Chapter 42: Problem 13

Question: In the spectrum of quaser 3C9, some of the familiar hydrogen lines appear but they are shifted so far forward toward the red that their wavelengths are observed to be 3 times as large as that observed in the light from hydrogen atoms at rest in the laboratory.

1.

Show that the Doppler equation, which assumes that light behaves like sound gives a velocity of recession greater than the speed of light.

2.

Assuming that the relative motion of 3C9 and the earth is entirely one of recession, find the recession speed predicted by the relativistic Doppler relation.

Solution: If light behaved like sound, we would have that

$$\lambda^\prime~=~\lambda { 1+{v\over c}\over 1 }$$

and for $\lambda^\prime~=~3\lambda$, we would need v = 2c. For light, however, we know that

$$\lambda^\prime~=~\lambda { 1+{v\over c}\over\sqrt{1-{v^2\over c^2}} }$$

from which we can show that

$${v\over c}~=~{ \left({\lambda^\prime\over\lambda}\right)^2 - 1\over \left({\lambda^\prime\over\lambda}\right)^2 + 1}$$

for pure recession, and hence we have that v = 0.8c.