На русском языке
Metric theory
of relativity
Metric theory of relativity (MTR) is the theory that describes transformation
of physical quantities
in the laws of motion,
including the laws of mechanics, electrodynamics and the covariant theory of gravitation, based on
the spatial-temporal 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 weak field and inertial reference frames, as well as the
general theory of relativity (GTR) in the part concerning 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
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Creation of MTR
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 Lorentz-invariant 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 4-dimensional 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, absolute speed of
the wave relative to the source is not the same in different sections of closed
path. To determine the absolute velocity of the wave relative to the moving
source it is necessary to consider non-closed paths during the propagation of
waves. Such are, for example, well-known experiments measuring the temperature
of cosmic microwave background radiation coming to Earth from different
directions, and showing, based on the Doppler effect, absolute speed of the
Earth's movement relative to this radiation, about 600 km/s in the direction of
the 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 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] [6]
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. [7]
[8] In these experiments, the measured speed of light varied
depending on the diurnal cycle of the Earth's rotation and 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: [9]
.
In acoustic Michelson-Morley experiment, [10] 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 sound wave returning to the starting point.
Postulates
MTR includes five postulates (assumptions accepted
without proving):
- Properties of the used spacetime
manifold in this or that reference frame depend on the properties of the
test bodies or waves, with the help of which the spacetime measurements
are carried out in this reference
frame. The most important property of test bodies or
waves is their propagation speed
, as it is used in the formulas as a measure of velocities of other
bodies and of delay of information in remote measurements. - Geometrical properties of
spacetime are fixed by corresponding mathematical object, which is a
function of spacetime coordinates of the reference frame. For a wide range
of reference frames the suitable mathematical
object is the non-degenerate four-dimensional symmetric metric tensor of
second rank
, the components of which are proportional to the scalar products of unit
vectors of coordinate axes of the chosen reference system. The tensor allows us to find any
invariants associated with 4-vectors and tensors. - The square of the interval
between two close events,
understood as contraction of the metric tensor with the product of
differentials 
of coordinates,
is an invariant, the measure of the proper dynamic
time 
of the moving particle, and does not
depend on the choice of the reference frame: 
The square of the interval for two close events is zero, if these
events are associated with the propagation of test bodies or waves, by
means of which the spacetime measurements and recording the metric are
carried out. - Physical
properties of ordinary matter and fields, including gravitational field,
in the given reference frame are specified by the corresponding
stress-energy tensors. There is a mathematical function of the metric
tensor
, which is found by certain rules and is proportional to the total
stress-energy tensor of the fields in the given reference frame. In the
simplest case for vector fields, this function is
the tensor, which is the left side of the equation for the metric: [11] 
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
stress-energy tensor 
of the general field can be written:
[12] [11] [13] [14]
where 
is
the electromagnetic stress-energy tensor, 
is the gravitational
stress-energy tensor, 
is the acceleration stress-energy tensor, 
is the pressure stress-energy tensor, 
is the dissipation stress-energy tensor, 
is the stress-energy tensor of the strong interaction
field, 
is the stress-energy 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 matter under the influence of the fields.
- There are additional conditions
which specify the required for calculations amount
of relations for displacements and rotations of the compared reference
frames, the velocities of their motion relative to each other, taking into
account symmetry properties of the reference frames.
Comparison of the postulates of MTR with the postulates
of GTR shows that the latter are a special case of the postulates of MTR. [15] [16]
Results
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 corresponding
physical quantities can be found.
For 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 non-inertial. At the same time,
a curvature of the motion of any test bodies and waves relative to a given
frame of reference occurs, since the action of the force (fields) on the frame
of reference and on the test bodies or waves is usually not the same due to the
difference in their velocities and accelerations. In this case velocity of test
bodies and waves in the accelerated reference frame becomes a function of time,
coordinates, and 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 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 well-known 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 four-dimensional vector-tensor 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 transformation
of any 4-vector (tensor) from one frame into the corresponding 4-vector
(tensor) of another reference frame. And according to axiom 5 additional
conditions should be introduced, which are necessary to derive transformations
of the coordinates and time from one frame to another. A typical condition is
that the velocity of inertial reference frame 1 relative to inertial reference frame
2 is equal, up to a sign, to the velocity of reference frame 2 relative to reference
frame 1.
Just as in GTR, according to axiom 4 the effective
spacetime curvature is determined by all the mass-energy sources, including the
mass-energy 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 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 a body is also the source of mass-energy
in determining the metric.
From the condition of equality of the interval to zero, axiom 3 allows us
to find dependence of the velocity of test bodies and waves used for space-time
measurements relative to an accelerated frame of reference, as a function of coordinates
and time of this reference frame. Since the axiom 4 implies the possibility of
determining the metric tensor through the known stress-energy tensor of the
frame, it leads to the ''principle of local equivalence of energy-momentum'':
"In accelerated reference frame the metric depends locally not on the type
of acting force, causing this acceleration, but on configuration of this force
in the spacetime of the reference frame, determined by the stress-energy
tensor".
This principle is generalization of Einstein equivalence principle
according to which based on equality of gravitational and inertial masses an acceleration
in gravitational field can be equated to the acceleration from inertial force
with respect to their influence on the physical processes. In the principle of local
equivalence of energy-momentum it is emphasized that the curvature of spacetime
is not simply the embodiment of 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 stress-energy 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 stress-energy 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 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, as soon as the particles in the system acquire charges and spins, the
metric previously found in GTR for uncharged particles near a massive charged
body becomes unsuitable for describing the free motion of charged and rotating
particles.
Effects of MTR
Since STR and GTR are part of MTR, then for MTR there are
classical relativistic effects, which include:
- time dilation in moving reference
frames (absolute effect).
- gravitational time dilation in reference
frames in gravitational field (absolute effect).
- reduction of linear sizes in
moving reference frames (apparent effect).
- reduction of sizes along the
gradient of gravitational field (apparent effect for external observer).
- change of the speed of light in non-uniform
gravitational field.
- gravitational redshift of wavelength.
- precession of perihelions of
planets.
- additional deflection of waves and
test particles near massive bodies due to the impact of gravitational
field on spacetime metric and its difference from the Euclidean form.
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 Lense-Thirring
effect, geodetic precession, etc.
Calculation of 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 so-called twin paradox more accurately.
References
1.
Fedosin S.G. Sovremennye problemy fiziki: v poiskakh
novykh printsipov. Moskva: Editorial URSS, 2002, 192 pages. ISBN
5-8360-0435-8.
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
978-5-9901951-1-0. (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. Штырков
E.И. Измерение
параметров движения Земли и Солнечной системы. Вестник КРАУНЦ. Серия науки
о Земле, 2005, № 2, Выпуск № 6 , стр. 135-143.
6. Eugene I.
Shtyrkov. Observation
of Ether Drift in Experiments with Geostationary Satellites. Proceedings of
the NPA, 2005, Vol.2, No 1, P. 201-205.
7.
Маринов С. Экспериментальные
нарушения принципов относительности, эквивалентности и сохранения энергии.
Физическая мысль России, 1995, № 2.
8.
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.
9.
Stefan Marinov (1983). "The
interrupted 'rotating disc' experiment". Journal of Physics A 16: 1885–1888. doi:10.1088/0305-4470/16/9/013.
Bibcode: 1983JPhA...16.1885M.
10. Feist N. Acoustic
Michelson-Morley Experiment. Proceedings of the NPA, 2010, Vol. 6, P. 1–4.
11. 11.0 11.1 Fedosin S.G .
About the cosmological constant, acceleration field, pressure field and energy.
Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). https://doi.org/10.5281/zenodo.889304.
12. Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, P. 1-15 (2016), e2459.
http://dx.doi.org/10.4236/oalib.1102459.
13. Fedosin S.G. Two components of the
macroscopic general field. Reports in Advances of Physical Sciences, Vol. 1,
No. 2, 1750002, 9 pages (2017). https://doi.org/10.1142/S2424942417500025.
14. Fedosin S.G. Lagrangian formalism in the theory of
relativistic vector fields. International Journal of Modern Physics A, Vol. 40,
No. 02, 2450163 (2025). https://doi.org/10.1142/S0217751X2450163X.
15.
Comments
to the book: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’
materii. – Perm, 2009, 844 pages. ISBN 978-5-9901951-1-0. (in Russian).
16.
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:
2250-0634, Vol.4, No. I
(2014), pp. 9-26.
See also
- Special relativity
- Extended
special theory of relativity
- General relativity
- Lorentz-invariant theory of gravitation
- Speed of gravity
- Mass–energy equivalence
- General field
- Dissipation field
- Acceleration
field
- Scale
dimension
External links
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