Relativistic Doppler Effect versus Time Dilation: Critical Inconsistency
Radwan M. Kassir
© June 2017
Abstract The period of a light wave emitted by an oscillating electron in a “stationary” reference frame was determined in a relatively moving frame using the Lorentz transformation applied on different event intervals; one being between two events on the wave propagation path, and the other between two co- local events on the oscillating source path. A critical inconsistency in the Special Relativity was revealed.
Keywords: Special Relativity, Relativistic Doppler shift, Time Dilation
Introduction the result will be the relativistic time dilation in terms of the source periods. Since the wave and 1 In his 1905 paper, Einstein derived the its source have the same period, inconsistency Lorentz transformation (LT) equations for the in the foresaid results is obvious. space and time coordinates on the basis of the relativity principle and the constancy of the Relativistic Doppler Shift speed of light. These equations resulted in the time dilation and length contraction. In In §3 of the above cited paper, 1 the time addition, transformation equations for the transformation equation converting event time electric and magnetic forces were then deduced between two inertial frames in relative motion from the Maxwell-Hertz and the LT equations. of velocity v, having the coordinate systems The obtained electrodynamics transformations Kx(, yzt , ,) and k(,ξ η , ζ , τ ) associated with applied on the wave equations for light led to what’s considered as “stationary” and the general relativistic Doppler shift formula. “moving” frame, respectively, is obtained as It follows that the coherence of the Doppler shift formula is vital for the Special Relativity vx reliability. In this paper, it is shown that when τ= γ t − , (1) 2 the LT time equation was applied on the c propagation of a light wave, the result will be where the relativistic Doppler formula in terms of the wave periods. Whereas, if the same LT transformation was applied on the wave source, Radwan M. Kassir © June 2017 2
−1 2 2 v γ =1 − v c , and c = speed of light T′ =γ T 1 + . (5) ( ) c in empty space. In §7 of the same paper, a light (electrodynamics waves) source, with given Relativistic Doppler Effect wave characteristics, is considered in the Contradiction with Time Dilation stationary system at a sufficiently far distance A critical inconsistency in the Special from the origin. The characteristics of these Relativity is the fact that the relation waves were to be determined when observed 1 from the moving frame. We quote the following T′ = T , (6) passage: v2 1 − From the equation for ω′ it follows c2 that if an observer is moving with velocity v relatively to an infinitely where T and T ′ are the light wave periods in distant source of light of frequency ν, the “stationary” source frame and the in such a way that the connecting line “moving” observer’s frame, respectively, is a “source-observer” makes the angle φ Special Relativity result. with the velocity of the observer In fact, consider an electron oscillating with referred to a system of co-ordinates a period T across the origin of the “stationary” which is at rest relatively to the source frame coordinate system along the Y - axis. of light, the frequency ν′ of the light Let λ be the wavelength of the emitted light perceived by the observer is given by wave, with respect to the “stationary” frame, the equation while the wave period must be equal to the oscillating electron period T.The observer rest 1− cosφ . v / c frame is moving at speed v relative to the ν′ = ν . (2) “stationary” frame. We are to determine the 1 − v2 c 2 wave period T ′ relative to the observer. Consider two consecutive events E1 and This is Doppler’s principle for any E2 occurring at two consecutive wave peaks. velocities whatever. When φ = 0 the These two events are separated by the spatial equation assumes the perspicuous form interval x =λ = cT , and the temporal interval t = T . From the perspective of the moving observer, the corresponding spatial ν′ 1 − v c interval is x′ =λ ′ = cT ′ , and the temporal = . (3) ν 1 + v c interval is t′ = T ′ . Applying the Lorentz transformation
Equation (3) can be written in the form 1 v x =t′ − t , 2 2 c v v 1 − ν′ = γν 1 − , (4) c2 c or in terms of the wave periods between these two events, from the observer’s perspective (the source is “moving” with the velocity −v), we get 2
3 Relativistic Doppler Effect versus Time Dilation: A Critical Inconsistency
1 v Conclusion T′ = T 1 + , (7) v2 c A critical contradiction in the Special 1 − c2 Relativity is revealed when the Lorentz transformation time equation is applied in accordance with SR Eq. (5) between two events on a light wave propagation Now, consider another two consecutive path, from one part, and between two events on events, E 3 and E 4, occurring when the the light wave emitting source (oscillating electron) path, from the other part. electron is at its maximum Y - coordinate for two consecutive times. Therefore, the temporal interval separating these events is the period of References the oscillating electron, which is the same as the wave period T. The spatial interval separating 1 A. Einstein, "Zur elektrodynamik these two events is zero. bewegter Körper," Annalen der Physik 322 Applying the Lorentz transformation (10), 891–921 (1905).
1 v x =t′ − t , 2 v2 c 1 − c2 between these two events, we get
1 T′ =() T − 0 ; v2 1 − c2
1 T′ = T . (8) v2 1 − c2
Hence, the above equation is ascertained to be for the wave period as observed from the moving frame, relative to its proper period. Yet, it contradicts the previous equation obtained as
1 v T′ T =1 + ; v2 c 1 − c2 both obtained under the Special Relativity predictions.
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