here - Department of Physics, HKU

CHAPTER 4. RELATIVISTIC KINEMATICS 28
N
= ct
λ 0 0
ct
vt
Figure 4.1: Relativistic Doppler effect.
This formula is valid for any object moving directly away or toward the
observer (v positive if it is receding and negative if it is approaching). The
formula for general direction of velocity is more complicated and not shown
here.
If v is much less than c, we can approximate the right hand side by 1+v/c
and the formula reduces to Eq. (4.1), because for small v S and v O ,
λ
λ 0
= 1 + v S/C
1 − v O /C ≈ (1 + v S/C)(1 + v O /C) ≈ 1 + (v S + v O )/C (4.6)
and v S + v O is the relative speed between them. If v is close to c, the shift
of wavelength will be infinite.
As an example, suppose two spaceships are moving toward each other,
with speeds v 1 and v 2 , relative to us. If the spaceship with speed v 1 shines a
laser with frequency f 1 in its frame to the other spaceship, what will be the
observed frequency in our frame, and the observed frequency in frame of the
other spaceship? Well, the observed frequency in our frame is simply
( ) 1/2
1 + v1 /c
f 1 . (4.7)
1 − v 1 /c