Question

Suppose you are viewing some red light coming from a galaxy far, far away. The wavelength of the red light is expected to be 6300 nm if the relative speed of the galaxy with respect to your world is zero. That is, if the galaxy and you are not moving toward or away from each other. However, you observe a wavelength that is longer than expected. How would you interpret this observation?     Recall the frequency dependence of the Doppler shift: f’ = f (1 ⁺ u/c).

Answer 1

If the galaxy and the observer are moving toward each other, then the Doppler shifted frequency is
   
And so, the wavelength of the Doppler shifted wave is

  
And so, we see that as u > 0, so,

So, when the observer and the galaxy move towards each other, the observed wavelength is smaller than the emitted wavelength.
On the other hand, if the the galaxy and the observer are moving away from each other, then the Doppler shifted frequency is
   
where, is the relative speed between the two and is by definition positive, i.e., .
And so, the wavelength of the Doppler shifted wave is

  
And so, we see that as |u| > 0, so,

So, when the observer and the galaxy move away from each other, the observed wavelength is greater than the emitted wavelength.
And so, in the given situation as the observed wavelength is longer than the emitted wavelength, so, the galaxy and the observer are moving away from each other. So, the observer will conclude that the galaxy is moving away from him.

And the relative velocity of recession can be calculated as