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3. DEPENDENCE OF LIGHT SPEED UPON A  SOURCE SPEED DERIVED FROM THE RELATIVITY PRINCIPLE

Then, any event  can be characterised by Galilean coordinates x, y, z , t in the reference frame A, and by Galilean coordinates x', y', z', t' in the reference frame B.

Suppose the analogous axes of cartesian spatial coordinates in the both reference  frames are parallel each other, the axes x and x' coinside, the reference frame B moves at a physical speed u in the positive direction of the axis x of the reference frame А, and the clocks, which are at rest in the origins of coordinates of the reference frames А and B, have zero indications at the moment, when the origins of spatial coordinates of the inertial reference frames А and B coinside. As we founded  our reasoning on the principle of full equality of inertial reference frames, we should consider equations  (2.13) and (2.14) to be correct absolutely. Let us introduce a denomination

С (R_А, G_А) = С (R_B, G_B) = c_o , (3.1)

where c_o is the speed of light in vacuum from an immovable source.

Then, in order to obtain a law of light propagation for a moving source, and if we wish that law to be the consequence of the principle of full equality of inertial reference frames, we should suppose that quantities, which are included in the equation  (2.14), depend upon speed and those dependences have the form

С (R_А, G_B) = c_o·Y(u),   (3.2)

С (R_B, G_А) = c_o·Y( - u), (3.3)

Y(u) = Y( - u), (3.4)

Y(0) = 1. (3.5)

where Y(u) is an unknown function (which is to be found) against speed u of movement of one inertial reference frame relatively to another one; Y(0) is the value of unknown functionY(u) when u = 0;  c_o = 299 792 458 m/s is the speed of light in vacuum from an immovable source [ [24]. Sazhin M. V. Light speed// Space Physics. Small Encyclopedia. Мoscow: Soviet Encyclopedia, 1986. p. 622 (in Russian)].

Any other information about properties or a form of function Y(u), besides properties (3.4) and (3.5), we can not obtain from equations (2.13) and (2.14). But this does not mean that the problem  to find a law of light propagation from a moving source, which should be a consequence from the principle of full equality of inertial reference frames (from the relativity principle), could not be solved.  Because equalities (2.13) and (2.14) are only partial consequences from the relativity principle for the process of light propagation and they do not exhaust the content of the relativity principle.

Indeed, Einstein formulated  the principle of relativity in the following wording [ [17]. Einstein A.  To electrodynamics of moving bodies. Collection of scientific tractates, v. 1. - Мoscow: Nauka, 1965. - pp. 7 - 35 (in Russian)]:

In the article [[3]. Einstein A. Zur Electrodynamik bewegten Korper // Annalen der Physik. - 1905. - B., 17. - s. 891 – 921.  “To electrodynamics of moving bodies”] this principle was formulated so:

And from this statement it follows that the principle of full equality of inertial reference frames ( the principle of relativity) does not result in any "time dilation" in moving reference frames. Namely this consequence from the principle of full equality of inertial reference frames (from the relativity principle) we shall use for obtaing the law of light propagation from a moving source

c_u = c_o Y(u), (3.6)

where c_u is  the speed of light in vacuum from a moving source; c_o is  the speed of light in vacuum from an immovable source;  Y(u) is some unknown function having properties (3.4) and (3.5).

Suppose that B_о is the origin of spatial coordinate system in the inertial reference frame B, А_o is the origin of coordinates of the inertial reference frame А. Suppose that a light source G_B, which is at rest in point B_o, at time moment t' = 0 sends light signal in the direction of axis y', which is perpendicular to the direction of movement of the inertial reference frame A relatively the inertial reference frame B. Suppose that on the axis y' of the reference frame B at a distance of  y_o' from point B_о a mirror B₁ is installed. The light signal reflects from this mirror and returns to the point B_o. Then (because either light source G_B, and mirror are at rest in the reference frame B) this light signal propagates in the reference frame B at a speed of c_o either when it moves from the point B_o to the mirror B₁, and when it moves from the mirror B₁ to the point B_o, as it is shown in fig. 3.1 а. As a result the light signal will return to the point B_o in time interval

Dt' = 2 y_o'/c_o (3.7)

after radiation of this light signal from the point B_o. And now let us consider propagation of the same light signal in the inertial reference frame А, with respect to which the light source G_B and the mirror  move to the right  together with the  reference frame B at a speed u.

Fig. 3.1. Propagation of light in two reference frames moving relatively each other:   а)  in that reference frame B, relatively to which the light source is at rest; b) in that reference frame A, relatively which this light source is moving.

At time moment t' = 0 points B_о and А_о coinside spatially. That is why in the inertial reference frame А radiation of this light signal takes place from point А_о. During the time interval, within which the light signal moves in the reference frame B from the point B_о to the mirror B₁, the reference frame B moving at a speed u relatively the reference frame А will move at a certain distance. Therefore reflection of light from the mirror B₁ in the inertial reference frame А will take place in the point В of fig. 3.1 b.  And during the time interval, within which the light signal moves in the reference frame B from the mirror B₁ to the point B_о, the reference frame B will also move at a certain distance and at the time moment, when the light signal will come to the point B_о in the reference frame B, the same point B_о of  the reference frame B will coincide with the point M of the reference frame A.

It is quite evident, that А_оВ = ВМ. It is also quite evident, that light signal path length in the reference frame A will be greater, than path length of the same light signal in the reference frame B.

If Dt is a time interval between a moment of light signal emission from the point А_о and a moment of the same signal reception in the point M of the reference frame A, then distance s passed by the light signal in the reference frame A from the point А_о to the point M can be determined using Pythagorean theorem 

s = 2 [ y_o² + ( 0,5 u Dt )² ]^1/2 . (3.8)

But in the reference frame A both the  light source and the mirror are moving at a speed u. Therefore we can suppose that the speed of light lignal propagation in the reference frame A along straight lines А_оВ and ВМ is determined by expression (3.6). As a consequence of this the time interval Dt between a moment of light signal emission in the point A and a moment of its reception in the point M in the reference frame A can be calculated dividing light signal path length s from equation (3.8) by the speed of light propagation from equation (3.6). We have

Dt  = 2 [ y_o² + (0,5 u Dt )² ]^1/2/ [c_o Y(u) ] . (3.9)

From expression (3.9)  we have

Dt  = 2 y_o/{ c_o [ Y²(u) - u²/c_o² ] ^1/2}. (3.10)

And now let us use the statement, which we have obtained earlier as a consequence of the principle of full equality of inertial reference frames: "The laws, according to which the indication of a clock change, do not depend upon which of two coordinate systems moving uniformly and rectilinearly each relatively other  these changes of indication are referred to".

According to this consequence from relativity principle, if in the points А_о and М of the inertial reference frame A there are two clocks synchronised each with other as mentioned above, and if in the point B_о also there is a clock of the same design, which at the moment t = t' = 0 has an indication equal with an indication of the clock, which is at rest in the point А_о, then at the moment of light signal reception in the point M the clock, which is at rest in the point B_о, should have an indication equal with indication of the clock, which is at rest in the point M. This means, that according to this consequence from the relativity principle we have the right to make the right-hand parts of equalities (3.7) and (3.10) equal each other, i. e. we have the right to write down

2 y_o'/c_o = 2 y_o/{c_o [ Y²(u) - u²/c_o²]^1/2}. (3.11)

In order to continue our consideration we should show, that lateral dimensions (dimensions in any direction perpendicular to the direction of movement) of a moving body do not depend upon a speed of body movement.

Let us perform such proof by contraries using an imaginable experiment with two simplest flat bodies: a ring and a disk, which are moving each to other along their mutual axis of symmetry and which have equal outer diameters.

Firstly, let us suppose that lateral dimensions of a moving body decrease, when the body speed increases.

Fig. 3.2. Lateral dimensions of moving bodies can not decrease at speed increasing . If lateral dimensions of a moving body decrease, then: a) after a meeting the disk and the ring will continue moving without damages; b) at the moment of meeting the disk and the ring will be destroyed.

If we shall consider the ring as an immovable body and the disk as a moving body, then according to the initial assumption (that lateral dimensions decrease at speed increasing) at a certain speed the outer diameter of the moving disk can become less than the inner diameter of the immovable ring and the disk will fly through the opening in the ring without damages and the both bodies will continue moving undamaged (see fig. 3.2 a). And if we shall consider the disk as an immovable body and the ring as a moving body, then according to the same initial assumption (that lateral dimensions decrease at speed increasing) the outer diameter of the moving ring will become less than the diameter of the immovable disk and a collision with distruction of the both bodies will take place at their meeting. But insomuch as according to the relativity principle any of the two bodies can be considered either as a moving body or as an immovable body, then the initial assumption about decreasing of lateral dimensions of a moving body when its speed increases results in appearance of contradiction of existence for moving bodies after their meeting. Indeed, it is impossible to imagine that two statements: 1) "after a meeting the disk and the ring exist undamaged" and 2) "after a meeting the disk and the ring are distroyed" can be true simultaneously. That means that the first initial assumption is wrong.

Secondly, let us suppose that lateral dimensions of moving bodies increase with speed increasing.

Fig. 3.3. Lateral dimensions of bodies can not increase at speed increasing. If lateral dimensions of a moving body increase, then: a) after a meeting the disk and the ring will continue existence without damages; b) the disk and the ring will be destroyed at their meeting.

Then, if we shall consider the ring as an immovable body and the disk as a moving body, in accordance with this second assumption the diameter of the moving disk will become greater than the outer diameter of the ring and as the result a collision of the both bodies will occur with distruction of the both bodies at their meeeting. And if we shall consider the disk as an immovable body and the ring as a moving body, then, in accordance with the same second assumption, at sufficiently great speed the both the innner diameter and the outer diameter of the moving ring can become greater than the outer diameter of the immovable disk and after the meeting of the both bodies they will continue moving without any damages. So, the second assumption (that lateral dimensions of moving bodies increase with speed increasing) result in contradiction of existance of the both boddies after their meeting. That means that the second assumption is also wrong.

But if lateral dimensions of moving bodies can not either increase or decrease with speed increasing, then it remains for us only  to consider that the statement about independence of lateral dimensions of moving bodies upon speed of bodies movement is prooved.

So, according to the above consideration we can take for granted that quantities y_o and y_o' from formula (3.11) are equal each other

y_o = y_o'. (3.12)

(because lateral dimensions of a moving body can not depend upon speed of the boddy movement).

Having now substituted equality (3.12) into expression (3.11) we have

Y(u) = (1 + u²/c_o²)^1/2. (3.13)

Ssubstituting then the expression (3.13) into the expression (3.6), we have the law (2.1) of light propagation from a moving source, which is a consequence of the principle of full equality of inertial reference frames (relativity principle)

c_u = c_o(1 + u²/c_o²)^1/2.   (3.14)

Indeed, it is not difficult to check, that the function (3.13) possesses properties (3.4) and (3. 5), and that equalities (2.13) and (2.14) are correct exactly..

The law (3.14) can be obtained also from invariance of the 4-dimensional interval ds².

Let us consider the inertial reference frame X', Y', Z', T', which moves at a constant speed V in the direction of positive values of coordinate X of the unprimed reference frame X, Y, Z, T. Then the expression for the square of 4-dimensional interval in the primed cartesian coordinates will be determined by the exxpression

dS² = c_o²·(dT)² - (dX')² - (dY')² - (dZ')². (3.15)

Let us apply  to expression (3.15)  the Galilean transformation

t = T',   x = X' + VT',   y = Y',   z = Z' . (3.16)

The reverse transformation has the form

T' =  t ,    X'= x  - Vt,   Y' = y,   Z' = z . (3.17)

Having taken differentials from the both parts of equalities (3.17) and having substituted them into expression(3.15), we have

dS² = c_o² (1 - V²/c_o²) dt² + 2 V dx dt - dx² - dy² - dz². (3.18)

In order to get rid of a cross member dx dt in the right-hand part  of the expression (3.18) let us form a full square in it. As a result the interval (3.18) takes the form

dS² = [c_o²/(1 - V²/c_o²)] [(1 - V²/c_o²) dt +(V/c_o²) dx]² - dx²/ (1 - V²/c_o²) - dy² - dz². (3.19)

Now, let us introduce new speed, which coincides with spatial component of the four-dimensional speed from the special relativity theory

u = V(1 - V²/c_o²)^-1/2,   (3.20)

new time

T = t(1 - V²/c_o²) + V·x/c_o². (3.21)

and new coordinates

X = x (1 - V²/c_o²) ^-1/2,   Y = y,   Z = z . (3.22)

Then expression (3.19) for the interval in this variables will have the form

dS² = [c_o²/(1 - V²/c_o²)] dT² - dX² - dY² - dZ². (3.23)

If we want to have invariant interval, the expression (3.23) should have the form

dS² = c_u² dT² - dX² - dY² - dZ². (3.24)

Changing the expression (3.23) by the expression (3.24) we can perform having introduced the new expression for light speed

c_u = c_o (1 - V²/c_o²)^-1/2.(3.25)

Let us determine V from the expression (3.20). We have

V  = u (1 + u²/c_o²)^-1/2. (3.26)

Now, let us substitute the expression (3.26) into formula (3.25). As a result we have the law of light speed dependence upon light source speed

c_u = c_o(1 + u²/c_o²)^1/2, (3.27)

which coincides  with expressions (3.14) or (2.1).

The law (3.27) of light speed dependence upon light source speed   was for the first time found  by P. M. Rapier [ [25]. Rapier P. M. An extension of Newtonian relativity to include electromagnetic phenomena // Proceedings of the IRE. - 1961. - V. 49. - P. 1691 - 1692; 1962.-v.50.-p. 229-230; Spectroscopy Letters. - 1971. - v. 4(9).-p-303-311.] This law allows to build up a new space-time  theory based upon the only principle of relativity. But before the beginning of such building up first of all it is necessary to ascertain, that all experiments on cheking   Einstein's second postulate, which have been carried out earlier, do not contradict the existence in nature of the law (3.27).

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