Note 1 to entry: Letter symbol $u$ is used to distinguish this quantity from the relative velocity $v$.

Note 2 to entry: Four-velocity is a four-vector analogous to velocity in three-dimensional space.

Note 3 to entry: $\underset{\_}{\underset{\_}{u}}:=\frac{\text{d}\underset{\_}{\underset{\_}{r}}}{\text{d}\tau }=\frac{\text{d}\underset{\_}{\underset{\_}{r}}}{\text{d}t}\frac{\text{d}t}{\text{d}\tau }=\gamma \left(\text{j}{c}_0;\dot{\overrightarrow{r}}\right)=\gamma \left(\text{j}{c}_0;\overrightarrow{v}\right)$ or $\underset{\_}{\underset{\_}{u}}:=\left(u_0,u_1,u_2,u_3\right):=\left(\gamma \text{j}{c}_0,\gamma v_{\text{x}},\gamma v_{\text{y}},\gamma v_{\text{z}}\right)=\gamma \left(\text{j}{c}_0;\overrightarrow{v}\right)$ where $\gamma$ is Lorentz factor, $\text{j}$ is the imaginary unit, and $c_0$ is speed of light in vacuum.

Note 4 to entry: The squared four-magnitude of four-velocity, i.e. ${\displaystyle {\sum }_{\mu }{u}_{\mu }}{u}_{\mu }$, is always equal to $-c_0^2$.

For example, if $\overrightarrow{v}=\left(\frac{4}{5}{c}_0;0;0\right)$, then $\beta =\frac{4}{5}$, $\gamma =\frac{5}{3}$ and $\underset{\_}{\underset{\_}{u}}=\left(\frac{5}{3}\mathrm{j}{c}_0;\frac{4}{3}{c}_0;0;0\right)$, so that ${\displaystyle {\sum }_{\mu }{u}_{\mu }}{u}_{\mu }=-c_0^2$.

Note 5 to entry: The coherent SI unit of four-velocity is metre per second, $\text{m}\cdot {\text{s}}^{-1}$.