Editing Four-vector (section)

====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>