SELF

58

S.B. Karavashkin and O.N. Karavashkina

4. The pattern of light propagation in the moving inertial reference frame from the point of relativistic mechanics

Above we showed: when ignored the difference in description of material points and field processes, this was the cause because of which relativists formulated the incorrect postulates of their conception. Basically, this lifts the necessity to continue the analysis of relativistic conception. None the less, in this section we will specially consider the distinctions of field processes description in passing between the inertial frames from the point of Lorentz transforms, in order to lift the speculation as if incorrect premises are able to bring us the physically associative results.

As the initial equations of equiphase surfaces excited by the source resting in the resting frame, we have all grounds to take the system (3.1) which we used above to find classically the direct coordinate transformation. Instead (3.2) used before, we have to write the trivial Lorentz transforms which will determine the passing between the inertial frames in the relativistic conception:

After manipulation, yield for the moving frame

The dynamic diagram plotted with (4.2) is shown in Fig. 4.1.

Fig. 4.1. Dynamic diagram of propagation of equiphase surfaces of light in the moving frame from the point of relativistic conception; the frame moves with the speed v = 0,8 c  with the positive values of the axis x of the resting frame

The diagram shows that with the Lorentz transforms, just as with the direct classical transform, the equiphase surfaces do not transform, but the plane of events inclines with respect to the moving coordinate system. To understand the consequences of such inclination, consider a 2-D pattern, how the equiphase surfaces propagate in the plane (x', ct' )  passing the world line of the source of light of Fig. 4.1. This projection is shown in Fig. 4.2.

Fig. 4.2. Dynamic diagram of propagation of equiphase surfaces of light in the plane (x', ct' ) of the frame from the point of relativistic conception; the frame moves with the speed v = 0,8 c   with the positive values of the axis x of the resting frame; the red markers show the projections of equiphase surfaces onto the plane; the world line of the source is denoted in green; the blue line is the world line of the observer resting with respect to the dotted frame

To make the diagram visual, we added the world line of the observer (blue line) and drew the projection of the plane of events of this observer (the green-blue line drawn in parallel to the axis x' ), to draw the Reader’s attention that the solutions yielded with the Lorentz transforms are inconsistent with the statement of problem on whose basis Einstein yielded these formulas. Actually, according to the statement of problem, the physical time was involved in both frames, which means synchronisation of all identical clocks in each frame: “Let through the resting clocks in the resting frame and with the help of light signals, in the way shown in section 1 we determine the time t of the resting frame for all points of this last in which the clocks are located. Let then the time of the moving frame to be determined in the same way for all points of this frame in which the clocks resting with respect to this last are located, as we did it in the section 1 – through the light signals among the points where these clocks are located” [4. p. 13, italics is ours].

From this, the clocks of all observers in the moving frame have to read the synchronised time. But when we apply this stipulation to the dynamic diagram in Fig. 4.2, it appears that all observers located on one plane of events with the selected observer will not see at all the source and equiphase surfaces, as all material objects, light waves, sources and observers will be located on the inclined plane (in this case on its projection (x', ct' ) ). Therefore, nothing can be located out of this plane at any chosen moment of time, i.e. namely in the regions through which the plane of events goes (the green-blue line) of the selected observer in Fig. 4.2.

True, we can premise that the diagram in Fig. 4.2 describes not the signal propagation in the physical time but is only some regularity characterising the signal propagation in the moving frame yielded in the transforms. If we look the whole diagram, the observers at each point, at some moment of time (out of the plane of events) will, nevertheless, see these equiphase surfaces. But then the Lorentz transforms will give us not final but intermediate solution which we yet have to agree additionally with the statement of problem.

To put the solution into agreement with the statement of problem, we have to recalculate the inclined plane of events produced in the Lorentz transform to the physical time introduced by Einstein after the statement of problem in this frame. This pass was necessary also because we do all these measurements in our moving frame which is at rest for us and generally moves for the observers in other frames. This means, as in our frame we use the physical time, the true properties of the Lorentz transforms we can reveal only passing to the dotted (in this case) frame.

To do so, let us do this transform from the solution (4.2) to the solution describing the propagation of equiphase surfaces in the units of physical time in the moving frame. For it, we have to transform x'_a(t, t'₀, _a) and y'_a(t, t'₀, _a) into x'_a(t, t'₀, '_a) and y'_a(t, t'₀, '_a). We will do it in two stages, transforming at the first stage the initial regularities into the regularities x'_a(t, t'₀, _a) and y'_a(t, t'₀, _a). With it, we will not transform the parameter _a  at the first stage, given it from the very beginning is ancillary and automatically accounts the change of propagation direction of the element of equiphase surface in passing to the moving frame in the Lorentz transforms.

To pass in these regularities from the time parameter t₀   characterising the moment when the equiphase surface was radiated from the point of resting frame to its equivalent t'₀  in the moving frame, pay attention that the stipulation t = t₀  has to correspond to the stipulation  t' = t'₀ . Hence, from the last equation of the system (4.2) we yield

or

The regularity (4.4) shows that the equal time intervals of the light radiation by the source in the resting frame will correspond to the equal time intervals in the moving frame. As we will show below, in the relativistic conception we may not conclude from it concerning the period of signals detected by the observer, as in this case we have to use the complete expression to transform the times in (4.2). And (4.4) is true only for the relation between the moments of radiation of the light equiphase surfaces. But these values keep proportion, and this makes the problem much easier.

Then, we may substitute (4.4) to the same equation describing the time transform in (4.2) and yield the relation between the physical time of resting and moving frames:

Basing on (4.4) and (4.5), we can find the relation between the time intervals of equiphase surfaces propagation from the moment of their radiation to the current time in the related frames:

We see from (4.6) that on one hand, for each given direction _a  the related time intervals are strongly proportional, but on the other hand, the fact that it remains to depend on _a  evidences the anisotropy of time transform, and just for the case of light beams propagation which arises in the Lorentz transforms and remains when recalculating to the physical time introduced in both frames. It follows from the anisotropy seen in (4.6) that the strong proportionality between the moments of radiation of equiphase surfaces in the relativistic interpretation does not mean that they are isotropic and that the functional interrelation between the intervals remains in general case. Actually, substituting to (4.6)

yield

At the same time, if we take two sequential moments of time t₀₁ , t₀₂ and t'₀₁ , t'₀₂ at which the equiphase surfaces were radiated, with (4.4) we will yield

We see from (4.8) and (4.9) that the right parts of these equations are equal only if cos _a = 0  in direction of wave propagation being transverse to the motion of source. In this direction the period T '  is more than T ( cos _a = -1), as it is determined by (4.9), and in the opposite direction it is less than (cos _a = 1). Thus, we see that in the moving frame, the equiphase surfaces are radiated with one periodicity and propagate in space with another, and this second periodicity is anisotropic. With it, if in classical physics the variation of wave period was not caused by the change of time flowing in this frame, in the relativistic conception we see a direct relation between the change of period and change of time as such; this relation is determined by (4.6), we have revealed it in our previous paper on the basic postulates of SRT: “in general case, too, the transformation of time interval, in accordance with Lorentz transformation, depends on the pattern of motion of material point, and the transformation coefficient can be more and less than 1, dependently on direction of motion of the material point” [11, p. 38]. So we may not attribute this relativistic effect to the Doppler effect – it reveals if the time flow in the frame remains and the frequency of moving source changes. In the relativistic conception both the frequency and the time interval change; we will not stop at these points until we finish the passing to the dotted frame and analysis of features of relativistic description, how the light propagates in inertial frames.

To finish the first stage of this passing to the dotted coordinates, we, substituting (4.4), (4.6) and (4.6) to the first two equations of the modelling system (4.2), yield the sought regularities of coordinates of equiphase surface against the time parameters of the moving frame:

Thus, by way of additional transforms, we yielded some system of equations (4.10) which without mediation of space-time parameters of the frame S describes the equiphase surfaces in the frame S' .

We can notice, these equations satisfy the light speed constancy. It is easy to check it, finding the derivatives of both equations (4.10) with respect to t' :

We also have to notice that expressions (4.10) satisfy the equation

that is equivalent to the Einsteinian 4-D interval. We can easily check it, too, given in accordance with (4.11)

and we can represent the expression for x'_a   in (4.10) as