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