Editing Lorentz transformation (section)

===Vector transformations===

{{Further|Euclidean vector|vector projection}}

[[File:Lorentz boost any direction standard configuration.svg|upright=1.75|thumb|An observer in frame {{mvar|F}} observes {{mvar|F′}} to move with velocity {{math|'''v'''}}, while {{mvar|F′}} observes {{mvar|F}} to move with velocity {{math|−'''v'''}}. {{According to whom|The coordinate axes of each frame are still parallel|date=November 2020}} and orthogonal. The position vector as measured in each frame is split into components parallel and perpendicular to the relative velocity vector {{math|'''v'''}}.<br />'''Left:''' Standard configuration. '''Right:''' Inverse configuration.]]

The use of vectors allows positions and velocities to be expressed in arbitrary directions compactly. A single boost in any direction depends on the full relative [[velocity vector]] {{math|'''v'''}} with a magnitude {{math|1={{abs|'''v'''}} = ''v''}} that cannot equal or exceed {{mvar|c}}, so that {{math|0 ≤ ''v'' < ''c''}}.

Only time and the coordinates parallel to the direction of relative motion change, while those coordinates perpendicular do not. With this in mind, split the spatial [[position vector]] {{math|'''r'''}} as measured in {{mvar|F}}, and {{math|'''r′'''}} as measured in {{mvar|F′}}, each into components perpendicular (⊥) and parallel ( ‖ ) to {{math|'''v'''}},
<math display="block">\mathbf{r}=\mathbf{r}_\perp+\mathbf{r}_\|\,,\quad \mathbf{r}' = \mathbf{r}_\perp' + \mathbf{r}_\|' \,, </math>
then the transformations are
<math display="block">\begin{align}
              t'  &= \gamma \left(t - \frac{\mathbf{r}_\parallel \cdot \mathbf{v}}{c^2} \right) \\
   \mathbf{r}_\|' &= \gamma (\mathbf{r}_\| - \mathbf{v} t) \\
\mathbf{r}_\perp' &= \mathbf{r}_\perp
\end{align}</math>
where {{math|·}} is the [[dot product]]. The Lorentz factor {{mvar|γ}} retains its definition for a boost in any direction, since it depends only on the magnitude of the relative velocity. The definition {{math|1='''β''' = '''v'''/''c''}} with magnitude {{math|0 ≤ ''β'' < 1}} is also used by some authors.

Introducing a [[unit vector]] {{math|1='''n''' = '''v'''/''v'' = '''β'''/''β''}} in the direction of relative motion, the relative velocity is {{math|1='''v''' = ''v'''''n'''}} with magnitude {{mvar|v}} and direction {{math|'''n'''}}, and [[vector projection]] and rejection give respectively
<math display="block">\mathbf{r}_\parallel = (\mathbf{r}\cdot\mathbf{n})\mathbf{n}\,,\quad \mathbf{r}_\perp = \mathbf{r} - (\mathbf{r}\cdot\mathbf{n})\mathbf{n}</math>

Accumulating the results gives the full transformations,

{{Equation box 1
|title='''Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
          t'  &= \gamma \left(t - \frac{v\mathbf{n}\cdot \mathbf{r}}{c^2} \right) \,, \\
  \mathbf{r}' &= \mathbf{r} + (\gamma-1)(\mathbf{r}\cdot\mathbf{n})\mathbf{n} - \gamma t v\mathbf{n} \,.
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The projection and rejection also applies to {{math|'''r′'''}}. For the inverse transformations, exchange {{math|'''r'''}} and {{math|'''r′'''}} to switch observed coordinates, and negate the relative velocity {{math|'''v''' → −'''v'''}} (or simply the unit vector {{math|'''n''' → −'''n'''}} since the magnitude {{mvar|v}} is always positive) to obtain

{{Equation box 1
|title='''Inverse Lorentz boost''' (''in direction '' {{math|'''n'''}} '' with magnitude '' {{mvar|v}})
|indent =:
|equation =
<math>\begin{align}
           t &= \gamma \left(t' + \frac{\mathbf{r}' \cdot v\mathbf{n}}{c^2} \right) \,, \\
  \mathbf{r} &= \mathbf{r}' + (\gamma-1)(\mathbf{r}'\cdot\mathbf{n})\mathbf{n} + \gamma t' v\mathbf{n} \,,
\end{align}</math>
|cellpadding
|border = 1
|border colour = black
|background colour=white}}

The unit vector has the advantage of simplifying equations for a single boost, allows either {{math|'''v'''}} or {{math|'''β'''}} to be reinstated when convenient, and the rapidity parametrization is immediately obtained by replacing {{mvar|β}} and {{math|''βγ''}}. It is not convenient for multiple boosts.

The vectorial relation between relative velocity and rapidity is<ref>{{harvnb|Barut|1964|page=18–19}}</ref>
<math display="block"> \boldsymbol{\beta} = \beta \mathbf{n} = \mathbf{n} \tanh\zeta \,,</math>
and the "rapidity vector" can be defined as
<math display="block"> \boldsymbol{\zeta} = \zeta\mathbf{n} = \mathbf{n}\tanh^{-1}\beta \,, </math>
each of which serves as a useful abbreviation in some contexts. The magnitude of {{math|'''ζ'''}} is the absolute value of the rapidity scalar confined to {{math|0 ≤ ''ζ'' < ∞}}, which agrees with the range {{math|0 ≤ ''β'' < 1}}.