Light emitted from a comoving galaxy located at g at time te will reach an observer situated at o 0 at a later time to, where
Equation 5 provides the relation among g, to, and te. For a comoving galaxy, g is unchanged so that differentiating eq. 5 leads to
This result relates the evolution of the universe (ao / ae) as the photon travels from emission to observation, to the change in its frequency () or wavelength (). As the universe expands (or contracts!), wavelengths expand (contract) and frequencies decrease (increase). The redshift of a spectral line is defined by relating the wavelength at emission (the "lab" or "rest-frame" wavelength e) to the wavelength observed at a later time to, o.
Since the energies of photons are directly proportional to their frequencies, as the universe expands photon energies redshift to smaller values: E = h E (1 + z)-1. For all particles, massless or not, de Broglie told us that wavelength and momentum are inversely related, so that: p -1 p (1 + z)-1. All momenta redshift; for non-relativistic particles (e.g. galaxies) this implies that their "peculiar" velocities redshift: v = p/M (1 + z)-1.