Editing Four-vector (section)

====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.