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2. ESSENSE OF EINSTEIN'S SECOND POSTULATE

The original formulation of the special relativity theory (SRT) was based upon evident or implicit use of two postulates: the relativity principle and the principle of light speed constancy. But according to Einstein [9] "The purpose of theoretical physics consists in creation of notions system based upon the possibly less quantity of logically independent hypotheses allowing to determine the causality interrelation of all complex of physical processes".

In accordance with this statement in recent years the repeated attempts were initiated to prove that the SRT is based upon "the minimal quantity of postulates" [10]. With this purpose in [11], for example, the both principles are united into one "Einstein's relativity principle", in [6] it is asserted that the SRT can be built up using the only postulate about pseudo-Euclidean structure of space-time and that the postulate about light speed constancy is a consequence of pseudo-Euclidean structure of space-time, and in [12] it is allegedly proved that Einstein's second postulate (the principle of light speed constancy) is a consequence of Einstein's first postulate (the relativity principle).

Einstein (one of creators of the SRT) also united two initial principles of the SRT. For example, in 1914 he wrote [13]: "Uniting the law of light speed constancy in vacuum with the relativity principle it is possible to come by means of purely deductive way to a theory, which is called nowadays the "relativity theory". Or in 1916 he wrote [16]: "So, the special relativity theory differs from classical mechanics not only by the relativity principle, but, mainly, by the postulate of light speed constancy in vacuum, from which, at its uniting with special relativity principle, the relativity of simultaneity, Lorentz transformations and laws connected with these transformations related to behavior of rigid bodies and clocks are known consequences". And even in 1950 Einstein wrote not about one, but about two principles lying in the basis of the SRT [15]: "Notwithstanding the fact that the both principle mentioned above are well proved by experiments they do not seem logically compatible. The special theory of relativity succeeded in making them compatible at the price of changing kinematics, or, at the price of changing physical notions about space and time". Speaking about uniting those two principles Einstein always implied their conciliation, logical compatibility, but he always considered them completely independent each on other [16]: "The theory named today as "relativity theory" is based upon two principles completely independent each on other, namely:

1) upon principle of relativity for uniform straight-line movement;

2) upon the principle of light speed constancy".

On page 93 in [6] it is asserted that "the postulate about speed light constancy can be formulated in inertial reference frame in any admissible space-time coordinates as partial consequence of pseudo-Euclidean structure of space-time on the basis of physical speed notion introduced earlier". From this statement it follows that the postulate about constancy of physical speed of light is allegedly the consequence of pseudo-Euclidean structure of the space-time.

But we can show (see chapters 4 and 5) that all experiments carried out till now do not refute the existence in reality of namely physical speed of light upon physical speed of source movement according to the formula

c_u = c_o·(1 + u²/c_o²)^1/2,   (2.1)

where c_o is the physical speed of light in vacuum from immovable source; u is the physical speed of a light source; c_u  is the physical speed of light in vacuum from a moving light source.

The dependence (2.1) does not contradict the main postulate of A. A. Logunov (see page 31 in [6]): "The physical processes take place in four-dimensional space, which geometry is pseudo-Euclidean".

Indeed, it can be shown, that if the events happen with a body, which is at rest in an inertial reference frame with Galilean coordinates х', y', z', t', which moves at a constant physical speed u with respect to another inertial reference frame with Galilean coordinates х, y, z, t in the direction of positive values of the coordinate х, then the coordinates of events happening with this body in these two inertial reference frames aare connected each with other by means of formulas

с_o t' = Г (c_u t - b x),  x' = Г (x - b c_u t),  y' = y,  z' = z, (2.2)

where b = u/c_u ;

    Г = (1 - b²)^-1/2 = (1 + u²/c_o²)^1/2 ; .(2.3)

c_u is determined by dependence (2.1).

Having taken differentials from the both parts of equations (2.2), we have

d(с_o t') = Г [d(c_u t) - b dx], dx' = Г [dx - b d(c_u t)], dy' = dy, dz' = dz. (2.4)

Let us substitute now the differentials (2.4) into the expression

ds² = [d(c_o·t')]² - (dx')² - (dy')²  - (dz')² (2.5)

for the interval in the primed inertial reference frame, in which the body under consideration is at rest. We have

ds² = Г²{[d(c_ut)]² - 2bdx d(c_ut) + b²dx² - b²[d(c_ut)]² + 2bdx d(c_ut) - dx²} - dy²  - dz². (2. 6)

After evident simplifications we have

ds² = Г²(1 - b²) [d(c_u t)]² -  Г² (1 - b²) dx² - dy²  - dz². (2.7)

Taking into account expression (2.3) the formula (2.7) looks like

ds² = [d(c_u·t)]² - dx² - dy²  - dz².   (2.8)

The non-zero components of the metric tensor of space-time in the primed inertial reference frame which are determined by the interval (2.5), are equal (see Appendix 1)

g_oo' = 1,   g₁₁' = g₂₂' = g₃₃' = - 1 . (2.9)

And the non-zero components of the metric tensor of space-time in the unprimed inertial reference frame, which are determined by the interval (2.8), are equal to (see Appendix 1)

g_oo = 1,   g₁₁ = g₂₂ = g₃₃ = - 1. (2.10)

From expressions (2.9) and (2.10) it follows that the structure of space-time geometry is pseudo-Euclidean not only in case, when the physical speed of light in the inertial reference frame does not depend upon speed of light source, but also in case, when the physical speed of light depends upon the speed of light source according to formula (2.1). That is why notwithstanding the fact that the postulate about constancy of light speed is not a consequence of pseudo-Euclidean structure of space-time, the main Logunov's postulate is valid also in case when physical speed of light depends upon source speed according to formula (2.1). As a consequence, the special relativity theory in book [6] is actually based not only on the main Logunov's postulate, but also on the implicit use of the second Einstein's postulate, which is formulated on page 93 in [6] as follows:

"The physical speed of light in an inertial reference frame in any admissible coordinate systems of space-time is always equal to с_o and does not depend upon direction of movement".

As for assertions available in [12] about dependence of the second postulate of the SRT on the first postulate and about possibility of building up the SRT without postulate about light speed constancy, such assertions are based on deduction of relativistic law of speeds u and V addition

w = (u + V)/(1+ k·u·V). (2.11)

given in [12] without use of the second Einstein's postulate (in expression (2.11) k is a positive constant, which does not depend upon neither u, nor upon V).

Indeed, the principle of light speed constancy is not used in [12] neither at deduction of law (2.11) by means of the first method (using mental experiment "competition in moving train") nor at deduction of law (2.11) by means of the second method (using mental experiment "ball-oscillator").

But in [12] it was proved convincingly that without use of the principle of light speed constancy the law (2.11) of speeds addition can be obtained only in case when speed V of relative movement of two inertial frames of reference is sufficiently less than the speed of light in vacuum (at limit V ® 0). Indeed, having designated the speed of one runner by letter s, the speed of the second runner by letter u, and the speed of the train by letter V, Mermin obtains in [12] the equality

(1/s²)·[1 - 1/h'(s)] = (1/u²)·[1 - 1/h'(u)], (2.12)

where h'(z) is a derivative against z from function h(z) introduced in [12]. And further Mermin reasons in the following manner: "As soon as the left-hand part of this equality depends only upon s and the right-hand part depends only upon u, each of these expressions should be equal to the same constant k, which does not depend upon s and u".

And possible, in principle, dependence of value k from equation (2.11) upon train speed V Mermin deletes in the process of logical argumentation at deduction of equality (2.12) by passage to the limit at V® 0 (see formulas (3.6) and (3.8) in article [12]).

And at deduction of equation (2.11) by means of the second method from the text of article [12] it also follows that the value k is also determined as a limit, to which one of expressions tends at V ® 0 (see formula (4.12) and the text after formula (4.11) in article [12]).

So, in article [12] it is proved, that speeds addition law (2.11), in which value k does not depend upon train speed, can be obtained without use of Einstein's second postulate only when V ® 0.

But whether or no it is possible to obtain law (2.11) without use of the postulate about light speed constancy at sufficiently greater magnitudes of train speed, in article [12] there is no answer on such a question.

That is why the purpose put into the name of article [12] (the article name is "Theory of relativity without postulate of light speed constancy") is not to be considered as achieved. Because the SRT and the law (2.11) as its component should be valid not only at V ® 0, but at any speeds not exceeding the speed of light in vacuum. So, all attempts initiated earlier to prove that the SRT is based on introducing the minimal quantity of postulates look bankrupt. But the only fact of such attempts performing proves that, notwithstanding almost century period of its existence, the real essence of the second Einstein's postulate is not completely clarified.

The second postulate of the SRT named by Einstein as "the principle of light speed constancy" reads:

"The light in vacuum always propagates at a certain speed c_o irrespective of the condition of movement of radiating body" (see page 7-8 in [17]) or:

"Each beam of light moves in "immovable" coordinate system at a certain speed c_o irrespective of its radiation whether by immovable or moving body" (see page 10 in [17]).

In [3] these formulations have the following wording, respectively:

“... Das Licht im leeren Raume stets mit einer bestimmten vom Bewegungszustande des emittierenden Kц rpers unabhд ngigen Geschwindigkeit V fortplanze”,

“Jeder Lichtstrahl bewegt soch im “ruhenden” Koordinatensystem mit der bestimmten Geschwindigkeit V, unabhд ngig davon, ob dieser Lichtstrahl von einem ruhenden oder bewegten Korper emittiert ist”.

For clarification of Einstein's second postulate essence let us consider the following situation. Let the inertial reference frame B to move evenly and rectilinearly with respect to "immovable" inertial reference frame A (as these two inertial frames of reference are completely equal, we can on the same grounds speak that inertial reference frame A moves evenly and rectilinearly with respect to "immovable" inertial reference frame B). Now let a light source (generator) and device for measuring light speed be placed in each of these reference frames. Let G_A be a light source, which is at rest in reference frame A, and G_B be a light source, which is at rest in reference frame B. Let R_A be a device (for measuring light speed), which is at rest in reference frame A, and R_B be a device (for measuring light speed), which is at rest in reference frame B.

Then, using these two light sources and these two devices for measuring light speed, we can obtain the following four values for the light speed (see Fig. 2.1):

С (R_А, G_А) - speed value measured by device R_А for light radiated by source G_А;

С (R_B, G_B) - speed value measured by device R_B for light radiated by source G_B;

С (R_А, G_B) - speed value measured by device R_А for light radiated by source G_B;

С (R_B, G_А) - speed value measured by device R_B for light radiated by source G_А.

Fig. 2.1.

In accordance with the SRT for these four values of light speed we can write down the following six equalities (according to the quantity of combinations from four elements by two):

С (R_А, G_А) = С (R_B, G_B), (2.13)

С (R_А, G_B) = С (R_B, G_А), (2.14)

С (R_А, G_A) = С (R_А, G_B), (2.15)

С (R_B, G_B) = С (R_B, G_A), (2.16)

С (R_B, G_B) = С (R_А, G_B), (2.17)

С (R_А, G_А) = С (R_B, G_А). (2.18)

It is not difficult to make oneself sure that from these six equalities only three equalities are independent each upon other. Indeed, equalities (2.13) and (2.14) are formalised records of the relativity principle (the principle of complete equality of inertial reference frames), according to which in each inertial reference frame the physical processes at identical initial conditions develop identically, in the context of light propagation. At that in equality (2.13) the identity of initial conditions for the process of light propagation consists in immovability of light sources with respect to devices, by means of which the speed of light is measured, and in equality (2.14) the identity of initial conditions for the light propagation process consists in identity of speeds of light sources movement with respect to devices, by means of which the speed of light is measured.

It is evident, that equalities (2.13) and (2.14) are independent one upon another. Because from validity of equality (2.13) the validity of equality (2.14) does not follow, and vice versa, from validity of equality (2.14) the validity of equality (2.13) does not follow also.

Now let us pay attention to the structure of equalities (2.15), (2.16), (2.17) and (2.18). In equality (2.15) the left-hand parts of equalities (2.13) and (2.14) are equalised. In equality (2.16) the right-hand parts of equalities (2.13) and (2.14) are equalised. In equality (2.17) the right-hand part of equality (2.13) is equalised to the left-hand part of equality (2.14). And, at last, in equality (2.18) the left-hand part of equality (2.13) is equalised to the right-hand part of equality (2.14). And from this it follows that logically independent each upon other are the equalities (2.13), (2.14) and any one of equalities (2.15), (2.16), (2.17), (2.18).

Because among four equalities (2.15), (2.16), (2.17) and (2.18) each equality individually is a formalised record of Einstein's formulation of the principle of light speed constancy.

Indeed, both in left-hand and in right-hand parts of equality (2.15) the light speed values measured by the same device R_А enter, but in the right-hand part the speed value corresponds to the light radiated by the source G_B, which is moving relatively to R_А, and in the left-hand part the speed value corresponds to the light radiated by the source G_A, which is at rest relatively R_А. But Einstein's second postulate reads, that "each light beam moves in the "immovable" coordinate system at a certain speed c_o, irrespective of whether this light beam is radiated by an immovable or by a moving body". That is why the equality (2.15) is a formalised record of the second Einstein's postulate in the "immovable" reference frame A, and the equality (2.16) is a formalised record of the second Einstein's postulate in the "immovable" reference frame B. Equalities (2.17) and (2.18) also can be considered as formalised record of Einstein's second postulate, if this formulation can be changed for the following: "Light beams from the same source move in any reference frame at the same speed c_o, irrespective of whether this reference frame is moving relatively to this source or it is at rest".

This changed formulation differs from Einstein's formulation only by a body, which is considered to be "immovable": in Einstein's formulation the "immovable" body is a device, by means of which the light speeds from different sources (moving and immovable) are measured, and in the changed formulation the immovable body is a light source and light speeds measurements are performed by different devices (moving and immovable). At that the equality (2.17) is a formalised record of the changed formulation of the second postulate if light source is at rest in inertial reference frame B, and equality (2.18) - if the light source is at rest in inertial reference frame А. As a consequence of this, from four formalised records of the second postulate - from equalities (2.15), (2.16), (2.17) and (2.18) - only any one record, for example the equality (2.15) is independent upon equalities (2.13) and (2.14). The presence of independent upon equalities (2.13) and (2.14) formalised record of the second Einstein's postulate is the basis for considering the second Einstein's postulate logically independent on the first Einstein's postulate.

Thus, the essence of Einstein's second postulate is disclosed by equality (2.15), and equalities (2.13) and (2.14) into the content of the second Einstein's postulate do not enter. That is why in many publications (see [18]...[23]) Einstein's second postulate is quite correctly named as the postulate about independence of light speed upon light source speed.

The sense content of other names of Einstein's second postulate, such as "the principle of light speed constancy" or "the principle of light speed invariance", contains actually all three equalities (2.13), (2.14) and (2.15), what forms the grounds for erroneous statements about logical dependence of Einstein's second postulate upon Einstein's first postulate. In order to delete such grounds, it is expedient in future to name Einstein's second postulate as a law of independence of light speed upon the source speed.

So, two logically independent each upon other statements lie in the basis of the special relativity theory: the principle of complete equality of inertial reference frames (always named as "relativity principle") and the law of independence of light speed upon source speed (named sometimes as "light speed constancy principle"). Therefore the further development of our notions about space and time within the framework of the SRT is not possible and it is expedient to look for the possibility of building up a new space-time theory based upon the only one initial principle.

So, let us consider whether it is possible to build-up a new space-time theory, if we shall put as the basis for consideration only one relativity principle (the principle of complete equality of inertial reference frames), and whether it is possible to obtain the law of propagation of light from a moving source as a consequence of this relativity principle.

Well, having clarified that Einstein's second postulate (equality (2.15)) is not a corollary from Einstein's relativity principle, let us try to replace equality (2.15) by such expression, which should be a corollary of relativity principle.

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