Redshift Calculator

What Is the Redshift Calculator?

The Redshift Calculator converts between redshift, wavelength, and velocity, and reveals what a given redshift means about an object's motion and the universe itself. Redshift, written z, is the dimensionless measure of how much light has been stretched to longer wavelengths, defined as z = (observed wavelength minus emitted wavelength) divided by the emitted wavelength. Enter a redshift, a pair of wavelengths, or a velocity, and the tool works out z, the wavelength stretch factor, the cosmological scale factor, and two distinct velocity interpretations. As the NASA Imagine the Universe primer explains, redshift is the single most important measurement in observational cosmology.

Most redshift tools hand back a single recession speed and stop there, which is exactly where the biggest misconceptions begin. This calculator instead shows the naive cz velocity beside the relativistic Doppler velocity, so the difference is impossible to miss, and it carries out a spectral line analysis showing where real emission lines land at your chosen z. Given that the same measured shift can come from motion, from cosmic expansion, or from gravity, the tool also includes a gravitational redshift explorer for compact objects. It even handles blueshift, the negative z of objects like Andromeda that are approaching rather than receding.

The Redshift Formula and the Scale Factor

At its core the calculation is simple: z = (λ_observed minus λ_emitted) divided by λ_emitted, which means the observed wavelength is just the emitted wavelength multiplied by (1 plus z). A line emitted at 500 nanometres and seen at 600 nanometres has z = 0.2. That said, the real power of redshift is what it tells you about cosmic history through the scale factor. The scale factor a, set to 1 today, was a = 1 divided by (1 plus z) when the light was emitted, so a galaxy at z = 1 emitted its light when the universe was half its present size.

This turns redshift into a clock as much as a speedometer. A quasar at z = 3 shows us the universe at one quarter of its current scale, billions of years in the past. The European Space Agency's overview of redshift describes how this single number lets astronomers pull out the epoch, distance, and expansion history of everything they observe. To carry the cosmological reading further and convert a redshift into an actual distance and lookback time, you can feed the same z into our Hubble law distance calculator, which performs the full expansion-model integration.

Three Kinds of Redshift: Doppler, Cosmological, and Gravitational

The same stretching of light arises from three physically different causes, and telling them apart is essential to reading a spectrum correctly. The table below summarises how they differ and where each one dominates.

Doppler redshift behaves like the familiar drop in pitch of a passing siren, and the relativistic Doppler formula keeps the implied speed below light speed for any finite z. Cosmological redshift, by contrast, comes from space expanding while the light is in transit, which is why its recession velocities can exceed c without breaking relativity. Gravitational redshift, the subject of the calculator's compact-object explorer, follows 1 plus z = 1 divided by the square root of (1 minus 2GM divided by Rc squared), and becomes infinite at a black hole's event horizon, a connection you can pursue further with the relationships behind our luminosity calculator.

Spectral Lines: Reading Redshift From Real Light

Astronomers do not measure redshift from a single wavelength but from the shifted pattern of known spectral lines. Hydrogen, calcium, and oxygen produce lines at precise laboratory wavelengths, and the amount those lines have moved gives z directly. The calculator's spectral line table makes this concrete by taking familiar rest-frame lines and showing where each one lands at your chosen redshift, along with the band it shifts into. At low redshift the lines barely move, but at high redshift the consequences are dramatic.

Consider the Lyman-alpha line, emitted at 122 nanometres in the ultraviolet. At z = 11 it is stretched to around 1450 nanometres, deep in the infrared, which is the fundamental reason the James Webb Space Telescope was built as an infrared observatory. Visible light from the earliest galaxies simply does not arrive as visible light. On top of that, the line pattern stays intact as it shifts, so even a spectrum displaced far into the infrared can be matched unambiguously to its rest-frame template, which is what makes redshift such a precise and trusted measurement.

Accuracy and Limitations

The conversions among redshift, wavelength, and velocity are exact algebraic relations, so the outputs are accurate to the precision of your inputs. The relativistic Doppler velocity uses the exact special-relativistic formula, and the gravitational redshift uses the exact Schwarzschild expression for a non-rotating mass. The scale factor relation a = 1 divided by (1 plus z) is a definition and holds in any expanding cosmology.

What the calculator deliberately does not do is convert a cosmological redshift into a distance or a lookback time, because those depend on the full expansion history and the assumed cosmological parameters, which the companion Hubble law tool handles. The naive cz velocity is provided specifically to illustrate the common misconception, not as a recommended value, and the relativistic Doppler reading should only be interpreted as a true speed for genuinely local, moving sources. For distant galaxies the honest statement is the redshift itself, since splitting a cosmological z into a unique velocity is model-dependent, a subtlety the Davis and Lineweaver analysis treats in full.

The Most Common Redshift Mistake: Treating cz as a Real Speed

In my experience the error that causes the most confusion is taking velocity equals c times z literally at high redshift and concluding that distant galaxies move faster than light through space. They do not. A galaxy at z = 7 has a naive cz of seven times light speed, but that figure is only valid as a cosmological recession velocity, the rate at which expanding space carries the galaxy away, and it tells you nothing about motion through space, which never exceeds c. With that in mind, always check which interpretation applies before quoting a speed: for a star or a jet use the relativistic Doppler value the calculator gives, and for a distant galaxy quote the redshift itself. This single distinction resolves the great majority of arguments about faster-than-light expansion, and it is the reason professional astronomers almost always report z rather than a converted velocity.