
Cimento B, Vol.v52, 136–41 (1967).
- John A. Macken. The Universe is Only Spacetime. Chapter 8. Analysis of
Gravitational Attraction. Gravitational Energy Storage. p. 8-25.

**See**** also**

**External**** links**

Source: http://sergf.ru/gpsen.htm