21. THE INFLUENCE OF MOTION OF THE RADIATION SOURCE AND THE RECEIVER ON LIGHT AND SOUND FREQUENCY (DOPPLER EFFECT)
 
21.1 The classical way of determining the Doppler effect
 
   The Doppler effect is well known in classical physics. In 1842 Doppler discovered that the motion of a radiation source influences the frequency of acoustic or light radiation. However, the motion of the radiation source and also the motion of the receiver of radiation influence the frequency which is registered by receiver.
   When a radiation source moves towards the observer the radiation frequency is increased, when it moves away this frequency is decreased. So, the radiation frequency is increased in the direction the source is moving, and decreased in the opposite direction.
   If we mark with the frequency, in relation to the system to which the source is connected, that is, the frequency of the source, and with - the frequency which the receiver receives, then

(21.1)

when the source moves away from the receiver and

(21.2)

when the source approach to the receiver.
   In case of receiver motion we have

(21.3)

when the receiver moves away from the source and

(21.4)

when the receiver approaches the source.
   In previous equations is the speed of the source, is the speed of the receiver and is the velocity of light or sound.
   The given equations can be applied for motion along the straight line "radiation source - receiver". When the motion is under some angle in relation to that straight line, then in expression we take . So, instead of Eqs. (21.1) and (21.2) as well as (21.3) and (21.4) we obtain

(21.5)

in case of source motion and

(21.6)

in case of receiver motion.
   In case of receiver and source motion in the same direction in relation to the environment we have

(21.7)

   When then . However, the change of frequency does not depend on the difference in speed but in general on and in relation to the environment.
   The above is a summary of how classical physics, based on experience and everyday measurements in the sphere of radar and laser technique, treats the Doppler effect, that is, the Doppler frequency shift.
 
21.2 The relativistic way of determining the Doppler effect
 
   The theory of relativity has another approach and other formulas for the calculation of the Doppler effect. Along with a longitudinal, there is also the transversal Doppler effect, which is not accepted by classical physics.
 
21.2.1 Determining the Doppler effect by use of equations of the Lorentz transformation
 
   The theory of relativity comes to the formulas for the Doppler effect by means of the Lorentz transformation equations. For that, this theory starts from the fact that the intensity of the plane light wave which propagates in vacuum in a system is proportional to

(21.8)

and the intensity of the same light wave in system is proportional to

(21.9)

where , , , , and are the cosine of orientation of the wave normal relatively to the corresponding coordinate system.
   According to the theory of relativity, expression (21.8) is invariant with respect to the transformation, so we then have

(21.10)

   Using the first and fourth equation of the Lorentz transformation in the first expression of Eq. (21.10) yields

(21.11)

   Comparison of the coefficients of in Eq. (21.10) and (21.11) we obtain the following relation

(21.12)

   In this way, according to the theory of relativity, we come to the equation (21.12), which is used for the calculation of the Doppler effect. In regard to this equation Einstein says [5]:
   Quotation: "Let us explain the formula for for two different possibilities: when the observer is moving but the infinitely distant source is at rest and opposite, when the observer is at rest but the source is moving.
   a) if the observer is moving at a speed relative to an infinitely distant light source with the frequency , so that the line "light source - observer" forms an angle with the observer's speed relative to the coordinate system which is at rest relative to the light source, then the frequency of light received by the observer will be given by equation

(21.13)

   b) If the light source which radiates light of frequency , in the system which is moving with it, moves so that the line "light source - observer" forms an angle with the speed of the light source relatively to the system that is at rest relatively to the observer, then the frequency , received by the observer is given by equation

(21.14)

Both these relations express the Doppler effect in a general form." End of quotation.
   The Eq. (21.14), which Einstein gave for the case of a radiation source in motion, cannot be correctly derived neither by the relativistic procedure nor by the classical. As such it is neither relativistic nor classical. The relativistic equation for the Doppler effect for the case of a source in motion, which is derived by the relativistic procedure as well as the Eq. (21.13), is useless, since it gives a result contrary to the well known reality. With the aim of proving this claim, let us derive the relativistic equation of the Doppler effect for the case of a radiation source in motion.
   In deriving this equation we shall use the same principle and procedure as in the derivation of Eq. (21.12), that is (21.13), for the case of a moving receiver. In that derivation the radiation source was at rest in the unmoving system , and the receiver was in the moving system . Thus, the receiver was moving together with the system relatively to the system and also to the radiation source. Under those condition the Lorentz transformation was applied to the Eq. (21.10), so that the coordinates of the system were transformed to the system ( and were expressed by means of and ), from which the observation was performed, that is the receiving of radiation.
   In case of motion of the radiation source relatively to the receiver, which is at rest, the source should be connected to the moving system , and the receiver to the unmoving system . So the source will move together with system relatively to the system and to the receiver which is at rest in that system. Since, in this case the observer is in system , then the transformation of coordinates is performed relative to that system, and Eq. (21.10) should take the following form

   Taking that and we finally get

(21.15)

   From this derived relativistic Eq. (21.15) and also from before mentioned Eq. (21.14) it turns out that the frequency of radiation, received by the observer, increases when the source of radiation moves away from observer, and decreases when the source of radiation approaches the observer. However, it is well known that in reality the opposite happens.
   From this example it can already be seen that the relativistic way of determining the Doppler effect is unsustainable. Nevertheless, it is interesting to show other fallacies and weaknesses of the relativistic way of determining the Doppler effect.
   If motion is along a straight line "light source - observer" = 0 and = 1, and then

(21.16)

in case of receiver motion and

(21.17)

in case of source motion.
   Eqs. (21.16) and (21.17) express the longitudinal Doppler effect.
   If the motion is normal to the straight line "light source - observer", then = 90° and = 0, so

(21.18)

in case of receiver motion and

(21.19)

in case of source motion.
   Eqs. (21.18) and (21.19) express, the so called, transversal Doppler effect.
   So, by using the Lorentz transformation which is derived for a spherical light wave, equations for the Doppler shift for a plane light wave motion are obtained. As mentioned earlier, with the Lorentz transformation the requirement for identical satisfaction of the invariability of equations for a plane light wave propagation is not achieved. Einstein himself required the invariability as can be seen in the quoted text: "The simple derivation of the Lorentz transformation", given in chapter 10.
   Why did Einstein chose the plane wave and not the spherical wave in deriving relativistic equations of The Doppler effect? Probably those equations cannot be derived by using the equation of a spherical wave. The transversal Doppler effect is a relativistic product. The assertion about its existence is unfounded, which can be seen from the following consideration.
   Let us take the case in Fig. 21.1 where is a radiation source of spherical light waves which is at rest and is a receiver which moves along the straight line . When it moves from point to point the receiver gets closer to the source () all the way to the point , so the frequency which the receiver receives is higher than the source frequency. In further motion, from the point towards the point , the receiver moves away from the source, so the frequency which it receives is lower than the source frequency. In transition from a higher to a lower frequency than that of the source radiation has to pass through the same frequency of the source radiation. In other words, on the way from plus to minus, zero must be crossed. This transition from the higher to lower frequency appears at point , which means that there is no the frequency shift at point . In other words, there is no a transversal Doppler effect, given by Eq. (21.18) and also by Eq. (21.19), because the same is valid for the light source motion, as well.

Fig. 21.1

   The relativistic equations for the Doppler effect are derived for the case of propagation of plane waves, which, necessarily means that they cannot be used for the propagation of spherical waves. However, the Lorentz transformation of coordinates was applied to plane waves, which does not satisfy the requirement for invariability of the equation for propagation of a plane wave. Judging by this, relativistic equations for the Doppler effect cannot be applied to the propagation of plane waves either.
   The relativistic equations for the calculation of the longitudinal Doppler effect, which is the only one that exists, can be used only when the speed of motion is small relative to the speed of light, and then, in essence, they give the same result as classical equations, whose form is simpler and easier to apply. For higher speeds, which approach the speed of light, and for which they are designed, relativistic equations are useless since the mistakes in determining the Doppler effect are unacceptably large. The proof of this is simple and can easily be derived in the following way.
   Fig. 21.2 shows one possible arrangement of devices for the performance of this proof: at point we have a radio transmitter which can emit radio pulses with a pulse repetition rate of 100 MHz; at point , at a distance of 0.27 km is the first radio receiver; at point , in the same direction and at a distance of 0.3 km is the second radio receiver and at point there is a starting device, which is connected with the said radio devices with cables of the same length and electric characteristics, which enable simultaneous switching on and off of all three radio devices.

Fig. 21.2

   A spatial distribution of radio pulses after = 10^-6 s from the time of the emission beginning is as in Fig. 21.3

Fig. 21.3

   The radio pulse, emitted from point , will travel the distance = 0.3 km and reach point in time = 10^-6 s. If, with the help of the starting device, all three radio devices are switched on at the same time for the duration of = 10^-6 s, then in that time the radio transmitter, from point , will emit 100 radio pulses, and the first radio pulse will reach the radio receiver at point . Ten pulses will pass and be registered by the radio receiver at point . The other 90 pulses will be on the way from point to the point .
   Let us assume that the first radio receiver from point was next to the radio transmitter at point at the moment when all the radio devices were switched on, and that from that moment it was moved at the speed of towards point (like the coordinate system , whose speed of motion was ). After the time of 10^-6 s from the moment of switching on it will arrive at the point . On that path from point to point ten radio pulses will pass by it, in the direction of point , at the speed . The first receiver will register these ten pulses in motion. The other 90 pulses will be in motion from the transmitter towards the first receiver at point , which is given in Fig. 21.3. Had the first receiver stayed at point , it would have registered all 100 pulses. Since it moved away from the radiation source at the speed it registered only 10 pulses, which is in accordance with the classical equation for the Doppler effect

   According to the relativistic Eq. (21.13), that frequency, because of the Doppler effect, should be

from which follows that the first radio receiver, on the path from the point to the point , should have registered 23 instead of 10 impulses. It means that between point and point , after 10^-6 s from the start, 23 impulses instead of 10 impulses would be arranged, which it certainly did not, and cannot be.
   From the given example we see that when a receiver moves away from the source of radiation at the speed of , the mistake in determining frequency according to the relativistic formula is as much as 130%. With the increase of speed, the mistake increases as well. Such major mistakes are certainly unacceptable, as is the relativistic way of determining the frequency of the Doppler shift.
   The relativistic formulas for the energy of electromagnetic waves are also unacceptable, since their form is based on the relativistic formulas for frequency. Einstein used these equations in, for example, deriving the Eq. (23.48) for kinetic energy.
   Earlier on it was stated that all coordinate transformations have the same value, if they satisfy the requirement for the invariability of the equation of the light wave propagation. Therefore let us see what will happen if we use equations of the transformation No. 2, No. 4 and No. 5 instead of the equations of the Lorentz transformation. The application of equations of transformations No. 4 and No. 5 is especially interesting, because they have been derived for the case of the plane wave, which is used in the theory of relativity to derive relativistic equations of the Doppler effect. By equations of these two transformations, as we know, the identical satisfaction of the invariability of the equation for plane wave propagation is achieved. Judging by this it should be that, at applying equations of these transformations, obtained results in the most real way would show the true value and steadiness of the relativistic way of determining that effect.
 
21.2.2 Determining the Doppler effect by use of equations of transformation No. 2
 
   Substitution of expressions for and from Eq. (12.22) into (21.10) and comparing the coefficient of from the expression so obtained and the corresponding expression in Eq. (21.10) in the same way as in previous case, we obtain

(21.20)

in case of receiver motion and

(21.21)

in case of light source motion.
   Eqs. (21.20) and (21.21), which are derived by use of the equations of transformation No. 2, express the Doppler effect in general form. As can be seen it greatly differs from Eqs. (21.13) and (21.14) from the previous case, that is from the adequate equations derived by using of equations of the Lorentz transformation.
   For motion along the line "radiation source - receiver" it is = 0 and = 1 so that

(21.22)

for receiver motion and

(21.23)

for source motion.
   Eqs. (21.22) and (21.23) express the longitudinal Doppler effect.
   When = 90°, that is, when motion is normal to direction of "radiation source - receiver", the so-called transversal Doppler effect appears. Then = 0 and for receiver motion Eq. (21.20) obtains the following form

(21.24)

and for source motion Eq. (21.21) obtains the form

(21.25)

   The transversal Doppler effect is expressed by Eqs. (21.24) and (21.25).
   Thus, using the equations of transformation No. 2 for derivation equations of The Doppler effect according to theory of relativity, both the longitudinal and transversal Doppler effect appear. However, they differ both in the form of the equations and in their value from the previous case, that is, when the equations of the Lorentz transformation are used.
 
21.2.3 Determining the Doppler effect by use of equations of transformation No. 4
 
   Substitution of equations for and from Eq. (12.24) into Eq. (21.10) yields

(21.26)

   Comparing the coefficient of from Eqs. (21.26) and (21.10) we obtain, for receiver motion

that is

(21.27)

and for source motion

(21.28)

   If the receiver or source motion is along the straight line "radiation source - receiver" then = 0 and = 1, so from Eq. (21.27) we obtain that and from Eq. (21.28) , which means that there is no longitudinal Doppler effect in both cases, for the motion of the receiver and that of the source, which runs counter to the well known reality.
   However, when the motion is normal to the direction of "radiation source - receiver", that is at = 90° and = 0, then in the case of receiver motion

(21.29)

and in case of source motion

(21.30)

   Eqs. (21.29) and (21.30) express the transversal Doppler effect.
   This means that when we apply transformation No. 4, in the relativistic procedure for determining the Doppler effect, we find that there is no longitudinal Doppler effect but only a transversal one and this, as we know runs contra to what was established long ago by experiment and is confirmed in everyday practice.
 
21.2.4 Determining the Doppler effect by use of the equations of the transformation No. 5
 
   By substitution of equations for and from Eq. (12.25) into Eq. (21.10) and by comparing the coefficient of from the equation thus obtained and the corresponding expression in Eq. (21.10) we find that, in the case of receiver motion

(21.31)

and in the case of source motion

(21.32)