Ch 05-04 The Doppler Effect and Redshifts

The Doppler effect of sound waves and EM waves

The EM wave Doppler effect

In the frame of a moving source
ν_s = 1/dt_s
In the observer's frame
   the emission interval
     dt_e = dt_s (1-β²)^-1/2          β = v/c
  (the time dilation from the special relativity)

   the reception interval
     dt_r = dt_e (1+β_r)                β_r = v_r/c
  (the usual kinematic Doppler effect)

     dt_r = dt_s (1+β_r) (1-β²)^-1/2
ν_obs / ν_s = dt_s/dt_r
= (1-β²)^1/2 / (1+β_r)

The redshift
z = (λ_obs-λ_s)/λ_s = (ν_s/ν_obs) - 1
- Doppler redshifts
  z = (1+β_r) (1-β²)^-1/2 - 1
  - for β² << 1 , z ~ β_r = v_r/c
  - for β = β_r , z = [(1+β_r)/ (1-β_r)]^1/2 - 1
    if β_r << 1 , z ~ β_r
- gravitational redshifts
- cosmological redshifts