На русском языке
Extended
special theory of relativity
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

Works on the axiomatics of STR [1]
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.^{
[14]}
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 (the wave speed).
Introduction
The analysis of the axioms and
the results of SRT gives the following:
 All
inertial reference frames in STR are completely equivalent in the sense,
that the kinematic characteristics of the physical processes in the moving
frame are not identical but are similar to the characteristics of the same
processes in the stationary reference frame. This means the Lorentz
covariance of the mathematical form of physical laws.
 All
the effects of SRT in the final analysis are the consequence of the fact
that the speed of light is limited.
 The
Lorentz transformations can be derived in different ways, in different axiomatics, including the use of representations of
mathematical groups.
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.
The postulates of ESTR
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:
 1)
The principle of relativity holds (if all the material bodies of the
physical system are brought to the state of free and uniform rectilinear
motion relative to the frame, conventionally called the rest or stationary
frame, then the phenomena in the moving reference frame for the comoving
observer would look as in the stationary reference frame for the fixed
observer).
 2)
There is such an isotropic reference frame in which the speed of light
propagation is equal in all directions and does not depend on the speed of
the light emitter.
 3)
The validity of symmetries with respect to rotations in the Euclidean
spacetime. In particular, during the motion of the reference frame the
coordinate axes are considered to remain parallel to the axes of the fixed
reference frame. Also the independence is implied of the speed of light on
its propagation direction in the transverse direction relative to the
velocity of motion of the inertial reference frame.
 4)
The validity of symmetries with respect to displacements in the Euclidean
spacetime. This means the linearity of transformations of coordinates and
time from one inertial frame to another (all the coordinates in the
transformations are included in the first degree,
the terms with higher degrees are absent). In addition it is assumed that
the transverse length of the rod does not depend on the sign of the velocity
of motion of the rod, and is determined by the magnitude of the velocity.
 5)
Spacetime measurements are carried out by means of electromagnetic or
other waves propagating at high speed.
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.
Proving the equal value of ESTR and STR
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.
The results of ESTR
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. ^{[11]} 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). ^{[12]} 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.
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. ^{[13]} 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.
References
 Fedosin S.G. Sovremennye problemy fiziki: v poiskakh novykh printsipov,
Moscow: Editorial URSS, 2002, 192 pages. ISBN 5836004358.
 von W. v. Ignatowsky, «Einige allgemeine Bemerkungen zum Relativitätsprinzip», Verh. d. Deutsch. Phys. Ges.
12, 78896, 1910 (русский
перевод)
 von Philipp Frank und Hermann Rothe «Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme», Ann. der Physik, Ser. 4, Vol. 34, No. 5, 1911, pp. 825—855.
 Pauli, Wolfgang (1981). Theory of Relativity. New
York: Dover Publications. ISBN 048664152X.
 Терлецкий
Я. П. — Парадоксы теории относительности, М.: Наука (1965)
 Mermin N.D. — «Relativity without light», Am.J.Phys., Vol. 52, No. 2 (1984) p. 119—124. Русский
перевод: Мермин Н. Д. — «Теория
относительности без постулата о постоянстве скорости света», Физика за рубежем. Серия Б. (1986)
 Lee A.R. Kalotas T.M. — «Lorentz transformations from the first
postulate», Am.J.Phys., Vol. 43, No. 5, (1975) p. 434—437.
 Achin Sen «How Galileo could have derived the special theory
of relativity» Am.J.Phys., Vol. 62, No. 2 (1994)
p. 157—162.
 Nishikawa S. — «Lorentz transformation without the direct use of
Einstein’s postulates» Nuovo Cimento,
Vol. 112B, No. 8 (1997) p. 1175—1187.
 А.
К. Гуц, «Аксиоматическая теория
относительности», УМН, 37:2(224) (1982), с. 39—79.
 Fedosin S.G. Electromagnetic
and Gravitational Pictures of the World. Apeiron,
Vol. 14, No. 4, P. 385413 (2007).
 Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. Perm,
20092011, 858 pages, Tabl. 21, Pic. 41, Ref. 293. ISBN 9785990195110. (in Russian).
 Alexander L. Kholmetskii. Empty spacetime, general
relativity principle and covariant ether theories. 12 Jan 2005. arXiv:physics/0501060v1.
 Luigi Maxmilian Caligiuri, Amrit Sorli, Special Theory of Relativity Postulated on Homogeneity of Space and Time and on Relativity Principle, American Journal of Modern Physics. Vol. 2, No. 6, 2013, pp. 375382.
See also
 Special
relativity
 Metric theory of relativity
 General
relativity
 Lorentzinvariant theory of
gravitation
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