Editing Four-vector (section)

===Four-vectors in a real-valued basis===

A '''four-vector''' ''A'' is a vector with a "timelike" component and three "spacelike" components, and can be written in various equivalent notations:<ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (BSA), 2006, {{ISBN|0-07-145545-0}}</ref>

<math display="block"> \begin{align}  
  \mathbf{A} & = \left(A^0, \, A^1, \, A^2, \, A^3\right) \\
  & = A^0\mathbf{E}_0 + A^1 \mathbf{E}_1 + A^2 \mathbf{E}_2 + A^3  \mathbf{E}_3 \\
  & = A^0\mathbf{E}_0 + A^i \mathbf{E}_i \\
  & = A^\alpha\mathbf{E}_\alpha
\end{align}</math>

where ''A<sup>α</sup>'' is the magnitude component and '''E'''<sub>α</sub> is the [[basis vector]] component; note that both are necessary to make a vector, and that when ''A<sup>α</sup>'' is seen alone, it refers strictly to the <em>components</em> of the vector.

The upper indices indicate [[Covariance and contravariance of vectors|contravariant]] components. Here the standard convention is that Latin indices take values for spatial components, so that ''i'' = 1, 2, 3, and Greek indices take values for space ''and time'' components, so ''α'' = 0, 1, 2, 3, used with the [[summation convention]]. The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or [[raising and lowering indices]].

In special relativity, the spacelike basis '''E'''<sub>1</sub>, '''E'''<sub>2</sub>, '''E'''<sub>3</sub> and components ''A''<sup>1</sup>, ''A''<sup>2</sup>, ''A''<sup>3</sup> are often [[Cartesian coordinates|Cartesian]] basis and components:

<math display="block"> \begin{align}  
  \mathbf{A} & = \left(A_t, \, A_x, \, A_y, \, A_z\right) \\
  & = A_t \mathbf{E}_t + A_x \mathbf{E}_x + A_y \mathbf{E}_y + A_z  \mathbf{E}_z \\
\end{align}</math>

although, of course, any other basis and components may be used, such as [[spherical polar coordinates]]

<math display="block"> \begin{align}  
  \mathbf{A} & = \left(A_t, \, A_r, \, A_\theta, \, A_\phi\right) \\
  & = A_t \mathbf{E}_t + A_r \mathbf{E}_r + A_\theta \mathbf{E}_\theta + A_\phi \mathbf{E}_\phi \\
\end{align}</math>

or [[cylindrical polar coordinates]],

<math display="block"> \begin{align}  
  \mathbf{A} & = (A_t, \, A_r, \, A_\theta, \, A_z) \\
  & = A_t \mathbf{E}_t + A_r \mathbf{E}_r + A_\theta \mathbf{E}_\theta + A_z \mathbf{E}_z \\
\end{align}</math>

or any other [[orthogonal coordinates]], or even general [[curvilinear coordinates]]. Note the coordinate labels are always subscripted as labels and are not indices taking numerical values. In general relativity, local curvilinear coordinates in a local basis must be used. Geometrically, a four-vector can still be interpreted as an arrow, but in spacetime - not just space. In relativity, the arrows are drawn as part of [[Minkowski diagram]] (also called ''spacetime diagram''). In this article, four-vectors will be referred to simply as vectors.

It is also customary to represent the bases by [[column vector]]s:

<math display="block">
  \mathbf{E}_0 = \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix} \,,\quad
  \mathbf{E}_1 = \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix} \,,\quad
  \mathbf{E}_2 = \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix} \,,\quad
  \mathbf{E}_3 = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}
</math>

so that:

<math display="block"> \mathbf{A} = \begin{pmatrix} A^0 \\ A^1 \\ A^2 \\ A^3 \end{pmatrix} </math>

The relation between the [[covariant vector|covariant]] and contravariant coordinates is through the [[Minkowski metric|Minkowski]] [[metric tensor]] (referred to as the metric), ''η'' which [[raising and lowering indices|raises and lowers indices]] as follows:

<math display="block">A_{\mu} =  \eta_{\mu \nu} A^{\nu} \,, </math>

and in various equivalent notations the covariant components are:

<math display="block"> \begin{align}  
  \mathbf{A} & = (A_0, \, A_1, \, A_2, \, A_3) \\
  & = A_0\mathbf{E}^0 + A_1 \mathbf{E}^1 + A_2 \mathbf{E}^2 + A_3  \mathbf{E}^3 \\
  & = A_0\mathbf{E}^0 + A_i \mathbf{E}^i \\
  & = A_\alpha\mathbf{E}^\alpha\\
\end{align}</math>

where the lowered index indicates it to be [[Covariance and contravariance of vectors|covariant]]. Often the metric is diagonal, as is the case for [[orthogonal coordinates]] (see [[line element]]), but not in general [[curvilinear coordinates]].

The bases can be represented by [[row vector]]s:

<math display="block">\begin{align}
  \mathbf{E}^0 &= \begin{pmatrix} 1 & 0 & 0 & 0 \end{pmatrix} \,, &
  \mathbf{E}^1 &= \begin{pmatrix} 0 & 1 & 0 & 0 \end{pmatrix} \,, \\[1ex]
  \mathbf{E}^2 &= \begin{pmatrix} 0 & 0 & 1 & 0 \end{pmatrix} \,, &
  \mathbf{E}^3 &= \begin{pmatrix} 0 & 0 & 0 & 1 \end{pmatrix},
\end{align}</math>
so that:
<math display="block"> \mathbf{A} = \begin{pmatrix} A_0 & A_1 & A_2 & A_3 \end{pmatrix} </math>

The motivation for the above conventions are that the inner product is a scalar, see below for details.