Relativistic doppler effect for light

[Samuelriesterer]

Problem statement, work done, relative equations:

I am unsure if I got this problem right, especially part (e)

A star is moving at 0.2c along the x axis. The star is moving away from observer A and toward observer B. The star emits light with a maximum intensity at wavelength 500nm.

(a) Calculate the spacing between wave crests for emitted light with λ= 500nm ahead of the star and behind it in the star's frame of reference.

$\lambda_{ahead} = \lambda (1 - \frac{v}{c})$
$\lambda_{behind} = \lambda (1 + \frac{v}{c})$

(b) Transform these lengths to the observers' frame of reference.

$L_{proper} = \gamma L$

(c) Calculate the frequency of the light in the star's frame of reference.

$f=\frac{v}{\lambda} = \frac{c}{500 nm}$

(d) Calculate the frequency measured by observers A and B. This would be the time interval between receiving two successive wave crests (no relativity needed).

$f' = \frac{v}{\lambda} = \frac{.2c}{L_{proper}}$

(e) Suppose observer A is moving toward the star (and Observer B) at 0.4c. Recalculate the frequency observer A measures. You will need to recalculate the length contraction given the new relative speed and then figure the time between encountering the wave crests. What you have done is calculate the Doppler shift for light.

$u’ = \frac{u+v}{1-\frac{uv}{c^2}}=\frac{.2c+.4c}{1-\frac{.2c*.4c}{c^2}}$
$\lambda” = \lambda (1 + \frac{u’}{c})$
$L” = \gamma L=L \frac{1}{\sqrt{1-\frac{u’^2}{c^2}}}$
$f” = \frac{u’}{L”}$

(f) Show that the formula for the red shift can be written as

$f’=f \sqrt{\frac{1 \pm \beta}{1 \pm \beta}}$

 

[mfb]

Question (a) is strange. In the reference system of the star, there is no "ahead" and "behind" as the star does not move.

(e) is fine - get the relative speed with relativistic velocity addition, do the same calculations as before.

 

[vela]

I'd say your answer for (a) is wrong. mfb's comment should give you a hint as to why if you don't already see why. But it seems perhaps your answer is what was intended. You should ask your teacher.