In the relativistic case, there is no group of transformations for which this is the velocity addition law, since it is impossible to independently rescale time and distance measurements. When c is not the speed of light, the velocity u is not the velocity of anything, just a false inferred velocity from the point of view of the moving ship. This is the relativistic analog of the Doppler velocity addition formula. Just as in the non-relativistic case, this is the velocity at which a source would have to be moving in order to make the Doppler shift factor for a moving receiver equal to the Doppler shift factor for the velocity u. It is also possible to determine, in the relativistic case, the actual velocity of a source, when a moving ship falsely determines it from a Doppler shift without taking its own motion into account. So that the relativistic Doppler shift of light is determined by the relativistic difference of the two velocities. Multiplying the two factors for the emitter and receiver gives the relativistic Doppler shift: From this, the received frequency can be read off: In this time, the receiver has moved (in the fluid frame) an amountĪnd the proper time between the two crest crossing isĪnd this is the time between crest-crossings as measured by the receiver. The fluid frame time between crest-crossings does not require changing frames and is the same as in the nonrelativistic case: If the receiver is moving with a velocity u through the fluid perpendicular to the wave fronts, the frequency received is determined by the proper time between the events where the receiver crosses crests. This fact was not understood until the development of quantum theory in the 1920’s, but it was known considerably earlier.
In the rest frame of the medium, the frequency emitted by a relativistic source moving with velocity v is decreased by the time dilation of the source: THE NONRELATIVISTIC DOPPLER SHIFT: It is a well-known fact that atoms emit and absorb radiation only at certain xed wavelengths (or equivalently, at certain xed frequencies). But when the speed of the wave c ≠ 1, meaning that the phase velocity of the wave is different from that of light, the relativistic Doppler shift formula does not depend only on the relative velocities of the emitter and receiver, but on their velocities with respect to the medium. In this case the frequency of the received waves can only depend on the relativistic sum of the velocities of the emitter and the receiver. In the theory of the relativistic Doppler shift, the case where the speed of the wave is equal to the speed of light is special, because then there is no preferred rest-frame.
0 kommentar(er)
