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Frequency ratios multiply when a photon is transformed first between a source and its local comoving frame, then propagated through an expanding universe. Thus the kinematic redshift and cosmological redshift factors multiply. Their redshifts themselves add only to first order when both are small.

For a radial null geodesic in the FRW metric, define $\chi (r)={\int }_0^r(1-k{r}^{\prime 2}{)}^{-1/2}d{r}^{\prime }$. Two successive wave crests emitted and received at the same comoving positions obeySubtracting and keeping the first order in the wave periods gives $\Delta t_0/a(t_0)=\Delta t_e/a(t_e)$. Thus the cosmological time dilation and the cosmological redshift have the same factor:Here the emitter and observer are comoving; an additional peculiar velocity would supply a separate kinematic redshift. The derivation uses the null geodesic propagation of the crests rather than interpreting a finite cosmological distance as an ordinary Doppler effect.

Take $v>0$ for recession. By the definition of kinematic redshift, $1+z_{\mathrm{kin}}={\nu }_e/{\nu }_0$. Inverting the supplied relativistic Doppler effect and expanding with $\beta =v/c$ givesThusA negative radial velocity gives a blueshift at this order.

A kinematic redshift is the relativistic Doppler effect arising from relative motion of emitter and receiver. In a local inertial frame, recession decreases the received frequency and approach increases it. A cosmological redshift instead compares photons emitted and received by comoving observers at different cosmic times: expansion of the scale factor stretches their wavelengths. It exists even when both observers have zero peculiar velocity. Both are frequency ratios measured by observers, but a large cosmological redshift should not be inserted into a flat-spacetime radial-speed formula to define the galaxy's unique recession speed.

Let ${\nu }_c$ be the photon frequency in the local comoving frame just after emission. The kinematic redshift gives ${\nu }_e/{\nu }_c=1+z_{\mathrm{kin}}$, and propagation in the FRW metric gives ${\nu }_c/{\nu }_0=1+z_{\mathrm{cosm}}$. Multiplication givesEquivalently, $z_{\mathrm{tot}}=z_{\mathrm{cosm}}+z_{\mathrm{kin}}+z_{\mathrm{cosm}}{z}_{\mathrm{kin}}$. Both redshifts being small is what permits their approximate addition. Here the observer's own peculiar velocity is assumed already corrected to the Hubble flow frame.