Lorentz Boost

The energy and momentum $(E^\prime,\vec{p}^\prime)$ viewed from a frame moving with velocity $\vec{\beta} \equiv \vec{v}/c$ is give by

$$\left( \begin{array}{c} E^\prime \\ p_{\vert\vert}^\prime \... ...t} \end{array}\right)\quad\textrm{and}\quad p_T^\prime = p_T ,$$ (3.16)

where $\gamma = (1 - \beta^2)^{-1/2}$, and $p_T$ and $p_{\vert\vert}$ are the components of $\vec{p}$ perpendicular and parallel to $\vec{\beta}$, respectively.

Since $\beta$ and $\gamma$ are related, we can define a single ``rapidity'' parameter, $\omega$, as

$$\beta \equiv \tanh\omega .$$ (3.17)

Using the properties of the hyperbolic functions we have

$$\gamma = \cosh\omega \quad\textrm{and}\quad \gamma\beta = \sinh\omega .$$ (3.18)

We can thus write

$$\left( \begin{array}{c} E^\prime \\ p_{\vert\vert}^\prime \end{array}\right) = \left( \begin{array}{rr} \cosh\omega & -\sinh\omega \\ -\sinh\om... ...d{array}\right) \left( \begin{array}{c} E \\ p_{\vert\vert} \end{array}\right)$$

$$= \cosh\omega \left( \begin{array}{cc} 1 & -\tanh\omega \\ -\tanh\... ...array}\right) \left( \begin{array}{c} E \\ p_{\vert\vert} \end{array}\right) .$$ (3.19)

Other 4-vectors, such as the space-time coordinates of events transform in the same way.

We see that the Lorentz transformation may be regarded as a rotation through an imaginary angle $i\omega$ in the $ict$-$z$ plane.