Metric theory of relativity (MTR) is the theory that describes the transformation
of physical quantities
in the laws of motion,
including the laws of mechanics, electrodynamics and the covariant theory of gravitation, based on
the spatialtemporal relations in arbitrary reference frames. The special cases
of MTR are the special theory of relativity (STR) and the extended special theory of relativity
(ESTR) in the limit of the weak field and the inertial reference frames, as
well as the general theory of relativity (GTR) in the part concerning the
transformation of physical quantities from one frame to another. The reference system in which the
rate of flow of time as the rate of similar movements differ due to shift along
the scale dimension also are included in
the scope of the MTR.
Contents

The analysis of the concept of
relativity in physics and the need of transformation of physical quantities
between different reference frames led first to the Galilean transformations in
mechanics, and then to the Lorentz transformations in STR, already suitable for
mechanical and for electromagnetic quantities. As it is shown in the Lorentzinvariant theory of gravitation
(LITG), in the inertial reference frames gravitational quantities satisfy the
Lorentz transformations.
The transition from the
relativity in STR to the relativity of arbitrary reference frames was done in
the beginning of GTR development. The feature of GTR is that the transformation
of quantities from one frame to another is done using the corresponding
4dimensional tensor transformation function, the components of which are the
partial derivatives relating the time and the coordinates of both reference
frames. GTR is a metric theory, so that in the transformations of the physical
quantities the metric tensor of the curved spacetime is involved.
The next step was development of
the extended special theory of relativity
(ESTR), which showed that the axiom of STR about the constancy of the speed of
light in all reference frames can be derived from a different set of initial
axioms. ^{[1]} In this case in ESTR the same formulas
are obtained as in STR.
MTR is described in the works of
Sergey Fedosin. ^{[2]} The characteristic feature of
MTR is that it introduces the concept of various wave representations of
phenomena. The events can be registered not only with the help of
electromagnetic, but also other waves, such as gravitational waves. ^{[3]} If these waves have different propagation speeds, it
should be taken into account in all the formulas of transformations of physical
quantities from one frame to another.
According to MTR, the constancy
of the speed of light, its independence on the speed of the light source and on
the observer's velocity is the consequence of the procedure accepted in SRT of
measuring the spatial and temporal parameters, such as the distance between the
points in space and time intervals between the events. Synchronization of
clocks at different points and measurements of the length in SRT are carried
out remotely by the waves and always imply that the wave returns to the point
from which it was originally sent as a signal. The wave passes a closed path in
space, so that no matter how the absolute velocity of the wave changes relative
to the moving source, being averaged on the way back and forth the wave speed
equals the speed of light in the medium at rest, in which the source is moving.
In fact, the wave speed in
different parts of the path is not equal relative to the source. To determine
the absolute velocity of the wave relative to the moving source it is necessary
to consider nonclosed paths during the propagation of waves. These are such
wellknown experiments on measuring with the help of Doppler effect the
temperature of the cosmic microwave background radiation coming to Earth from
different directions, and showing the absolute velocity of the Earth’s motion
relative to this radiation, about 600 km/s in direction of constellation Leo. ^{[4]}
It is possible to use a
geostationary satellite, which emits waves in one direction towards the
telescope on the ground, with a fixed distance between the transmitter and the
receiver. The Earth with this satellite revolves around the Sun, the Solar
system revolves around the galactic center, and the Galaxy itself is moving
relative to an isotropic reference frame in which the speed of light is the
same in all directions. In the observation of the satellite we should consider
the effect of aberration, which is dependent of the absolute velocity of the
Earth relative to the isotropic frame. Experiments show the orbital velocity of
the Earth around the Sun and the absolute velocity of the Solar system, about
600 km/s, the speed and direction of which is sufficiently close to the axis of
the dipole microwave background radiation. ^{[5]}
Another example is the
experiments of Stefan Marinov on
measuring the speed of light from lasers with the help of rotating discs with
small holes in them. ^{[6]} In these
experiments, the measured speed of light varied depending on the diurnal cycle
of the Earth's rotation and the corresponding change of position measuring
system in space and change the direction of light propagation. For the absolute
velocity of the Earth, there were obtained about 360 km/s.
The analysis of the Michelson–Morley experiment based on
the ideas of MTR brought Fedosin to the following dependence of the absolute
speed of light in the interferometer: ^{[2]}
where is the angle between the direction of the
light beam velocity and the direction of the velocity of motion relative to the interferometer
of the isotropic reference frame in which the speed of light is equal in all
directions. The velocity is not equal to zero, because the
interferometer moves with the Earth relative to the isotropic reference frame,
and there is expected the aether wind. For comparison, Marinov under the same
conditions of motion for the speed of light found the following: ^{[7]}
.
In acoustic MichelsonMorley
experiment, ^{[8]} the ultrasonic range finder was used
to measure the distances and there were sound waves in the work instead of
electromagnetic waves. The range finder was mounted on a moving car and the incoming
air has the ability to change the speed of sound as it propagates back and
forth. The experimental results can be interpreted in the same way as in the
special theory of relativity, that is, involving the effect of reducing the
longitudinal dimensions and time dilation. Both effects are the result of
measurements using the reflected wave returning to the starting point.
MTR includes five postulates
(assumptions accepted without proving):
where is the Ricci tensor, which is the trace of the Riemann curvature tensor, is the scalar curvature, is the metric tensor with contravariant indices, is the coefficient to be determined in the comparison with the experiment (in GTR this coefficient is equal to 1), is the gravitational constant. For the stressenergy tensor of the general field can be written: ^{[9] }
^{}
where are some
coefficients, is the electromagnetic
stressenergy tensor, is the gravitational stressenergy tensor, is the acceleration stressenergy tensor, is the pressure stressenergy tensor, is the dissipation stressenergy tensor, is the
stressenergy tensor of the strong interaction field, is the
stressenergy tensor of the weak interaction field.
The equation for the metric
realizes connection between the geometrical properties of the used spacetime
manifold, on the one hand, and the physical properties of the available matter
and the acting fields, on the other hand. The covariant derivative acting on
the both sides of the equation for the metric, makes them vanish. This fixes
the properties of the tensor in the left side of the equation for the metric,
and at the same time specifies the equation of motion of the matter under the
influence of the fields.
Comparison of the postulates of
MTR with the postulates of GTR shows that the latter are a special case of the
postulates of MTR. ^{[}^{10}^{]} ^{[1}^{1}^{]}
The postulates of MTR are
intended to summarize all the possible ways by which in the framework of metric
theories of spacetime the relations between different reference frames can be
realized and the corresponding physical quantities can be found.
For the inertial reference
frames, by definition moving at a constant velocity, the metric tensor is
constant and independent on the coordinates and time. Under the influence of
forces and fields acting on the reference frame accelerations arise in it and
the frame becomes noninertial. At the same time the curvature of motion of any
test bodies and the waves relative to this reference frame takes place, since
the action of forces (fields) on the reference frame and the test bodies or the
waves as a rule is not equal because of the differences in their velocities and
accelerations. In this case the velocity of test bodies and the waves in the
accelerated reference frame becomes the function of time, coordinates, and the
internal parameters of the reference frame such as mass and the sizes of the
frame. The curvature of motion of the test bodies or the wave, by means of
which the measurements are made, is perceived by the observer in the
accelerated reference frame as the curvature of spacetime. It can be described
as the dependence of the metric tensor on the coordinates and time.
On the basis of axiom 3 of MTR we
can compare different reference frames with wellknown metric tensors with each
other. Thus, the comparison of two different inertial reference frames allows
us to obtain the formula of velocities addition and the Lorentz transformations
for the coordinates and time in STR. This is sufficient to develop the special
theory of relativity. From the perspective of the fourdimensional
vectortensor formalism, from the equation of the squared interval, expressed
through the time and coordinates of two different reference frames, we can
determine the matrix for the transformation of any 4vector (tensor) from one
frame into the corresponding 4vector (tensor) of another reference frame. And
according to axiom 5 the additional conditions should be introduced, which are
necessary to derive transformations of the coordinates and time from one frame
to another. The typical condition is that the velocity of the reference frames
relative to each other, is up to sign the same in each frame.
Just as in GTR, according to
axiom 4 the effective spacetime curvature is determined by all the massenergy
sources, including the massenergy of different types of fields which take
place in the frame (because they first of all affect the propagation of test
bodies and waves). Accordingly, in each reference frame from the equations for
the metric we can find its metric tensor. Since the test bodies and the waves,
used for spacetime measurements, can differ significantly from each other, then
according to axioms 1 and 2 the metric tensor becomes the function of the
properties of test objects or waves. For example, the metric can be presented
as depending on the electric charge of test bodies, or on the speed of the
wave. In contrast to GTR, in MTR the gravitational field of the body is also
the source of massenergy in determining the metric.
Axiom 3 allows us from the
condition the equality of the interval to zero to find the dependence of the
velocity of test bodies and the waves, used for spacetime measurements,
relative to the accelerated reference frame, depending on the coordinates and
the time of the reference frame. Since the axiom 4 implies the possibility of
determining the metric tensor through the known stressenergy tensor of the
frame, it leads to the principle of local equivalence of the energymomentum:
"In the accelerated reference frame the metric depends locally not on the
type of the acting force, causing this acceleration, but on the configuration
of this force in the spacetime of the reference frame, determined by the
stressenergy tensor".
This principle is the
generalization of Einstein equivalence principle according to which based on
the equality of the gravitational and the inertial masses the acceleration in
the gravitational field can be equated to the acceleration from the inertial
force with respect to their influence on the physical processes. In the
principle of the local equivalence of the energymomentum it is emphasized that
the curvature of spacetime is not simply the embodiment of the gravitational
field, as it is considered in GTR. The effective curvature of spacetime is
rather the consequence of any fields and forces available in the reference
frame, and the metric itself exists even in the absence of gravitation. Besides
it is shown that to describe the equivalent phenomena we should use not the
equality of acting forces or accelerations, as in GTR in the equivalence
principle, but the equality of the form of stressenergy tensors, present in
the compared frames and leading to the same form of the metric tensor.
In MTR the gravitational field is
only one of the types of fields existing in nature and causing the change of
the metric in comparison with its form in the inertial reference frames. In MTR
it is also possible to make the spacetime measurements in this or that
reference frame with the help of the proper test bodies or waves. This implies
that for mathematical determination of the metric in this representation it is
necessary to express the effective stressenergy tensor in terms of the
characteristics of the given test bodies or waves. For example, for the
electromagnetic wave the main characteristics are its speed in the vacuum, the
equality of the rest mass, charge and magnetic moment of the wave to zero. The
momentum and the angular momentum of the wave are neglected in the measurements
carried out in the frame.
The most important conclusion of
MTR is that the dependence of the metric on the type of the used test bodies
and waves means the absence of the unified spacetime for each reference frame.
Thus the reduction of the physical system only to one chosen geometry of
spacetime is incomplete for the description of the system – the geometry can
not completely replace the physics of the phenomena. Therefore the geometry of
GTR, based on the measurements by means of electromagnetic waves, can not be
considered sufficient for complete description of physical systems. For
example, once the particles in the system obtain charges or spins, then the
previously found in GTR metric for uncharged particles becomes unsuitable for
describing the free motion of charged and rotating particles.
Since STR and GTR are part of
MTR, then for MTR there are classical relativistic effects, which include:
Improvement of measurement
techniques and the use of satellites allowed us to measure the number of
orbital and spin effects predicted earlier, such as spin and orbital precession
in the LenseThirring effect, geodetic precession, etc.
The calculation of the metric
inside the uniformly accelerated reference frame leads to new effects. In such
a reference frame the rate of clock becomes dependent on the location of the
clock and on the duration of the acceleration, which changes the velocity. ^{[2]} In the accelerated reference frame it is
possible that the time runs faster than in the inertial reference frame.
Besides, there is visual extension of bodies along the direction of
acceleration, shortening of the transverse dimensions and deformation of the
body shape. With this in mind, it becomes possible to describe the socalled
twin paradox more accurately.
1.
Fedosin S.G. Sovremennye problemy fiziki: v poiskakh
novykh printsipov. Moskva: Editorial URSS, 2002, 192 pages. ISBN
5836004358.
2.
^{2.0} ^{2.1}
^{2.2} Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’
materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN
9785990195110. (in Russian).
3.
Fedosin S.G. Electromagnetic and Gravitational
Pictures of the World. Apeiron, 2007, Vol. 14, No. 4, P. 385 – 413.
4.
George F. Smoot. Cosmic
microwave background radiation anisotropies: their discovery and utilization.
Nobel Lecture, December 8, 2006.
5.
Eugene I. Shtyrkov. Observation
of Ether Drift in Experiments with Geostationary Satellites. Proceedings of
the NPA, 2005, Vol.2, No 1, P. 201205.
6.
Marinov S (2007). "New Measurement
of the Earth's Absolute Velocity with the Help of the Coupled Shutters
Experiment". Progress in Physics 1: 31–37.
7.
Stefan Marinov (1983). "The
interrupted 'rotating disc' experiment". Journal of Physics A 16: 1885–1888. doi:10.1088/03054470/16/9/013.
Bibcode: 1983JPhA...16.1885M.
8.
Feist N. Acoustic
MichelsonMorley Experiment. Proceedings of the NPA, 2010, Vol. 6, P. 1–4.
9.
Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, P. 115
(2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.
10.
Comments to the
book: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii.
– Perm, 2009, 844 pages. ISBN 9785990195110. (in Russian).
11.
Fedosin S.G. The General Theory of Relativity, Metric
Theory of Relativity and Covariant Theory of Gravitation: Axiomatization and
Critical Analysis . International Journal of Theoretical and Applied Physics (IJTAP), ISSN:
22500634, Vol.4, No. I
(2014), pp. 926.

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