20.4 Relativistic addition of velocities  (Page 3/7)

Doppler shift

Although the speed of light does not change with relative velocity, the frequencies and wavelengths of light do. First discussed for sound waves, a Doppler shift occurs in any wave when there is relative motion between source and observer.

In the Doppler equation, ${\lambda }_{\text{obs}}$ is the observed wavelength, ${\lambda }_s$ is the source wavelength, and $u$ is the relative velocity of the source to the observer. The velocity $u$ is positive for motion away from an observer and negative for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written

Notice that the – and + signs are different than in the wavelength equation.

The relativistic Doppler shift is easy to observe. This equation has everyday applications ranging from Doppler-shifted radar velocity measurements of transportation to Doppler-radar storm monitoring. In astronomical observations, the relativistic Doppler shift provides velocity information such as the motion and distance of stars.

Section summary

• With classical velocity addition, velocities add like regular numbers in one-dimensional motion: $\text{u=v+u}\prime$ , where $v$ is the velocity between two observers, $u$ is the velocity of an object relative to one observer, and $u\prime$ is the velocity relative to the other observer.
• Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:
  $\text{u=}\frac{\text{v+u}\prime }{1+\frac{v\text{u}\prime }{c^2}}$
• An observer of electromagnetic radiation sees relativistic Doppler effects    if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation
  ${\lambda }_{\text{obs}}{\text{=λ}}_s\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$
  ${\lambda }_{\text{obs}}$ is the observed wavelength, ${\lambda }_s$ is the source wavelength, and $u$ is the relative velocity of the source to the observer.