Any fast enough light source has a red Doppler shift

Perhaps for many it will be a surprise to learn that as the speed of an approaching source increases, its radiation first “turns blue” and then “turns red”. This is illustrated in the figure below. The geometrical location of the points of the hodograph of the velocity of the Source with a constant ratio of the wavelengths of the Receiver and the Source equal to n is an ellipsoid as in the figure below.

The velocity vector β , directed to the right as a whole, as it grows, first crosses ellipsoids with shorter wavelengths (n <1) of light, and then begins to cross ellipsoids with increasingly long (n> 1) waves.

The author would appreciate comments.


If the end of the velocity vector (hodograph) for an approaching source abuts at point B for n = 1,618 , as in the figure, then, considering the source simply receding, we assume that its end abuts at point B ' . In this case, trying to determine the speed of the source by the magnitude of its “red” shift, we will determine its speed of “removal” is significantly less than it actually has a speed of approach. For a source with a velocity at point C, we can even assume that it is motionless, i.e. as it has a speed at point C ' . Let’s figure out how it turns out, and you won’t need to dive into the wilds of the service station. And, by the way, all the derived formulas can be used in real practice.

Let at some moment the source emits an electromagnetic wave 1 ' . And after a period of time T ₁ - wave 2 . By this time, the wavefront 1 ' will occupy position 1 . But during the same time, the Source will move in the direction of the Receiver by a distance V _1X · T ₁ , where V _1X = V ₁ · Cos (ψ) . Thus, the front of wave 2 will be separated from the front of wave 1 by a distance L ₁ .

Let the receiver at some point receive wave 1 . Wave 2will catch up with him after a period of time T ₂ , but during this time the receiver will move in the direction of wave propagation to a distance V _2X · T ₂ , where V _2X = V ₂ · Cos (φ) .

Since the wave is plane and its front is perpendicular to the beam, only the inclination of the velocity vectors to the light ray plays a role, and their circular relative orientation is indifferent.

The above relations can be written as a system of equations (1).



Its solutions will be equalities (2). Note that L ₁ is the wavelength of light ( λ ₁) emitted by the source in the direction of the receiver in the coordinate system of the external observer.

The time intervals T ₁ and T ₂ in the observer ISO will correspond to the intervals T ₁₀ and T ₂₀ in units of proper time in the source and receiver ISO in accordance with relations (3). This just corresponds to the Lorentz transformations in SRT. In the proper units of the moving ISO, relations (4) are valid. At the same time, we use that in our own ISO the speed of light is c . Substituting (3) and (4) into formulas (2), we obtain relation (5) in which the wavelengths λ ₂₀ and λ ₁₀are indicated already in the own ISO of the Receiver and Source.

If we assume that the ISO of the Receiver is conditionally fixed, then expression (5) can be written in the form (6). In this form, the formula for the Doppler effect completely coincides with its form in the SRT ( L.D. Landau and E.M. Lifshits Field Theory, §48) But there it was deduced by recalculating the 4-vector of the components of the electromagnetic field to the coordinates of the ISO moving in Minkowski space. And we deduced it according to Euclidean geometry in Newtonian space simply by assuming that phenomena such as time dilation and Lorentz contraction are, as it were, actually realized in moving bodies. This "technique" allows us to consider relativistic phenomena as if occurring in a trivial 3-dimensional space, but, as they say, "truth is invariant with respect to the way it is received."

Let us replace the variables according to expressions (7). Then expression (6) is written as expression (8). Omitting the intermediate analytical calculations, from expression (8) we can go to expression (9).

This is the equation of a family of ellipsoids compressed along the X axishaving a common point in the coordinates {1,0} , and Y ² _max = n ² / (n ² +1) at X = 1 / (n ² +1) .

A series of these ellipsoids with n = λ ₂₀ / λ ₁₀ multiple of 1.618 (golden ratio) is shown in the first figure.

Unfortunately, in the original version of the article, the author made the wrong conclusion that the reason could be “that as the speed of the source approaches light, the increase in speed is no longer expected. And due to the incident of the source on the waves emitted by it, their length in the propagation medium is almost not reduced. ” This conclusion of the author is incorrect, which he was rightly pointed out in the first comments, for which the author sincerely thanks. But the error did not affect the derivation of the formulas and the result.

_{Bibliography:}
^{1.L.D. Landau, E.M. Lifshits Field Theory, 4th ed., 1962}