Doppler Effect for Light
When an ambulance races past you, the pitch of its siren drops noticeably — that is the Doppler effect for sound. Light obeys an analogous principle, but with an important twist: because light always travels at the same speed c in a vacuum regardless of who is moving, the relativistic version of the Doppler effect applies. When a light source moves away from an observer, the detected wavelength is stretched toward the red end of the spectrum — a phenomenon called redshift. When the source approaches, wavelengths are compressed toward the blue end, producing blueshift. These shifts are not merely curiosities; they are among the most powerful tools in modern astrophysics. By measuring the spectrum of a distant galaxy and comparing the observed wavelengths of known spectral lines (such as hydrogen's Balmer series) with their laboratory values, astronomers can determine how fast the galaxy is moving relative to Earth. Edwin Hubble used exactly this technique in 1929 to show that almost every galaxy is receding from us — the observational cornerstone of the expanding-universe model and modern cosmology. For everyday astronomical speeds much smaller than the speed of light (v ≪ c), a convenient non-relativistic approximation is accurate enough for most calculations. This article focuses on that approximation, explains the physics behind redshift and blueshift, and shows how to apply the Doppler formula to real problems involving light.
- Why light is different from sound
- Defining the fractional shift Δλ/λ
- The non-relativistic approximation (v ≪ c)
- The full relativistic formula
- Spectral lines as cosmic rulers
Why light is different from sound
For sound, the Doppler formula depends separately on the motion of the source and the motion of the observer relative to the medium (air). Light needs no medium, and its speed c is the same for every inertial observer (a consequence of special relativity). This means only the relative velocity between source and observer matters — there is no separate 'source moving' versus 'observer moving' case for light.
Defining the fractional shift Δλ/λ
Suppose a source emits light of wavelength λ₀ (the rest wavelength, measured in the source's own frame). An observer who is moving away from the source at relative speed v detects wavelength λ_obs. The fractional change in wavelength is defined as z = (λ_obs − λ₀) / λ₀. A positive z means the wavelength has increased (redshift); a negative z means it has decreased (blueshift). The quantity z is called the redshift parameter and appears throughout cosmology.
The non-relativistic approximation (v ≪ c)
When the relative speed v is much less than c, the full relativistic formula simplifies beautifully. The fractional wavelength shift equals v/c: z ≈ v/c, so λ_obs ≈ λ₀(1 + v/c) for recession and λ_obs ≈ λ₀(1 − v/c) for approach. Equivalently, in terms of frequency f, recession gives f_obs ≈ f₀(1 − v/c) and approach gives f_obs ≈ f₀(1 + v/c). This approximation is valid to better than 1% for speeds below about 0.14c (~42,000 km s⁻¹), which covers most stars and nearby galaxies.
The full relativistic formula
For sources moving at a significant fraction of c, the exact relativistic Doppler formula must be used: f_obs = f₀ √[(1 − β)/(1 + β)], where β = v/c and the sign convention is that v > 0 for recession. Equivalently, λ_obs = λ₀ √[(1 + β)/(1 − β)]. Expanding this to first order in β reproduces the non-relativistic approximation. Quasars, the most distant objects observed, can have redshift parameters z > 7, meaning wavelengths are stretched by more than a factor of eight — deep into the relativistic regime.
Spectral lines as cosmic rulers
Every element has a unique fingerprint of spectral lines at fixed rest wavelengths. Hydrogen's Hα line, for example, is emitted at λ₀ = 656.3 nm in the rest frame. If a galaxy's spectrum shows this line at 700 nm, the redshift z = (700 − 656.3)/656.3 ≈ 0.0666, giving a recession speed of approximately 0.0666c ≈ 20,000 km s⁻¹. This elegantly converts a wavelength measurement into a velocity — no radar gun required.
The Key Equation
- z — Redshift parameter (dimensionless): Fractional change in wavelength; positive for recession (redshift), negative for approach (blueshift).
- \lambda_{\text{obs}} — Observed wavelength (metres (m) or nanometres (nm)): Wavelength measured by the observer in their own frame.
- \lambda_0 — Rest wavelength (metres (m) or nanometres (nm)): Wavelength emitted by the source in the source's rest frame; known from laboratory measurements.
- v — Relative recession speed (metres per second (m s⁻¹)): Speed at which source and observer move apart; positive for recession, negative for approach.
- c — Speed of light in vacuum (metres per second (m s⁻¹)): Universal constant, approximately 3.00 × 10⁸ m s⁻¹.
Worked Example
Problem. A distant star emits the hydrogen Hα spectral line at a rest wavelength of λ₀ = 656.3 nm. An astronomer observes this line at λ_obs = 669.4 nm. (a) Calculate the redshift parameter z. (b) Hence determine the speed at which the star is moving relative to Earth, and state whether it is approaching or receding. (Assume v ≪ c.)
Answer. The star is receding from Earth at approximately 6,000 km s⁻¹ (2% of the speed of light).
Origins · 1 minChristian Doppler and how this was discovered
Christian Doppler first described the effect that bears his name in his 1842 paper 'On the Coloured Light of the Double Stars', in which he proposed that the observed frequency of waves depends on the relative motion between the source and the observer. He originally — and incorrectly — suggested this could explain the colour differences of binary stars, but the underlying wave principle was correct. His ideas were experimentally verified for sound waves in 1845 and later extended to light, forming the foundation of spectroscopic velocity measurements in astronomy. Doppler predicted his effect would change the colour of starlight, but the speeds of stars are far too small for the human eye to perceive any colour change — spectroscopic instruments are needed to detect the tiny wavelength shifts.
Related Concepts
- Doppler Effect for Sound — The acoustic Doppler effect introduces the core idea of frequency shift due to relative motion; understanding it builds the intuition needed before tackling the relativistic version for light.
- Wave Speed, Frequency and Wavelength — The relationship v = fλ underpins all Doppler calculations; you must be confident converting between wavelength and frequency to interpret redshift and blueshift data.
- Electromagnetic Spectrum — Redshift and blueshift shift spectral lines across the electromagnetic spectrum; knowing the wavelength ranges of each region helps you interpret what a given shift means physically.
- Superposition and Interference — Spectral line identification — essential for measuring Doppler shifts — relies on the wave nature of light and the principle that different wavelengths superpose independently.