# Doppler Shift for light

1. Nov 12, 2013

### SteveDC

How do Doppler shifts work for light if, according to special relativity, light is constant velocity for all observers? So if c is unchanged, then surely wavelength and frequency don't change.

I appreciate that I must be misunderstanding something, because redshift on stars occurs, but I am struggling to explain why.

Thanks in advance for any help :)

2. Nov 12, 2013

### WannabeNewton

The frequency of a light wave corresponds to the time-like component of the wave 4-vector of a light wave, and components of 4-vectors are not invariant under non-trivial Lorentz transformations. More generally, energy is a frame dependent quantity. Why should wavelength and frequency of light waves be unchanged between inertial frames just because $c$ is unchanged between inertial frames? They can scale inversely under Lorentz transformations so as to cancel out any change in their ratio.

3. Nov 12, 2013

### Mentz114

$c=\lambda\nu$ is the equation relating velocity and wavelength and frequency. So both can change while keeping c constant.

[edit. got it wrong first time]

4. Nov 12, 2013

### Bill_K

One thing which is the same in all rest frames is the phase of the wave. That is, for example, the wave crest always remains a wave crest. A typical wave is exp(i(kx - ωt), where the phase is kx - ωt, or equivalently since ω = ck, the phase is k(x - ct). This quantity must be an invariant.

Under a Lorentz transformation,

x' = γ(x - vt)
t' = γ(t - v/c2 x)

implying that x - ct just picks up an overall factor:

x' - ct' = γ(1 - v/c)(x - ct) = √((1 - v/c)(1 + v/c)) (x - ct)

Invariance requires that the wave vector k picks up the inverse factor:

k' = √((1 + v/c)(1 - v/c)) k

which represents the Doppler shift.