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Relativistic Doppler Calculator

Find the relativistic Doppler-shifted frequency.

How it is calculated

For light travelling along the line of sight the tool uses f_obs = f × √((1 + β) / (1 − β)). Here f is the frequency emitted by the source in MHz (the result is labelled MHz) and β = v / c is the source speed as a fraction of the speed of light. Positive β means the source is approaching, so the observed frequency is higher (blueshift); negative β means it is receding, so the frequency is lower (redshift). β must lie strictly between −1 and 1, and frequency cannot be negative; otherwise there is no result. The answer has two decimals. The formula covers motion directly toward or away from the observer only, not sideways motion.

Worked example

A source emits at 1420.41 MHz and approaches at 0.6 c. Step 1: (1 + 0.6) / (1 − 0.6) = 1.6 / 0.4 = 4. Step 2: √4 = 2. Step 3: 1420.41 × 2 = 2840.82. The observed frequency is 2,840.82 MHz. If the same source were receding at 0.6 c, the factor would be 0.5 and the observed frequency 710.21 MHz.

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Common mistakes

  • Entering the speed in km/s or m/s; β is a fraction of c, so 0.6 means 60% of light speed, and 180000 would be rejected.
  • Getting the sign backwards: positive β is approaching and raises the frequency, negative β lowers it.
  • Using the classical formula f × (1 + v/c) at high speed; the relativistic factor gives a different answer.
  • Entering the frequency in Hz or GHz when the result is labelled in MHz.
  • Using the formula for a source moving perpendicular to the line of sight; that case needs a different, purely time-dilation expression.

About this calculator

This calculator applies the relativistic Doppler effect for light: f_obs = f_source·√((1+β)/(1−β)), where β = v/c is the source's speed as a fraction of the speed of light, positive for an approaching source and negative for a receding one. Unlike the classical Doppler formula for sound, this version accounts for time dilation as well as the light-travel-time change, so it holds even when the source moves directly toward or away from the observer at relativistic speeds — a regime where the ordinary approximation breaks down.

Astronomers use this exact relation to convert the observed redshift or blueshift of spectral lines from stars, quasars, and receding galaxies into recession velocities, while particle physicists apply it to radiation from relativistic beams and accelerator experiments. It also appears in special relativity coursework as the standard example of how motion changes the frequency — and hence the color and energy — of light.

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Comments (1)

  • Editorial team

    A quick sanity check at everyday astronomical speeds: when β is tiny, the fractional shift is almost equal to β. At β = 0.001, a 1000 MHz signal reads about 1,001.00 MHz, a 0.1 percent rise. The relativistic factor only departs noticeably from that at a good fraction of c. If your data is a wavelength, convert it to frequency first. What source are you modelling?

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