## Redshift and velocity

One this page I attempt to explain a few issues arising around the concept of redshift in astronomy. As usual, I cannot guarantee that the information on this page is correct and accurate, so please be critical of what you read below.

### Definition of redshift

Redshift is originally defined as the ratio of the difference between observed and emitted wavelength over emitted wavelength of electromagnetic radiation, hence

\begin{equation} z \equiv \frac{\lambda_{\rm obs} - \lambda_{0}}{\lambda_{0}} = \frac{f_{0} - f_{\rm obs}}{f_{\rm obs}} \label{eqn_redshift} \end{equation}

where $\lambda_{0}$ and $f_{0}$ denote the wavelength and frequency, respectively, of the emitted radiation in the rest frame of the emitter, whereas $\lambda_{\rm obs}$ and $f_{\rm obs}$ denote the same (but redshifted) quantities in the rest frame of the observer. Note that we made use of the relation between wavelength and frequency of an electromagnetic wave, $\lambda f = \mathrm{c}$, where $\mathrm{c}$ denotes the propagation speed (speed of light). In the following, we will only consider frequency rather than wavelength, as the former is the relevant parameter when dealing with radio astronomical data.

The historic choice of using frequency shift rather than frequency in the definition of redshift is somewhat unfortunate, as the resulting relation between redshift and frequency ratio following from the definition in Eq. $\ref{eqn_redshift}$ is given by

\begin{equation} \frac{f_{0}}{f_{\rm obs}} = 1 + z . \end{equation}

As a consequence, redshift correction factors appearing in physical relations are usually of the form $(1 + z)^{n}$, where $n$ is a small integer number.

### Redshift components

In astronomy, the observed redshift of a source normally consists of contributions from several different physical effects that have little or nothing to do with each other. The most important contributions are:

**Cosmological redshift**– This component stems from the so-called “cosmological expansion” of the universe. It is by far the dominant redshift contribution for galaxies beyond the local volume. There is still an ongoing debate about the nature and correct physical interpretation of cosmological redshift (Davis & Lineweaver 2004; Francis et al. 2007; Chodorowski 2007; Bunn & Hogg 2009; Braeck & Elgarøy 2012; Kaiser 2014).**Peculiar redshift**– This component stems from the peculiar velocity of a galaxy in the local gravitational field. It can simply be interpreted as a relativistic Doppler effect due to the velocity of a galaxy with respect to the observer and is the dominant contribution for objects within the Local Group and in the nearby universe.**Gravitational redshift**– This component is caused by time dilation in a gravitational potential as predicted by the theory of general relativity. It is usually of significance only in situations where a strong gravitational field is present, e.g. in the vicinity of black holes, and is otherwise usually neglected.

The observed redshift of a source can therefore be written as the product of the different contributing redshift components. From the definition of redshift in Eq. $\ref{eqn_redshift}$ we immediately derive

\begin{equation} 1 + z_{\rm obs} = (1 + z_{\rm cos}) \times (1 + z_{\rm pec}) \times (1 + z_{\rm grav}) . \label{eqn_decomposition} \end{equation}

Note that the relation $z_{\rm obs} = z_{\rm cos} + z_{\rm pec} + z_{\rm grav}$ occasionally encountered in the literature is wrong and inconsistent with the definition of redshift as a multiplicative parameter rather than an additive one. At best, it can serve as an approximation to Eq. $\ref{eqn_decomposition}$, but only in situations where all redshift components are small ($z_{i} \ll 1$). In some situations it may also be necessary to include an additional $1 + z$ factor in Eq. $\ref{eqn_decomposition}$ to account for the peculiar motion of the observer with respect to the Hubble flow.

It is obvious from Eq. $\ref{eqn_decomposition}$ that it is not generally possible to decompose an observed redshift into its individual components. It is therefore not possible to assign a radial velocity to an observed redshift, as there will be a (potentially large) contribution from the “cosmological expansion” of the universe. Velocities can only be assigned to the peculiar redshift component in certain circumstances where the cosmological redshift component is known (see below).

### Peculiar velocity

#### Relativistic case

As mentioned before, the peculiar redshift component of a galaxy can be attributed to the galaxy’s velocity relative to the observer and is caused by the **relativistic Doppler effect**. If the emitter is moving at an **arbitrary angle** with respect to the observer, the resulting redshift is given by

\begin{equation} 1 + z_{\rm pec} = \gamma \, [1 + \beta \cos(\vartheta)] \label{eqn_reldopgen} \end{equation}

where $\beta \equiv v_{\rm pec} \, / \, \mathrm{c}$ is the normalised peculiar velocity, $\gamma \equiv (1 - \beta^{2})^{-1/2}$ is the Lorentz factor, and $\vartheta$ is the angle between the direction of motion of the source and the line-of-sight from the observer to the source at the time when the radiation was emitted. For pure line-of-sight motion we can set $\vartheta = 0$ in Eq. \ref{eqn_reldopgen} to obtain the redshift caused by the relativistic velocity of an emitter **along the line-of-sight** to the observer:

\begin{equation} 1 + z_{\rm pec} = \sqrt{\frac{1 + \beta}{1 - \beta}} \, . \label{eqn_reldop} \end{equation}

Note, however, that this relation is only correct if the motion of the emitter is entirely along the line-of-sight. If a significant transverse motion is present, Eq. \ref{eqn_reldop} will no longer be valid. In a similar fashion we can derive the redshift of an emitter moving entirely in the **plane of the sky**, but without any line-of-sight component, by setting $\vartheta = 90^{\circ}$ in Eq. \ref{eqn_reldopgen}:

\begin{equation} 1 + z_{\rm pec} = \frac{1}{\sqrt{1 - \beta^{2}}} = \gamma \end{equation}

where $\gamma$ is the Lorentz factor as defined above. Note that this transverse Doppler effect is a phenomenon predicted by special relativity and does not occur in classical physics. In practice, we usually have no information on the transverse velocity component, making it impossible to derive the line-of-sight velocity for sources that move at relativistic speeds.

#### Non-relativistic case

In the **non-relativistic case**, where $\beta \ll 1$, the transverse Doppler effect vanishes, and the line-of-sight Doppler effect of Eq. \ref{eqn_reldop} can be simplified to

\begin{equation} z_{\rm pec} = \beta = \frac{v_{\rm pec}}{\rm c}. \label{eqn_nonreldop} \end{equation}

This non-relativistic approximation of the Doppler effect along the line-of-sight (transverse motion does not contribute to the non-relativistic Doppler effect) is equal to the so-called “optical definition” of radial velocity commonly used in astronomy. Note, however, that this equation only makes sense for peculiar velocities/redshifts. The commonly encountered definition of $v_{\rm obs} = \mathrm{c} \, z_{\rm obs}$ (referred to as the “recession velocity”) to characterise the radial velocities of distant galaxies is not particularly meaningful, as it does not constitute a velocity because the redshift of distant galaxies is not due to their motion with respect to the observer, but largely due to the “cosmological expansion” of the universe.

#### Recovery of velocity difference

Lastly, even at higher redshift it is still possible to recover the peculiar velocity difference (along the line-of-sight) between two objects under the assumption that both objects are at the same cosmological redshift. In this case we can combine Eq. $\ref{eqn_decomposition}$ with the non-relativistic Doppler effect in Eq. $\ref{eqn_nonreldop}$, under the additional assumptions that the peculiar velocity between the two objects is non-relativistic and the gravitational redshift component can be neglected, to derive

\begin{equation} \frac{\Delta v_{\rm pec}}{\rm c} = \frac{\Delta z_{\rm obs}}{1 + z_{\rm cos}} \end{equation}

where $\Delta v_{\rm pec}$ is the difference between the peculiar velocities of the two sources (along the line-of-sight), and $\Delta z_{\rm obs}$ is the difference between the observed redshifts of the objects. This equation can, e.g., be used to convert an observed frequency width of a spectral line into the correct velocity width in the rest frame of the source at any given redshift.

The relation can also be expressed in terms of frequency width, in which case we obtain the following approximation:

\begin{equation} \frac{\Delta v_{\rm pec}}{\rm c} \approx \frac{1 + z_{\rm cos}}{f_{0}} \Delta f_{\rm obs} . \end{equation}

Here, $f_{0}$ is the rest frequency of the observed line transition while $\Delta f_{\rm obs}$ is the observed frequency width or frequency difference.