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Relativistic Doppler formula.

Posted by peeterjoot on July 6, 2009

Transform of angular velocity four vector.

It was possible to derive the Lorentz boost matrix by requiring that the wave equation operator

\begin{aligned}\nabla^2 = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \boldsymbol{\nabla}^2 \end{aligned}

retain its form under linear transformation ([1]). Applying spatial Fourier transforms ([2]), one finds that solutions to the wave equation

\begin{aligned}\nabla^2 \psi(t,\mathbf{x}) = 0 \end{aligned}

Have the form

\begin{aligned}\psi(t, \mathbf{x}) = \int A(\mathbf{k}) e^{i(\mathbf{k} \cdot \mathbf{x} - \omega t)} d^3 k \end{aligned}

Provided that $\omega = \pm c {\left\lvert{\mathbf{k}}\right\rvert}$. Wave equation solutions can therefore be thought of as continuously weighted superpositions of constrained fundamental solutions

\begin{aligned}\psi &= e^{i(\mathbf{k} \cdot \mathbf{x} - \omega t)} \\ c^2 \mathbf{k}^2 &= \omega^2 \end{aligned}

The constraint on frequency and wave number has the look of a Lorentz square

\begin{aligned}\omega^2 - c^2 \mathbf{k}^2 = 0 \end{aligned}

Which suggests that in additional to the spacetime vector

\begin{aligned}X = (ct, \mathbf{x}) = x^\mu \gamma_\mu \end{aligned}

evident in the wave equation fundamental solution, we also have a frequency-wavenumber four vector

\begin{aligned}K = (\omega/c, \mathbf{k}) = k^\mu \gamma_\mu \end{aligned}

The pair of four vectors above allow the fundamental solutions to be put explicitly into covariant form

\begin{aligned}K \cdot X = \omega t - \mathbf{k} \cdot \mathbf{x} = k_\mu x^\mu \end{aligned}

\begin{aligned}\psi = e^{-i K \cdot X} \end{aligned}

Let’s also examine the transformation properties of this fundamental solution, and see as a side effect that $K$
has transforms appropriately as a four vector.

\begin{aligned}0 &= \nabla^2 \psi(t,\mathbf{x}) \\ &= {\nabla'}^2 \psi(t',\mathbf{x}') \\ &= {\nabla'}^2 e^{i(\mathbf{x}' \cdot \mathbf{k}' - \omega' t')} \\ &= -\left(\frac{{\omega'}^2}{c^2} - {\mathbf{k}'}^2 \right) e^{i(\mathbf{x}' \cdot \mathbf{k}' - \omega' t')} \\ \end{aligned}

We therefore have the same form of frequency wave number constraint in the transformed frame (if we require that
the wave function for light is unchanged under transformation)

\begin{aligned}{\omega'}^2 = c^2 {\mathbf{k}'}^2 \end{aligned}

Writing this as

\begin{aligned}0 = {\omega}^2 - c^2 {\mathbf{k}}^2 = {\omega'}^2 - c^2 {\mathbf{k}'}^2 \end{aligned}

singles out the Lorentz invariant nature of the $(\omega, \mathbf{k})$ pairing, and we conclude that this pairing
does indeed transform as a four vector.

Application of one dimensional boost.

Having attempted to justify the four vector nature of the wave number vector $K$, now move on to application of a boost along the x-axis to this vector.

\begin{aligned}\begin{bmatrix}\omega' \\ c k' \\ \end{bmatrix}&=\gamma\begin{bmatrix}1 & -\beta \\ -\beta& 1 \\ \end{bmatrix}\begin{bmatrix}\omega \\ c k \\ \end{bmatrix} \\ &=\begin{bmatrix}\omega - v k \\ c k - \beta \omega\end{bmatrix} \end{aligned}

We can take ratios of the frequencies if we make use of the dependency between $\omega$ and $k$. Namely, $\omega = \pm c k$. We then have

\begin{aligned}\frac{\omega'}{\omega}&= \gamma(1 \mp \beta) \\ &= \frac{1 \mp \beta}{\sqrt{1 - \beta^2}} \\ &= \frac{1 \mp \beta}{\sqrt{1 - \beta}\sqrt{1 + \beta}} \\ \end{aligned}

For the positive angular frequency this is

\begin{aligned}\frac{\omega'}{\omega}&= \frac{\sqrt{1 - \beta}}{\sqrt{1 + \beta}} \\ \end{aligned}

and for the negative frequency the reciprocal.

Deriving this with a Lorentz boost is much simpler than the time dilation argument in wikipedia doppler article ([3]). EDIT: Later found exactly the above boost argument in the wiki k-vector article ([4]).

What’s missing here is putting this in a physical context properly with source and reciever frequencies spelled out. That would make this more than just math.

References

[1] Peeter Joot. Wave equation based Lorentz transformation derivation [online]. http://sites.google.com/site/peeterjoot/geometric-algebra/lorentz.pdf.

[2] Peeter Joot. Fourier transform solutions to the wave equation [online]. http://sites.google.com/site/peeterjoot/math2009/wave_fourier.pdf.

[3] Wikipedia. Relativistic doppler effect — wikipedia, the free encyclopedia [online]. 2009. [Online; accessed 26-June-2009]. http://en.wikipedia.org/w/index.php?title=Relativistic_Doppler_effect&o%ldid=298724264.

[4] Wikipedia. Wave vector — wikipedia, the free encyclopedia [online]. 2009. [Online; accessed 30-June-2009]. http://en.wikipedia.org/w/index.php?title=Wave_vector&oldid=299450041.