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6. TIME AND COORDINATES TRANSFORMATIONS IN CASE OF NON-INVARIANT LIGHT SPEED

           If the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 of light speed upon light source speed exists in nature, then the well-known Lorentz's transformations will be not exact transformations, but only approximate transformations  valid only for not very great speeds of relative movement of inertial reference frames, at which this dependence of light speed upon source speed may be neglected. But then a question emerges immediately: what are the exact transformations consistent with the  law  of light propagation from a moving source having the form c_u = c_o(1 + u²/c_o²)^1/2.

             The simpliest way for obtaining an answer to this question is to use Logunov's method of derivation ( [6]. Logunov A. A. Lectures on theory of relativity and gravitation. Current analysis of the problem. 3-d edition, - Moscow: Issue of Moscow State University, 1985).

            Let us consider the same two reference frames A and B moving relatively each other, which we have considered in chapter 3. Let some object be at rest in the inertial reference frame B, which is moving at a speed u in the positive direction of X axis of inertial reference frame А.

              Then during derivation of transformations of coordinates and time of events, which happen to this object, we must take into consideration, that at infinitesimal change of this object condition caused by an event happening to it, the position and the state of movement  of charges, which are components of that object, change, in the result of such changes an electromagnetic wave is formed and begins propagating in   circumambient vacuum. (Hereinafter we shall call this electromagnetic wave for short as "light"). This light in the inertial reference frame B, with respect to which this object is at rest, propagates at a speed of c_o, and in the reference frame А, with respect to which this object moves at a speed of u, this light propagates at a speed c_u, which is determined by the expression c_u = c_o(1 + u²/c_o²)^1/2. As a result, the expression for the interval in galilean coordinates of the reference frame A will be as follows

ds^{2 =} c_u²dt²- dx² - dy² - dz², (6.1)

where c_u = c_o(1 + u²/c_o²)^1/2.

             Let us subject the expression (6.1) to galilean transformation

x'' = x - u t, t'' = t, y'' = y, z'' = z . (6.2)

With this purpose let us write down transformations inverse to transformations (6.2)

x = x'' + u t'', t = t'', y = y'', z = z'' . (6.3)

where x,  y,  z,  t are galilean coordinates of the same event in the inertial reference frame A.

              Having found differentials from the both parts of equalities (6.3) and having substituted dx,  dy,  dz,  dt  into the expression  (6.1) for the interval, we shall have

ds^{2 =} c_u²(dt'')²(1 - u²/c_u²) - 2 u dx'' dt'' - (dx'')² - (dy'')² - (dz'')².       (6.4)

The criss-cross member  dx'' dt'' in the expression (6.4) can be removed. For this purpose let us isolate in the expression (6.4) the perfect square. As a result the interval (6.4) will have the form

ds^{2 =} c_o²[y(u) dt''(1 - u²/c_u²)^1/2 - u dx''(1 - u²/c_u²)^-1/2(c_oc_u)^-1]² - (dx'')²/(1 - u²/c_u²) - (dy'')² - (dz'')²,   (6.5)

where y(u) = (1 + u²/c_o²)^1/2, that means that y(u) is determined by the expression (3.13).

              Now let us introduce new time

t' = y(u) t'' (1 - u²/c_u²)^1/2 - u x''(1 - u²/c_u²)^-1/2(c_oc_u)^-1 (6.6)

and new coordinates

x' = x'' (1 - u²/c_u²)^-1/2,  y' = y'' , z' = z''. (6.7)

Then the expression (6.5) will have the form

ds^{2 =} c_o²(dt')²- (dx')² - (dy')² - (dz')². (6.8)

                But expression (6.8) is the expression for the interval in the galilean coordinates of the inertial reference frame B.

               Thus, having used successively transformations (6.2) and  transformations (6.6) - (6.7), we have passed from the interval (6.1) in the inertial reference frame A  to the interval (6.8) in the inertial reference frame B. That means, that having substituted the expression (6. 2) into the expressions (6.6) and (6.7), we shall have transformations of coordinates and time of events from the  reference frame A to the inertial reference frame B

c_ot' = Г (c_ut   - b x) ,   x' = Г (x - b c_ut ) ,     y' = y ,   z' = z , (6.9)

where  b = u/c_u ,  Г = (1 - b ²)^-1/2.

               The inverse transformations have the form

c_ut = Г (c_ot' + b x'),  x = Г (x' + b c_ot' ), y = y', z = z', (6.10)

where  b  = u/c_u;  Г = (1 - b ²)^-1/2; c_u = c_o(1 + u²/c_o²)^1/2 .

              Expressions (6.9) and (6.10) are direct and inverse transformations of coordinates and time of events  from one reference frame to another one for that particular case, when events happen to an object, which is at rest in the inertial reference frame B.

                Similarly it can be shown, that if events happen to objects, which are at rest in the reference frame A,   the direct and the inverse transformations of   coordinates and time for  these events are as follows

c_ot = Г (c_ut' + b x'),   x = Г (x' + b c_ut' ),   y = y',   z = z', (6.11)

c_ut' = Г (c_ot   - b x),   x' = Г (x - b c_ot ),    y' = y,   z' = z , (6.12)

where still  b  = u/c_u ,  Г = (1 - b ²)^-1/2,    c_u = c_o(1 + u²/c_o²)^1/2.

                It is not difficult to see, that if the dependence of light speed upon light source speed can be neglected (at low speed of source motion as compared with speed of light c_o), then transformations (6.9), (6.10), (6.11) and (6.12) are converted into Lorentz's transformations from the special theory of relativity

c_ot' = Г (c_ot - b x),   x' = Г (x - b c_ot ),    y' = y,   z' = z , (6.13)

c_ot = Г (c_ot' + b x'),   x = Г (x' + b c_ot' ),   y = y',   z = z', (6.14)

where  Г = (1 - b ²)^-1/2;   b = V/c_o ; V is the speed of motion of one reference frame relatively other one, which can not exceed c_o.

             Transformations (6.9)...(6.12) are valid for such particular case of mutual disposition and motion of reference frames A and B, at which analogous coordinate axes of these two inertial reference frames are parallel each other, axes x and x' coincide and the reference frame B moves in the positive direction of x axis of reference frame A.

             Now let us find using Logunov's method [ [6]. Logunov A. A. Lectures on theory of relativity and gravitation. Current analysis of the problem. 3-d edition, - Moscow: Issue of Moscow State University, 1985] formulas for coordinates and time transformation  from one reference frame to another one in a case, when reference frames move at a constant speed but in arbitrary direction. First of all let us suppose that events are happening to an object, which is at rest in the primed reference frame B.

            Let   x, y, z denote the components of vector in the inertial reference frame A. Then the expression (6.1) for the interval in this reference frame A will have the form

(6.15)

Let us subject the expression (6.15) to galilean transformation

(6.16)

For this purpose let us write down transformations inverse to transformations (6.16)

(6.17)

Having found differentials from the both parts of equalities (6.17) and having substituted and   d t  into the expression (6.15), we have

or

, (6.18)

where

.

Our purpose is to find such new variables t' and , for which the expression (6.18) can be written in the form

(6.19)

Therefore let us at first introduce into expression (6.18) such designation

(6.20)

or

(6.21)

Substituting into the right-hand part of the expression (6.21) galilean transformations (6.16), we have

or

(6.22)

The negative part of the interval (6.18) can also be written using variables t and

. (6.23)

The first two members from the righ-hand part of the expression (6.23) can be written as a square of some vector , i. e.

(6.24)

Then the expression (6.23) can be written in the form

 (6.25)

But the right-hand part of the expression (6.25) is a square of a vector. That is why the expression (6.25) can be written in the form

(6.26)

Now let us introduce a designation

(6.27)

Then the space-like part of the interval (6.18) will have the form

(6.28)

So, from the expression (6.18) we have obtained the expression (6.19) and simultaneously we have also obtained the formulas (see expressions (6.22) and (6.27)) for transformations of coordinates and time for events happening  to an object , which is at rest in the inertial reference frame B, for the general case of reference frame B motion at a constant speed in an arbitrary direction

(6. 29)

In the same way we can show, that if events happen to an object, which is at rest in the inertial reference frame A, then instead of transformations (6.29) we must use the transformations

(6.30)

So, if the square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 of physical speed of light upon light source speed exists in nature, then instead of direct and inverse lorentz transformations (6.13) and (6.14) from the special theory of relativity we must use in the new space-time theory:

          a) Direct and inverse transformations (6.11) and (6.12) if the events happen to objects, which are at rest in the inertial reference frame A;

          б) Direct and inverse transformations (6.9) and (6.10) if the events happen to objects, which are at rest in the inertial reference frame B.

          Transformations  (6.9) were obtained for the first time (except for designations) by G. A. Kotelnikov  [ [52]. Kotelnikov G. A. About invariance of light speed in the special theory of relativity// Vestnik of Moscow University. Physics, astronomy. Moscow: Moscow State University, 1970, No. 4, pp. 371 - 374.]. But the real physical sense of the transformations obtained by him (see formulas (2) in [52]) G. A. Kotelnicov failed to find. The difference between these transformations and lorent transformations from the special relativity theory Kotelnikov explained by the absence of synchronisation of clocks, which are at rest in inertial reference frames moving relatively each other. On this basis G. A. Kotelnikov made an errenous conclusion that  the transformations obtained by him result in  the same kinematic and dynamic effects, which are known from the special relativity theory.

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