# 20.4 Relativistic addition of velocities  (Page 3/7)

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## 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.

## Relativistic doppler effects

The observed wavelength of electromagnetic 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 towards the observer.

${\text{=λ}}_{\text{obs}}{\text{=λ}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}.$

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

${f}_{\text{obs}}{\text{=f}}_{s}\sqrt{\frac{1-\frac{u}{c}}{1+\frac{u}{c}}}.$

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

## Career connection: astronomer

If you are interested in a career that requires a knowledge of special relativity, there’s probably no better connection than astronomy. Astronomers must take into account relativistic effects when they calculate distances, times, and speeds of black holes, galaxies, quasars, and all other astronomical objects. To have a career in astronomy, you need at least an undergraduate degree in either physics or astronomy, but a Master’s or doctoral degree is often required. You also need a good background in high-level mathematics.

## Calculating a doppler shift: radio waves from a receding galaxy

Suppose a galaxy is moving away from the Earth at a speed $\text{0.825}c$ . It emits radio waves with a wavelength of $0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}\text{m}$ . What wavelength would we detect on the Earth?

Strategy

Because the galaxy is moving at a relativistic speed, we must determine the Doppler shift of the radio waves using the relativistic Doppler shift instead of the classical Doppler shift.

Solution

1. Identify the knowns. $\text{u=}0\text{.}\text{825}c$ ; ${\lambda }_{s}=0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}m$
2. Identify the unknown. ${\lambda }_{\text{obs}}$
3. Choose the appropriate equation. ${\lambda }_{\text{obs}}{\text{=λ}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$
4. Plug the knowns into the equation.
$\begin{array}{lll}{\lambda }_{\text{obs}}& =& {\text{λ}}_{s}\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}\\ & =& \left(0.525 m\right)\sqrt{\frac{1+\frac{0\text{.}\text{825}\text{c}}{c}}{1-\frac{0\text{.}\text{825}\text{c}}{c}}}\\ & =& \text{1.70 m.}\end{array}$

Discussion

Because the galaxy is moving away from the Earth, we expect the wavelengths of radiation it emits to be redshifted. The wavelength we calculated is 1.70 m, which is redshifted from the original wavelength of 0.525 m.

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.

Suppose a space probe moves away from the Earth at a speed $0\text{.}\text{350}c$ . It sends a radio wave message back to the Earth at a frequency of 1.50 GHz. At what frequency is the message received on the Earth?

${f}_{\text{obs}}{\text{=f}}_{s}\sqrt{\frac{1-\frac{u}{c}}{1+\frac{u}{c}}}=\left(1\text{.}\text{50 GHz}\right)\sqrt{\frac{1-\frac{0\text{.}\text{350}\text{c}}{c}}{1+\frac{0\text{.}\text{350}\text{c}}{c}}}=1\text{.}\text{04 GHz}$

## 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.

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