Redshift Calculator

Astronomers identify spectral lines — characteristic wavelengths emitted or absorbed by specific elements — in the light from distant objects. By comparing the observed wavelength $$\lambda_{\text{obs}}$$ to the known laboratory (rest-frame) wavelength $$\lambda_{\text{emit}}$$, they compute the redshift:

$$z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}} = \frac{\Delta\lambda}{\lambda_{\text{emit}}}$$

The physical interpretation depends on the context:

• Doppler redshift ($$z < 0.01$$): For nearby objects, the shift is due to the source's peculiar motion. The velocity is $$v \approx zc$$. Stars in our galaxy can show either redshift or blueshift depending on their motion relative to Earth.
• Cosmological redshift ($$z > 0.01$$): For distant galaxies, the redshift is caused by the expansion of space itself. Photon wavelengths stretch as the universe expands: $$1 + z = a_{\text{now}}/a_{\text{emit}}$$, where $$a$$ is the cosmic scale factor. This is distinct from a velocity-based Doppler shift, though they give similar results at low $$z$$.
• Gravitational redshift: Light climbing out of a gravitational well loses energy and is redshifted. This is a separate effect, predicted by general relativity and confirmed in the Pound-Rebka experiment.

The exact relativistic Doppler formula relating redshift to velocity is:

$$1 + z = \sqrt{\frac{1 + v/c}{1 - v/c}}$$

Solving for velocity: $$\frac{v}{c} = \frac{(1+z)^2 - 1}{(1+z)^2 + 1}$$

This ensures $$v < c$$ for any finite $$z$$, even though $$z$$ can be very large. For cosmological distances, the relationship between redshift and distance requires a specific cosmological model (e.g., ΛCDM with parameters $$H_0$$, $$\Omega_m$$, $$\Omega_\Lambda$$).

A positive $$z$$ (redshift) means the source is receding and the wavelength has been stretched. A negative $$z$$ (blueshift) means the source is approaching and the wavelength has been compressed. The approximate velocity ($$v = zc$$) is valid for $$z \ll 1$$ but overestimates the speed at higher redshifts. The exact relativistic velocity accounts for the full Lorentz transformation and always yields $$v < c$$. For cosmological applications, note that galaxies with $$z > 1.5$$ have recession velocities that formally exceed $$c$$ in the Hubble flow — this is allowed because it reflects the expansion of space, not motion through space.