The relativistic Doppler effect describes how the wavelength or frequency of light changes when a source and an
observer are in relative motion. In ordinary everyday sound problems, the Doppler effect is often treated with a
classical formula. Light is different, because the speed of light is the same in every inertial frame, so the correct
analysis must use special relativity.
For light, the observed shift depends on the velocity component along the line of sight and on time dilation. If a
source is moving directly away from the observer, the observed wavelength increases and the light is redshifted.
If the source is moving directly toward the observer, the observed wavelength decreases and the light is
blueshifted.
Longitudinal relativistic Doppler formulas.
\[
\begin{aligned}
\lambda_{\mathrm{obs}} &= \lambda_0 \sqrt{\frac{1+\beta}{1-\beta}}
\qquad \text{(receding source)},\\[6pt]
\lambda_{\mathrm{obs}} &= \lambda_0 \sqrt{\frac{1-\beta}{1+\beta}}
\qquad \text{(approaching source)},
\end{aligned}
\]
where
\[
\beta=\frac{v}{c}.
\]
These formulas are exact for motion along the line joining the source and the observer. They differ from a classical
Doppler formula because the Lorentz factor is built into the square-root expression. At low speeds, the relativistic
formula approaches the familiar approximation \(z \approx \beta\) for recession, but the exact expression becomes
noticeably different as the speed grows.
Redshift parameter
Astronomers and physicists often describe the shift using the dimensionless redshift parameter
\[
\begin{aligned}
z = \frac{\lambda_{\mathrm{obs}}-\lambda_0}{\lambda_0}.
\end{aligned}
\]
If \(z>0\), the wavelength has increased and the light is redshifted. If \(z<0\), the wavelength has decreased and
the light is blueshifted. Since frequency and wavelength are related by \(f = c/\lambda\), an increase in wavelength
corresponds to a decrease in frequency, and vice versa.
Frequency relation.
\[
\begin{aligned}
f_{\mathrm{obs}} = \frac{c}{\lambda_{\mathrm{obs}}}.
\end{aligned}
\]
Transverse Doppler effect
A particularly important relativistic result is the transverse Doppler effect. Suppose the source has no
radial motion toward or away from the observer at the instant of observation, so the velocity is effectively sideways.
Classically, you might expect no shift. Relativity predicts a shift anyway, because moving clocks run slow. The light
emission process is tied to the source’s time, so the observed frequency is reduced purely by time dilation.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}},\\[4pt]
\lambda_{\mathrm{obs}} &= \gamma \lambda_0,\\[4pt]
f_{\mathrm{obs}} &= \frac{f_0}{\gamma}.
\end{aligned}
\]
This means transverse motion produces a redshift even though the source is not directly receding along the line of
sight. That is one of the clearest signatures that the effect is genuinely relativistic.
Sample interpretation
For a galaxy receding at \(v = 0.1c\) with rest wavelength \(500\ \mathrm{nm}\), the exact relativistic factor is
\[
\sqrt{\frac{1+0.1}{1-0.1}} \approx 1.106.
\]
Therefore the observed wavelength is about
\[
\lambda_{\mathrm{obs}} \approx 500 \times 1.106 \approx 553\ \mathrm{nm},
\]
which is a modest redshift, corresponding to \(z \approx 0.106\). A low-speed approximation would say \(z \approx 0.1\),
which is close but not exact.
Doppler shift versus cosmological redshift
A very important distinction is that cosmological redshift is not the same as special-relativistic Doppler
shift. Cosmological redshift comes from the expansion of space in general relativity, not simply from a source moving
through space relative to an observer in flat spacetime. For nearby galaxies, the two ideas may give similar small-shift
estimates, but they are conceptually different models.
That is why this calculator is best understood as a special-relativistic light-shift tool. It is ideal for
studying redshift, blueshift, transverse Doppler shift, and the connection between wavelength, frequency, and the
Lorentz factor.
Advanced direction
At a more advanced university level, the next extensions are aberration of light, four-wave-vectors, and
gravitational redshift in curved spacetime. Those effects go beyond this calculator, but they build on the same idea:
the observed light signal depends on spacetime geometry and on how motion changes the relation between emission and
observation.
