Editing Four-vector (section)

=== Lorentz transformation ===

{{main|Lorentz transformation}}

Given two inertial or rotated [[frame of reference|frames of reference]], a four-vector is defined as a quantity which transforms according to the [[Lorentz transformation]] matrix&nbsp;'''Λ''':
<math display="block">\mathbf{A}' = \boldsymbol{\Lambda}\mathbf{A}</math>

In index notation, the contravariant and covariant components transform according to, respectively:
<math display="block">{A'}^\mu = \Lambda^\mu {}_\nu A^\nu \,, \quad{A'}_\mu = \Lambda_\mu {}^\nu A_\nu</math>
in which the matrix {{math|'''Λ'''}} has components {{math|Λ''<sup>μ</sup><sub>ν</sub>''}} in row&nbsp;{{math|''μ''}} and column&nbsp;{{math|''ν''}}, and the matrix {{math|('''Λ'''<sup>−1</sup>)<sup>T</sup>}} has components {{math|Λ''<sub>μ</sub><sup>ν</sup>''}} in row&nbsp;{{math|''μ''}} and column&nbsp;{{math|''ν''}}.

For background on the nature of this transformation definition, see [[tensor#Definition|tensor]]. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see [[Special relativity#Transformations of physical quantities between reference frames|special relativity]].

====Pure rotations about an arbitrary axis ====

For two frames rotated by a fixed angle {{math|''θ''}} about an axis defined by the [[unit vector]]:

<math display="block">\hat{\mathbf{n}} = \left(\hat{n}_1, \hat{n}_2, \hat{n}_3\right)\,,</math>

without any boosts, the matrix '''Λ''' has components given by:<ref>{{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| page=[https://archive.org/details/mcgrawhillencycl1993park/page/1333 1333]| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park/page/1333}}</ref>

<math display="block">\begin{align}
                 \Lambda_{00} &= 1 \\
  \Lambda_{0i} = \Lambda_{i0} &= 0 \\
                 \Lambda_{ij} &= \left(\delta_{ij} - \hat{n}_i \hat{n}_j\right) \cos\theta - \varepsilon_{ijk} \hat{n}_k \sin\theta + \hat{n}_i \hat{n}_j
\end{align}</math>

where ''δ<sub>ij</sub>'' is the [[Kronecker delta]], and ''ε<sub>ijk</sub>'' is the [[three-dimensional]] [[Levi-Civita symbol]]. The spacelike components of four-vectors are rotated, while the timelike components remain unchanged.

For the case of rotations about the ''z''-axis only, the spacelike part of the Lorentz matrix reduces to the [[rotation matrix]] about the ''z''-axis:

<math display="block">
  \begin{pmatrix}
    {A'}^0 \\ {A'}^1 \\ {A'}^2 \\ {A'}^3
  \end{pmatrix} =
  \begin{pmatrix}
    1 & 0 & 0 & 0 \\
    0 & \cos\theta & -\sin\theta & 0 \\
    0 & \sin\theta &  \cos\theta & 0 \\
    0 & 0 & 0 & 1 \\
  \end{pmatrix}
  \begin{pmatrix}
    A^0 \\ A^1 \\ A^2 \\ A^3
  \end{pmatrix}\ .
</math>

====Pure boosts in an arbitrary direction====

[[File:Standard conf.png|right|thumb|300px|Standard configuration of coordinate systems; for a Lorentz boost in the ''x''-direction.]]

For two frames moving at constant relative three-velocity '''v''' (not four-velocity, [[#Four-velocity|see below]]), it is convenient to denote and define the relative velocity in units of ''c'' by:

<math display="block"> \boldsymbol{\beta} = (\beta_1,\,\beta_2,\,\beta_3) = \frac{1}{c}(v_1,\,v_2,\,v_3) = \frac{1}{c}\mathbf{v} \,. </math>

Then without rotations, the matrix '''Λ''' has components given by:<ref>Gravitation, J.B. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISAN 0-7167-0344-0</ref>
<math display="block">\begin{align}
                 \Lambda_{00} &= \gamma, \\
  \Lambda_{0i} = \Lambda_{i0} &= -\gamma \beta_{i}, \\
  \Lambda_{ij} = \Lambda_{ji} &= (\gamma - 1)\frac{\beta_{i}\beta_{j}}{\beta^2} + \delta_{ij} = (\gamma - 1)\frac{v_i v_j}{v^2} + \delta_{ij}, \\
\end{align}</math>
where the [[Lorentz factor]] is defined by:
<math display="block">\gamma = \frac{1}{\sqrt{1 - \boldsymbol{\beta}\cdot\boldsymbol{\beta}}} \,,</math>
and {{math|''δ<sub>ij</sub>''}} is the [[Kronecker delta]]. Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.

For the case of a boost in the ''x''-direction only, the matrix reduces to;<ref>Dynamics and Relativity, J.R. Forshaw, B.G. Smith, Wiley, 2009, ISAN 978-0-470-01460-8</ref><ref>Relativity DeMystified, D. McMahon, Mc Graw Hill (ASB), 2006, ISAN 0-07-145545-0</ref>

<math display="block">
  \begin{pmatrix}
    A'^0 \\ A'^1 \\ A'^2 \\ A'^3
  \end{pmatrix} =
  \begin{pmatrix}
     \cosh\phi &-\sinh\phi & 0 & 0 \\
    -\sinh\phi  & \cosh\phi & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1 \\
  \end{pmatrix}
  \begin{pmatrix}
    A^0 \\ A^1 \\ A^2 \\ A^3
  \end{pmatrix} 
</math>

Where the [[rapidity]] {{math|''ϕ''}} expression has been used, written in terms of the [[hyperbolic function]]s:
<math display="block">\gamma = \cosh \phi</math>

This Lorentz matrix illustrates the boost to be a ''[[hyperbolic rotation]]'' in four dimensional spacetime, analogous to the circular rotation above in three-dimensional space.