Note 1 to entry: Four-vector symbols can be written using two different forms of presentation:

1. a light face single letter in italics with a double underscore, which is that form mostly used in the special theory of relativity (STR) when the first component is imaginary, by analogy with the underscoring of symbols of complex quantities, e.g. $\underset{\_}{\underset{\_}{x}}$;
2. a light face single letter in italics with a subscript (denoting the covariant component) or a superscript (denoting the contravariant component), which can or cannot be enclosed in braces (curly brackets), and which is that form mostly used in theoretical physics in both special theory of relativity and general theory of relativity (GTR), e.g. $\left\{x_{\text{μ}}\right\}$ or $\left\{x^{\text{μ}}\right\}$, $x_{\mu }$ or $x^{\mu }$.

Note 2 to entry: In STR, the time-related component can be expressed as an imaginary quantity, using symbol $\mathrm{j}$ as the imaginary unit. Then, pseudo-Euclidean metric can be used with rules of Euclidean metric but allowing negative magnitudes $\left|\underset{\_}{\underset{\_}{x}}\right|<0$ and zero magnitudes $\left|\underset{\_}{\underset{\_}{x}}\right|=0$ even for $\underset{\_}{\underset{\_}{x}}\ne 0$. See IEV 113-07-18.

In case time-related component is real, it is denoted as the fourth component $x_4=ct$ and the space-related components are $x_1,x_2,x_3$. The corresponding components of the metric tensor yielding the four-scalar product and squared four-magnitude have opposite signs, e.g., for flat space-time in STR $g_{11}=g_{22}=g_{33}=1$, $g_{44}=-1$ or $g_{11}=g_{22}=g_{33}=-1$, $g_{44}=1$. In GTR, the non-diagonal metric tensor is used.

Note 3 to entry: The representations used in this part of IEC 60050 are $\underset{\_}{\underset{\_}{x}}=(x_0,x_1,x_2,x_3)=\left\{x_{\mu }\right\}=\left(x_0,\left\{x_m\right\}\right)$, where $x_0$ is the time-related component and $x_m$ are the space-related components. In three-dimensional space, components of three-dimensional vectors are denoted using lowercase Latin letters for indices $(i,j,k,l,m,\dots )$.

In four-dimensional space, components of four-dimensional vectors are denoted using lowercase Greek letters for indices, $\left(\text{ι},\text{κ},\text{λ},\text{μ},\text{ν},\dots \right)$. In STR, indices range usually from 0 to 3, where 0 is used for the imaginary time-related component, and in GTR, indices range usually from 1 to 4 where 4 is used for the real time-related component.

Examples in STR are the position four-vector $\underset{\_}{\underset{\_}{x}}:=(x_0,x_1,x_2,x_3)=(\mathrm{j}{c}_{0}t,x,y,z)$ and the electromagnetic four-potential $\underset{\_}{\underset{\_}{A}}=\left(\mathrm{j}V/c_0\text{;}{A}_x,A_y,A_z\right)=\left(\mathrm{j}V/c_0\text{;}\overrightarrow{A}\right)$.

Note 4 to entry: If there is no risk of misunderstanding, “free index symbolic” is used, e.g. a component $x_{\mu }$ instead of full vector $\left\{x_{\text{μ}}\right\}$. Index $\text{μ}$ is then called “free index”.