The extended special
theory of relativity (ESTR) is the special theory of relativity
(STR), derived in other axiomatics. The main
difference of ESTR from STR is replacement of the postulate of the constancy of
the speed of light and its independence on the motion of the sources of light
and on the motion of the observer, by the postulate of the existence of an
isotropic reference frame in which the speed of light is constant, depends
neither on the direction of its propagation, nor on the velocity of the source
of light. ESTR was developed by Sergey Fedosin in 2002 and is a special case of
the metric theory of relativity. ^{[1]}
Contents

In 1910 at the meeting of German naturalists and doctors
the Russian scientist Vladimir Ignatowski
made a report "Some general remarks to the principle of relativity":^{[2]}
Now I raise a question for myself, what relations or, more precisely, equations of transformation we can arrive at, if we put in the top of the study only the principle of relativity.
Ignatowski showed that based on linear transformations,
the principle of relativity and the isotropy of space we can derive the Lorentz
transformations. In this derivation the second Einstein postulate of invariance
of the speed of light was not used.
In the next 1911 year, in Annalen
der Physik the work was published by Philipp Frank
and Hermann Rothe: "On the transformation of the
spacetime coordinates from the fixed into the moving reference frames", ^{[3]} in which the approach of Ignatowski
received significant development. Based on the group analysis, Frank and Rothe in the class of linear functions found the most
general transformations between the inertial reference frames. They turned out
depending on two fundamental constants with the dimension of velocity. Adding
the axiom of space isotropy converts these transformations into the Lorentz
transformations, and the axiom of time absoluteness – into the Galilean
transformations. Frank and Rothe also were,
apparently, the first who noted that the most general transformations between
two inertial reference frames were the fractionallinear functions.
Despite the fundamental importance of these works for the
questions of physics foundations, they remained practically unnoticed. Most of
the educational literature up to the present time is based on the Einstein’s
axiomatic approach. Among the few references to the works of Ignatowski, Frank and Rothe we
can mention the textbook by Wolfgang
Pauli "The Theory of Relativity." However, in connection with
these works he wrote: ^{[4]}:
From the theoreticgroup considerations we can obtain only the form of the transformation formulas but not their physical content.
This assumes that the fundamental speed constant, which
occurs in the Lorentz transformations, can not be
interpreted as the speed of light, without involving additional hypotheses.
We shall note that the idea, that in order to justify STR
Einstein's second postulate is not required, has been repeatedly rediscovered, ^{[5]} ^{[6]} ^{[7]}
^{[8]} ^{[9]}, however, usually
without reference to the fundamental works of 19101911 years. An overview of
the works on the axiomatization of STR (in the framework of chronogeometry) can
be found in the work by Gutz ^{[10]}
in “Advances in Mathematical Sciences”. Among recent works there is the article
Caligiuri and Sorli.^{
[11]}
The important difference of ESTR from the above works is
that not only the axiom of space isotropy is used in it, but also the procedure
of spacetime measurements by means of electromagnetic (or other) waves. This
allows us to automatically determine the value of the theory’s constant, which
has the dimension of speed, and to equate it to the speed of light (more
precisely, to the speed of the corresponding wave).
The analysis of the axioms and the results of SRT gives
the following:
It is easy to see that the standard axiomatics
of SRT is too rigid. It is extremely relativistic, bringing the principle of
relativity of inertial reference frames to the absolute. From its postulates it
is impossible to imagine the existence of at least one somehow preferred
inertial frame. And the principle of independence of the speed of light is very
illsuited for the role of the basic axiom of STR. Indeed, the axiom as a rule
is considered a statement which does not require proving due to its
obviousness. But from the start the principle of independence of the speed of
light on the observer's velocity was hard to understand and hardly agreed with
the principle of relativity.
At the same time the true reason of constancy of the
speed of light in the vacuum still remains unknown and the structure of the
physical vacuum, in which electromagnetic waves propagate, is still a mystery.
Are the light quanta independent autonomous objects with intrinsic wave
properties, moving by inertia in the empty space, or do they transfer their
energy and momentum through the oscillations of the vacuum medium by means of
wave interaction? However that may be, the theory must be able to consider any
effects of interaction of the vacuum as the medium with the electromagnetic
field. The cross effects are also possible during the motion of bodies in the
vacuum, when the electromagnetic wave is propagating inside these bodies, and
the matter of bodies is interacting with the vacuum and changes the conditions
of the motion of the waves. However, the standard axiomatics
of STR does not allow considering these effects – for justification of STR the
existence of the ether is not required, and therefore the properties of the
ether or vacuum, as such, are not considered. Therefore in STR it is accepted
not to speak about the possible effect of the vacuum on the properties of the
moving bodies during the propagation of electromagnetic waves.
The purpose of development of the new axiomatics
of STR was to eliminate the above drawbacks – to find the internally
consistent, coherent theory axioms, to overcome the relativism absolutization,
to expand the possibilities of the theory in describing the reality, while
retaining all the previously achieved in STR results, repeatedly proven in
practice. The result of this search was determining such postulate of the
theory, which would replace the postulate of the constancy of the speed of
light for all observers.
Both SRT and ESTR use the PoincareEinstein principle of
relativity and the electromagnetic waves for connection between the events in
different inertial reference frames, provided that they take place in the
vacuum or in the medium which does not affect the propagation of light. In both
theories there are five axioms, that is, such basic assumptions which are
accepted without proving. If in STR one of the basic axioms is the constancy of
the speed of light and its independence on the motion of the light sources and
on the motion of the observer, then in ESTR the axiom is used instead of it
about the existence of the isotropic reference frame in which the speed of
light is constant and does not depend on the direction of its propagation and
on the velocity of the source of light.
The system of axioms of ESTR has the following form:
The measuring instruments can include electronic clock,
measuring light grids, etc., which are the standards for ordinary mechanical
rulers and clocks of any type. Synchronization of one clock with another is
done by circulating of the wave taking into account the time of its delay at
certain distance. In other words, the signal from the first clock should reach
the second clock and come back, then the observer at the first clock will be
able to give to the observer at the second clock the instruction to set the
second clock with the time shift equal to half of the time of the signal’s way
back and forth (the standard procedure of synchronization). Direct measurement
of length is possible only in the stationary reference frame, and in case of
motion of the object its length is determined indirectly by the light signals
sent from the ends of the object at the same time to the fixed measuring light
grid.
The logical scheme of ESTR is as follows. There is a
fixed isotropic reference frame S_{0} in the vacuum, in which the speed
of light, by definition, is always equal to с.
Then, we consider the motion of the reference frame S^{’} at the
constant velocity V_{0} relative to S_{0} along the axis OX
(all the axes of the two frames are parallel to each other.) In S^{’}
the light is propagating at the speed с_{1}
against the axis OX and at the speed с_{2}
along the axis OX, and it is unknown beforehand whether these speeds are equal
to each other. In S^{’} one light detector is located at the origin of
coordinates and two sources of light in different sides from the origin of
coordinates. These sources of light are moving along the axis OX at some
velocity V^{’} relative to S^{’}. The periods are calculated of
the waves which fall into the detector from the both light sources. After that,
the situation is considered again in the frame S_{0}. By comparing the
results, taking into account the recalculation of time intervals in different
frames, we obtain two equations.
In the next step the length of the body is calculated by
means of calculating the time required for the light to move to the end of the
body and back, in the fixed reference frame S_{0} and in the moving
reference frame S^{’}. Two quantities are introduced, one of which is
equal to the ratio of the time calculations, and the other is equal to the
ratio of the measured lengths in both frames. The result is one more equation.
In the third step the system of the three obtained
equations is solved. It is assumed that in each frame the observer carries out
spacetime measurements by the same procedure. As a result, at first, the
formula of speed addition of STR is obtained, the equality of speeds с_{1} and с_{2} to the speed of light с is proved, and the relation is derived
for the recalculation of the Lorentz factor from one inertial frame to another.
Based on the principle of relativity the effects of length contraction and time
dilation are found. Thus, the formulas of STR and the postulate of constancy of
the speed of light for all observers are derived in other axiomatics.
In order to understand the difference
between ESTR and STR, we shall consider the propagation of light inside the
moving bodies. In the reference frame S^{’}, where the body is at rest,
the speeds of light inside the body с_{3}
and с_{4} in the opposite
directions of the axis OX depend on the absolute refractive index, and
theoretically could also depend on the direction and the magnitude of the velocity of motion of the body in
the isotropic reference frame S_{0}. This follows from the fact that
the motion of the body in S_{0} can change the speed of light
propagation inside the body, for example, similarly to the ether drag effect.
From the point of view of S_{0}, the speeds of light inside the body
will be equal to с_{5}
and с_{6}. From
calculations we obtain the relations between the directed in one way speeds с_{4} and с_{6}, с_{3} and с_{5}.
These relations, in case of simplifying assumptions, turn into the standard
formulas of speeds addition in the Fizeau experiment, when the moving water
drags the light and effectively increases its speed.
But if no simplifications are made, ESTR implies the
possibility of emergence of additional effects due to the inequality of the
speeds с_{3} and с_{4}. Such inequality of speeds
is possible at high velocities or acceleration of the body in the isotropic
frame. STR can not make similar predictions due to
direct declaration of constancy of the speed of light in its axioms.
In contrast to STR, ESTR predicts the possibility of the
influence of the properties of the physical vacuum, in which the material
bodies are moving, on the propagation of electromagnetic waves inside these
material bodies. This influence is possible when the bodies are moving or being
accelerated relative to the isotropic reference frame. Since the phase speed of
light inside the material bodies depends on the absolute refractive index , then the influence of the physical
vacuum must be revealed through this index. In the theory of ESTR the
transformations of the coordinates and time are as follows:
In the substance the refractive index depends on the
angular frequency of
the wave according to the formula:
and the wave number
is also the function of (where is the wavelength). In the general case, in the Lorentz transformations
instead of the speed of light propagation in the vacuum, we must substitute the
group speed of light in the substance, which taking into account (2) equals:
This leads to transformations (1), different from the
partial Lorentz transformations by introduction of the absolute refractive
index and its derivative with respect to the angular
frequency of the wave , in order to take into account the speed of
the electromagnetic wave in the substance of any kind.
The absolute refractive index in
the general case depends not only on the properties of the substance, but also
on the state of motion of this substance relative to the isotropic reference
frame. Due to the influence of the physical vacuum, the measurements in the
moving and accelerating bodies can lead to different results as compared with
the external measurements of the time intervals and the lengths of the same
bodies, and compared with the measurements inside of the bodies at rest in the
isotropic reference frame. It must be taken into account also that in STR the
measurements are commonly used which are external relative to the body, and in
ESTR there is a possibility to carry out the spacetime measurements of the
standards of length and time also by means of electromagnetic waves inside the
material bodies.
The advantage of ESTR is that all the results of the
special theory of relativity are derived based on more intuitively
comprehensible system of axioms. In ESTR it is possible to carry out the
spacetime measurements not only by means of electromagnetic, but also any other
waves (e.g. gravitational), provided that the used standards of length and time
will be constructed on the basis of these waves. ^{[12]}
Accordingly, in all formulas the speed of light should be replaced by the
propagation speed of the used wave. ESTR is the basis for the Lorentzinvariant theory of gravitation.
In gravitational fields ESTR is replaced by the metric theory of relativity (MTR). ^{[13]} In ESTR it becomes possible to overcome the
absolutization of relativity of the reference frames of STR, which is
unacceptable from different points of view, including the philosophical point
of view.
Special relativity is a special case
with respect to the relativity in the curved spacetime. Just as in the general
theory of relativity, in the covariant theory of gravitation
the speed of electromagnetic signals near massive bodies changes under the
action of gravitation and becomes less than the speed of light. In this case,
the reference frame located at infinity from massive bodies can be considered
as the preferred reference frame, so that in this reference frame the signal
speed for spacetime measurements will be equal to the speed of light. In this
reference frame other special properties can be distinguished, such as
isotropy, and the ESTR approach can be further applied.
The analysis of the acceptable spacetime coordinate
transformations based on the principle of general relativity applied to the
inertial reference frames shows that the ether theories, in which there is an
isotropic reference frame with the same speed of light in all directions, are
subject to the Lorentz transformations for the physical quantities in the case
of parallel motion of bodies or light signals. ^{[14]}
Thus, the general theory of relativity, one of the particular cases of which is
the special theory of relativity, does not prohibit the existence of the ether.
ESTR uses the method of STR, by which for the maximum simplification of the
procedure of measurements, the physical scales of length and time of real
bodies, on the one hand, and the standards of their measurement based on the
waves, on the other hand, are identified (after this the physical rulers and
clocks are completely replaced by the wave standards of length and time). From
this point of view ESTR is the simplest form of the theory of relativity, which
includes STR and admits the existence of the ether or force vacuum field.
According to
the logic of the axiomatic approach, there is no need to substantiate the
axioms of the theory while the conclusions of the theory correspond to
experiments and are proved in practice. However, in order to move over to a
deeper level of scientific knowledge, the analysis of the essence and origin of
axioms is required. This also applies to the axioms of the theory of relativity
in all its forms, including STR, ESTR, the metric theory of relativity, etc.
From this point of view it is possible to consider the force vacuum field,
generating the gravitational and electromagnetic fields, which leads to
interaction of the massive charged bodies in the framework of the mechanism of
the Le Sage's theory of gravitation. ^{[15]}
^{[13]} ^{[16]}
In the Infinite Hierarchical
Nesting of Matter, the force vacuum field is a
multicomponent field consisting of relativistic particles. Each basic level of
matter generates its own component, which makes contribution to the vacuum
field. Thus, it is assumed that the fluxes of praons generated by nucleons create the
electromagnetic force ^{[17]} and the ordinary gravitation ^{[18]} as the force of attraction between
macroscopic objects. In this case, the fluxes of relativistic praons are
assumed to be similar in their properties to the cosmic ray
fluxes, which arise mainly near neutron stars. In the chain “neutron star –
nucleon – praon – graon”, each subsequent object is
the main constituent of the previous object: if the neutron star consists of
nucleons, then the nucleon consists of praons, and the praon consists of
graons, etc. This chain can be continued not only in the direction of
decreasing of the objects’ sizes, masses and energies, but also in the
direction of increasing. All these objects, accelerated to relativistic
velocities, are included into the force vacuum field and generate fundamental
interactions at each level of matter. In addition, it can be argued that the
photons, observed in standard experiments, also consist of praons, ^{[19]} and apparently this also applies to
neutrinos.
In view of the stated above, the
postulates of the STR that the speed of light depends neither on the velocity
of the light source, nor on the velocity of the observer or the receiver of the
light signal, can be understood in the following way. At the moment of a photon
formation in the light source, the fluxes of praons, moving at almost the speed
of light and having a very large Lorentz
factor, pass through the excited atom and are modulated by its
electromagnetic field, that is, the field of the nucleus and excited electrons.
This leads to formation of a rigid periodic structure of the emerging photon,
in which praons are held near each other by the strong gravitation.
When a photon is formed, its linear velocity almost does not differ from the
average velocity of the fluxes of praons, the only difference is that the
praons inside the photon get additional rotation under the action of the atom’s
field. It is the rotational energy of the photon’s particles and its
electromagnetic field that is fixed by the light receiver, leading to
excitation of electrons, while the energy of motion of the photon as a whole is
not determined in the experiment.
If the light source is moving relative
to the isotropic reference frame, this does not influence the speed of the
praons’ fluxes and the speed of the emerging photons. It’s only the photon’s
periodic structure and the corresponding length of their waves that changes,
which is equivalent to the red or blue shift of the photons’ frequency, depending
on the direction of the velocity of the light source relative to the stationary
light receiver. This allows us to explain the Doppler
effect due to the motion of the light source and the independence of the
speed of light from the motion of the light source.
In the opposite situation, the light
source is stationary, and the receiver is moving relative to the isotropic
reference frame, and therefore it detects the Doppler effect as a change in the
frequency in each of the photons reaching the receiver. In fact, the wavelength
of the photon, as the distance between two adjacent wave crests inside the
photon, does not change at all. If we multiply this constant wavelength by the
wave frequency in the moving receiver, we will obtain the wave velocity that is
not equal to the speed of light. However, according to the postulate of STR,
the speed of light does not depend on the velocity of the receiver. This leads
to the paradox of STR about contraction of the sizes of moving bodies (Lorentz
contraction), including the wavelength of the photon reaching the receiver. In
the ESTR, according to the fifth postulate, all measurements are made with the
help of the light signals and using the same procedure. In this case, agreement
between the results of the sametype measurements in different inertial
reference frames is achieved only in the case when the speed of light is
assumed to be constant in all reference frames. Hence we can see that the main
goal of STR is not the correct description of events, but the correct
interpretation of the results of relativistic experiments, regardless of the
arising paradoxes. The role of the ESTR reduces to explaining this from other
perspectives than in the STR, based on a more understandable system of axioms.
The theoretical approach developed by
Robertson, Mansouri and Sexl
(the references are provided in the article Robertson–Mansouri–Sexl framework) is close enough to ESTR. This approach
suggests the possibility of existence of a preferred (privileged) reference frame
as a reference frame with special properties (see the article preferred
frame) and is used as a basis for verifying the conclusions of the special
theory of relativity. Robertson also considered isotropic reference frames, in
which the speed of light is the same in all directions.
A. M. Chepick called isotropic reference frames
“absolute reference frames” and analyzed transformations of coordinates and
time between the moving inertial reference frames and absolute reference
frames. ^{[20]} In this case he used an additional
postulate: ^{[21]} “In any inertial reference frame in
the vacuum, the time of motion of the light signal along the closed linear
contour does not depend on the position of this contour.” This led him to the
STR formulas, including the Lorentz transformations.
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