Doppler Effect for Light Calculator

The relativistic Doppler effect combines two phenomena: the classical Doppler shift from relative motion and the time dilation of the moving source. For a source moving radially toward the observer at velocity $$v$$:

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

This can be derived by considering a source emitting light at frequency $$f_0$$ in its rest frame. Due to time dilation, the period between wave crests as measured by the observer is modified by the Lorentz factor, and the Doppler compression (or stretching) of wavelengths adds the classical velocity-dependent factor.

Key features of the relativistic Doppler effect:

• Blueshift (approaching): The Doppler factor $$D > 1$$, so $$f_{\text{obs}} > f_0$$ and $$\lambda_{\text{obs}} < \lambda_0$$. Light shifts toward shorter wavelengths (blue/ultraviolet).
• Redshift (receding): $$D < 1$$, so $$f_{\text{obs}} < f_0$$ and $$\lambda_{\text{obs}} > \lambda_0$$. Light shifts toward longer wavelengths (red/infrared).
• Transverse Doppler effect: Even when motion is purely transverse (perpendicular to the line of sight), there is a redshift due to time dilation alone: $$f_{\text{obs}} = f_0/\gamma$$. This has no classical analog and is a purely relativistic phenomenon.
• Redshift parameter: Astronomers use $$z = \frac{\lambda_{\text{obs}} - \lambda_0}{\lambda_0} = \frac{1}{D} - 1$$ for receding sources, where $$z > 0$$ means redshift.

For low velocities ($$\beta \ll 1$$), the relativistic formula reduces to the classical approximation $$\Delta f / f_0 \approx \pm v/c$$.

The Observed Frequency and Observed Wavelength are what the observer actually detects. Positive frequency shift means blueshift; negative means redshift. The Doppler Factor is the multiplicative ratio — at $$\beta = 0.5$$ approaching, $$D \approx 1.732$$, so the frequency nearly doubles. The Redshift Parameter z is the standard astronomical measure: nearby galaxies have $$z \sim 0.001-0.1$$, distant quasars reach $$z > 6$$, and the cosmic microwave background has $$z \approx 1089$$.