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.

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.

=λobs=λs1+uc1−uc.=λobs=λs1+uc1−uc. size 12{λ rSub { size 8{"obs"} } ital "=λ" rSub { size 8{s} } sqrt { { {1+ { {u} over {c} } } over {1 - { {u} over {c} } } } } } {}

(6) In the Doppler equation, λobsλobs size 12{λ rSub { size 8{"obs"} } } {} is the observed wavelength, λsλs size 12{λ rSub { size 8{s} } } {} is the source wavelength, and uu size 12{u} {} is the relative velocity of the source to the observer. The velocity uu size 12{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

fobs=fs1−uc1+uc.fobs=fs1−uc1+uc. size 12{f rSub { size 8{"obs"} } ital "=f" rSub { size 8{s} } sqrt { { {1 - { {u} over {c} } } over {1+ { {u} over {c} } } } } } {}

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

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.

Suppose a galaxy is moving away from the Earth at a speed
0.825
c
0.825
c
. It emits radio waves with a wavelength of 0.525m0.525m size 12{0 "." "525"" 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*

- Identify the knowns.
u=
0
.
825
c
u=
0
.
825
c
size 12{ ital "u="0 "." "825"c} {}
; λs=0.525mλs=0.525m size 12{λ rSub { size 8{s} } =0 "." "525"`m} {}
- Identify the unknown. λobsλobs size 12{λ rSub { size 8{"obs"} } } {}
- Choose the appropriate equation. λobs=λs1+uc1−ucλobs=λs1+uc1−uc size 12{λ rSub { size 8{"obs"} } ital "=λ" rSub { size 8{s} } sqrt { { {1+ { {u} over {c} } } over {1 - { {u} over {c} } } } } } {}
- Plug the knowns into the equation.
λobs
=
λs1+uc1−uc
=
(0.525 m)1+0.825 cc1−0.825 cc
=
1.70 m.
λobs
=
λs1+uc1−uc
=
(0.525 m)1+0.825 cc1−0.825 cc
=
1.70 m.

(8)

*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.

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