SELF

2

Supplement to the study of classical transverse Doppler effect in respond to received criticism

Dear Tim,

To begin with, we am touched by your Russian language, but in order Western colleagues to understand your claim and our respond, it would be preferable to write in English. So, if you are not against, we will first translate your words:

************

<< Absolutely incorrect. See, e.g., the paper below,

http://www.mathpages.com/rr/s2-04/2-04.htm   >>

************

Besides, for convenient analysis, we have to numerate formulas in your work. So when below we will refer to your formulas, we will operate with this through numeration, which you can see in the copy of your web page here in our e-journal:

Doppler Shift for Sound and Light.files

Now let us analyse your post. In it you are unambiguously stating that our solution is fully incorrect and that you have correct solution in your paper. Don't be in hurry with so wholesale statement, as you gave no one point to prove mistake out. In absence of your specific claims to our work, let us consider your solution. This will be a lesson for you, how you had to analyse. True, for it you would have to be aware not only of the phenomenon and calculation methods, you would have to understand the heart of phenomenon and the crux of methods. In this last you, seemingly, show your feebleness.

Let us start from the statement of problem. You are writing in your paper the following:

<< This sometimes gives the impression that relativity requires us to apply a different set of kinematical rules to the propagation of sound than to the propagation of light, but of course that is not the case. The kinematics of relativity apply uniformly to the propagation of all kinds of signals, provided we give the exact formulae. >>

Well, this is inconsistent with the SR postulates. In accordance with these postulates, effect of frequency transformation received by moving observer is in straight relation with transformation of space-time metric. While, judging by your statement, similar transformations occur in acoustic either hydrodynamic field. Cross yourself and better learn the phenomenology of Relativity. In this view, if you try to solve Doppler problem in acoustic field using Minkowski light cones, you would have to premise 4 D Einsteinian metric true in this acoustic field, which is simply absurd. But this is just the method you are using to solve the problem, and you are making roughest mistakes showing your full unawareness in the subject.

Before starting analysing your solution, we have to limit your computations to light (EM) waves, where 4 D metric is admissible in the view of Relativity. Well, let us see the graph in space-time coordinates from your paper, we conveniently copy it and show as Fig. 1.

Fig. 1

First of all we see that at the instant of pulse emission, your world line of receiver coincides (anyway, in axis x) with the world line of source. Do they move in your case from one point? Either in crossing trajectories? If from one point, your graph describes the LONGITUDINAL Doppler effect for the source and observer moving oppositely, so your further comparison of transverse effects becomes incorrect. If the source and observer move along crossing trajectories, your formulas (2)- (5) are untrue, since at different values of initial position in the axis y, the relationship will differ from yours. You can easily check it if you build your world lines spatially (plane problems allow do to so), having given the initial location of bodies in the plane xy and different inclinations of the world lines IN SPACE, corresponding to the observer's motion, as it is shown, for example, in Fig. 2.

Fig. 2

To the point, the world line of the source in this space-time volume will be directed strongly along the time axis, which will correspond to the statement of problem as Einstein and other authors solved it, but we will return to this aspect below. We can say, this scheme also is incomplete for solving the problem even in relativistic limits. The full scheme, noting the motion of source and receiver, as it is considered in your work, you can see in Fig. 3.

Fig. 3

In this figure we show the projection of spatial construction onto the plane yOt. In this projection, motion of source and receiver will be reflected by vertical lines, as in your problem they both move along the axis x. But even not this is important but that the instants at which the observer B detects the signal will not coincide with the projection of light cone onto this plane, which relativists so much like to draw. And they will not coincide with these projections in the plane xOt. This is quite obvious, as the trajectory of receiver will cross the light cone not on the plane yOt but in space. To determine the instants, when receiver detected the signal, you would have to find just these points of body B trajectory crossing with cones. And not by one cone, as you made in your work, but by two, as it is shown in Fig. 3 of our post. Additionally noting that in the source motion these cones are also shifted both along x and along the time axis together with the source. You would have to determine the time intervals not over AB but over the projections of sections A_0tA_1t B_0tB_1t onto the axis t. This is already other mathematics and solutions will be basically other. After this, we may stop analysing your derivation, but we continue, you to see the full amount of your mistakes.