Redshift is how astronomers measure how fast — and in which direction — a distant object is moving relative to us, just by comparing the wavelength of its light to what that light looks like at rest in a lab. This calculator converts between redshift, wavelength shift, and recession velocity, using both the simple classical approximation and the exact relativistic formula.
What Redshift Actually Measures
Every element emits and absorbs light at specific, well-known wavelengths — hydrogen's famous H-alpha line sits at 656.3 nanometers when measured at rest in a laboratory. When astronomers observe that same line in a distant galaxy's spectrum, it's almost always shifted to a longer wavelength. The fractional size of that shift is the redshift, z = (λ_observed − λ_emitted) / λ_emitted. A positive z means the wavelength stretched (the source is receding); a negative z (blueshift) means it compressed (the source is approaching).
Classical vs. Relativistic Recession Velocity
The simplest way to turn a redshift into a speed is the classical formula, v ≈ c·z — multiply the redshift by the speed of light. It's a good linear approximation when z is small (under roughly 0.1), which is why it's taught first. But it breaks down badly at higher z: plug z = 1 into the classical formula and you get v = c, the speed of light itself, which no massive object can reach. The exact relativistic Doppler formula, v = c·((1+z)²−1)/((1+z)²+1), never produces a velocity at or above c no matter how large z gets — it asymptotically approaches c as z grows without bound, which is the physically correct behavior.
Solving in Reverse: From Velocity to Redshift
The relativistic formula can be inverted algebraically to solve for z given a known velocity: 1+z = √((1+β)/(1−β)), where β = v/c. This calculator's From Velocity tab uses that inverse directly, so entering the recession velocity from the earlier examples recovers the original redshift exactly. A negative velocity (motion toward the observer) correctly produces a negative z, matching the blueshift convention. Note that this calculator solves the Doppler-shift relationship directly from velocity and redshift — it does not incorporate a distance or the Hubble constant (v = H0·d), which is a separate, complementary relationship astronomers use to estimate distances once a recession velocity is known.