The relativistic Doppler effect describes how the observed frequency of light or any periodic signal changes when the
source and observer move at speeds that are a significant fraction of the speed of light. Unlike the ordinary
nonrelativistic Doppler shift, the relativistic version must also include time dilation. That is why the final
frequency depends not only on whether the source is approaching or receding, but also on the Lorentz factor
associated with the motion.
The key speed variable is
Speed fraction. Express the relative speed as a fraction of the speed of light.
\[
\begin{aligned}
\beta &= \frac{v}{c}.
\end{aligned}
\]
The associated Lorentz factor is
Lorentz factor. This controls the time-dilation part of the shift.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}}.
\end{aligned}
\]
Longitudinal approaching case
When the source moves directly toward the observer, the received wave crests are compressed. The relativistically
correct observed frequency is
Approaching source. This gives a blueshift.
\[
\begin{aligned}
f' &= f\sqrt{\frac{1+\beta}{1-\beta}}.
\end{aligned}
\]
Because the square-root factor is greater than 1, the observed frequency is larger than the emitted frequency.
Longitudinal receding case
When the source moves directly away from the observer, the received wave crests are stretched farther apart. The
observed frequency becomes
Receding source. This gives a redshift.
\[
\begin{aligned}
f' &= f\sqrt{\frac{1-\beta}{1+\beta}}.
\end{aligned}
\]
Here the square-root factor is less than 1, so the observed frequency drops below the emitted frequency.
Transverse Doppler effect
A special relativistic effect appears even when the source moves transversely rather than toward or away from the
observer. In that case there is no ordinary geometric compression or stretching of the wavefront spacing along the
line of sight. The shift comes only from time dilation:
Transverse case. The source clock runs slow, so the observed frequency is reduced.
\[
\begin{aligned}
f' &= \frac{f}{\gamma} \\
&= f\sqrt{1-\beta^2}.
\end{aligned}
\]
This is a purely relativistic effect. A classical treatment would not predict such a shift for purely transverse
motion.
Worked example: source moving toward the observer
Take the sample values \(f = 500\ \mathrm{Hz}\) and \(\beta = 0.8\), with the source moving toward the observer.
Then
Step 1. Build the Doppler factor.
\[
\begin{aligned}
D &= \sqrt{\frac{1+\beta}{1-\beta}} \\
&= \sqrt{\frac{1+0.8}{1-0.8}} \\
&= \sqrt{\frac{1.8}{0.2}} \\
&= \sqrt{9} \\
&= 3.
\end{aligned}
\]
Step 2. Multiply the emitted frequency by the relativistic factor.
\[
\begin{aligned}
f' &= f \cdot D \\
&= 500 \cdot 3 \\
&= 1500\ \mathrm{Hz}.
\end{aligned}
\]
So the physically correct approaching-source result is \(1500\ \mathrm{Hz}\). This is a strong blueshift.
Worked receding comparison
Using the same \(\beta = 0.8\) but now with the source moving away,
Step 1. Build the redshift factor.
\[
\begin{aligned}
D &= \sqrt{\frac{1-\beta}{1+\beta}} \\
&= \sqrt{\frac{0.2}{1.8}} \\
&= \sqrt{\frac{1}{9}} \\
&= \frac{1}{3}.
\end{aligned}
\]
Step 2. Compute the observed frequency.
\[
\begin{aligned}
f' &= 500 \cdot \frac{1}{3} \\
&\approx 166.7\ \mathrm{Hz}.
\end{aligned}
\]
This is the corresponding relativistic redshift for the same speed magnitude.
Wavelength and period
Once the observed frequency is known, the observed period and wavelength follow immediately from
Related wave quantities. These help interpret the shift physically.
\[
\begin{aligned}
T' &= \frac{1}{f'}, \\
\lambda' &= \frac{c}{f'}.
\end{aligned}
\]
A higher observed frequency means a shorter observed period and wavelength, while a lower observed frequency means a
longer observed period and wavelength.
How to interpret the calculator
The calculator reports the relativistic Doppler factor, the observed frequency, the observed period, and the observed
wavelength. It also compares the result with a simple classical estimate so the relativistic correction is easy to
recognize. The visualization shows the physical meaning directly: approaching motion compresses the wave pattern,
receding motion stretches it, and the transverse case shifts frequency without the usual longitudinal geometry.
Core summary formulas. These are the main relations used by the calculator.
\[
\begin{aligned}
f'_{\text{toward}} &= f\sqrt{\frac{1+\beta}{1-\beta}}, \\
f'_{\text{away}} &= f\sqrt{\frac{1-\beta}{1+\beta}}, \\
f'_{\text{transverse}} &= f\sqrt{1-\beta^2} = \frac{f}{\gamma}, \\
\gamma &= \frac{1}{\sqrt{1-\beta^2}}, \\
T' &= \frac{1}{f'}, \\
\lambda' &= \frac{c}{f'}.
\end{aligned}
\]
At a more advanced university level, these ideas connect to relativistic beaming, aberration, and astrophysical
redshift analysis. The one-dimensional formulas used here are the most direct starting point.
