Last updated: 2026-06-16

Redshift and Recession Velocity

When light from a distant galaxy reaches us, its wavelengths are stretched toward the red end of the spectrum. The amount of stretching is the redshift:

$z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}$

Redshift is one of the most important observables in astronomy because it relates directly to how fast a source is receding and, through the Hubble law, to how far away it lies.

Two Formulas

For small redshifts the recession velocity is simply proportional to z:

$v \approx c\,z \qquad (\text{valid for } z \ll 1)$

For larger redshifts this overestimates the velocity and eventually exceeds the speed of light, which is unphysical for a Doppler interpretation. The special-relativistic Doppler formula corrects this:

$v = c\,\frac{(1+z)^2 - 1}{(1+z)^2 + 1}$

This always returns a velocity below $c$, no matter how large $z$ becomes.

Worked Example

A galaxy is observed at a redshift of $z$ = 0.1.

Even at this modest redshift the two formulas differ by about 5 %. By a redshift of 1 the classical formula reaches the speed of light, while the relativistic formula gives only about 60 % of it.

Cosmological versus Doppler Redshift

It is worth remembering what the velocity actually represents. A classical Doppler shift comes from motion through space. Cosmological redshift, however, arises because space itself expands while the light is in transit, stretching the wave. The relativistic formula above provides a useful equivalent recession velocity, but for high-redshift objects astronomers work instead with the scale factor and lookback time, which follow from a full cosmological model rather than a simple velocity. To turn the recession velocity into a distance, pair this calculator with the Hubble's law calculator.