Editing Wave vector (section)

===Lorentz transformation===
Taking the [[Lorentz transformation]] of the four-wavevector is one way to derive the [[relativistic Doppler effect]].  The Lorentz matrix is defined as
:<math>\Lambda = \begin{pmatrix}
           \gamma & -\beta \gamma & \ 0 \ & \ 0 \ \\
    -\beta \gamma &        \gamma & 0     & 0 \\
                0 &             0 & 1     & 0 \\
                0 &             0 & 0     & 1
 \end{pmatrix}</math>

In the situation where light is being emitted by a fast moving source and one would like to know the frequency of light detected in an earth (lab) frame, we would apply the Lorentz transformation as follows.  Note that the source is in a frame {{math|''S''<sup>s</sup>}} and earth is in the observing frame, {{math|''S''<sup>obs</sup>}}.
Applying the Lorentz transformation to the wave vector
:<math>k^{\mu}_s = \Lambda^\mu_\nu k^\nu_{\mathrm{obs}} </math>

and choosing just to look at the <math>\mu = 0</math> component results in
:<math>\begin{align}
             k^{0}_s &= \Lambda^0_0 k^0_{\mathrm{obs}} + \Lambda^0_1 k^1_{\mathrm{obs}} + \Lambda^0_2 k^2_{\mathrm{obs}} + \Lambda^0_3 k^3_{\mathrm{obs}} \\[3pt]
  \frac{\omega_s}{c} &= \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma k^1_{\mathrm{obs}} \\
                     &= \gamma \frac{\omega_{\mathrm{obs}}}{c} - \beta \gamma \frac{\omega_{\mathrm{obs}}}{c} \cos \theta.
\end{align}</math>

where <math>\cos \theta </math> is the direction cosine of <math>k^1</math> with respect to <math>k^0, k^1 = k^0 \cos \theta. </math>

So
:{|cellpadding="2" style="border:2px solid #ccccff"
|<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - \beta \cos \theta)} </math>
|}

====Source moving away (redshift)====
As an example, to apply this to a situation where the source is moving directly away from the observer (<math>\theta=\pi</math>), this becomes:
:<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 + \beta)} = \frac{\sqrt{1-\beta^2}}{1+\beta} = \frac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta} = \frac{\sqrt{1-\beta}}{\sqrt{1+\beta}} </math>

====Source moving towards (blueshift)====
To apply this to a situation where the source is moving straight towards the observer ({{math|1=''&theta;'' = 0}}), this becomes:

:<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - \beta)} = \frac{\sqrt{1-\beta^2}}{1-\beta} = \frac{\sqrt{(1+\beta)(1-\beta)}}{1-\beta} = \frac{\sqrt{1+\beta}}{\sqrt{1-\beta}} </math>

====Source moving tangentially (transverse Doppler effect)====
To apply this to a situation where the source is moving transversely with respect to the observer ({{math|1=''&theta;'' = ''&pi;''/2}}), this becomes:
:<math>\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - 0)} = \frac{1}{\gamma} </math>