Relativistic Doppler Effect

The Relativistic Doppler Effect

The classical Doppler effect shifts sound frequencies when source and observer move relative to each other. Light has a similar effect, but with a crucial difference: because of time dilation, there is a frequency shift even for transverse (sideways) motion.

Longitudinal Doppler

For a source moving directly away from the observer at velocity $v$ (recession), the observed frequency is:

$f_{\text{obs}} = f_0\sqrt{\frac{1 - \beta}{1 + \beta}}$

For a source moving directly toward the observer (approach):

$f_{\text{obs}} = f_0\sqrt{\frac{1 + \beta}{1 - \beta}}$

where $\beta = v/c$ and $f_0$ is the emitted frequency.

Behaviour

Notice that at $\beta = 0.6$: $\sqrt{0.4/1.6} = \sqrt{1/4} = 0.5$, so the observed frequency is halved.

Symmetry Property

For any speed $v$, the product of the receding and approaching factors equals $1$:

$f_{\text{recede}} \times f_{\text{approach}} = f_0\sqrt{\frac{1-\beta}{1+\beta}} \times f_0\sqrt{\frac{1+\beta}{1-\beta}} = f_0^2$

This is a useful sanity check.

Cosmological Redshift

Galaxies receding from us (due to cosmic expansion) have their light redshifted. The redshift parameter $z = (f_0 - f_{\text{obs}})/f_{\text{obs}}$ quantifies this. A galaxy at $z = 1$ has its light frequency halved.

Your Task

Implement:

• doppler_receding(f0, v) — observed frequency when source moves away
• doppler_approaching(f0, v) — observed frequency when source moves toward observer

Use $c = 299792458.0$ m/s, defined inside each function.