SQUARE45

For two inertial frames

S

and

S'

moving relative to each other with a constant relative velocity

\mathbf{v}

, the transformation of coordinates

x^{\mu} = (ct, x, y, z)

is given by the Lorentz transformation matrix

\Lambda^{\mu}{}_{\nu}

:\n

x'^{\mu} = \Lambda^{\mu}{}_{\nu} x^{\nu}

\nFor a boost along the

x

-axis, the matrix elements are:\n

\Lambda = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\ -\gamma \beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}

\nWhere

\beta = v/c

and

\gamma = \frac{1}{\sqrt{1 - \beta^2}}

Lorentz Transformations