Lorentz transformations

A fundamental class of coordinate transformations are the Lorentz transformations \(\Lambda\): \[\begin{equation} \require{physics} x'^{\mu} = \Lambda^\mu_\nu x^{\nu} % \thinspace , \end{equation}\] which are transformations that leave the space-time interval \(\dd{s}^2\) (cfr. equation \(\eqref{eq:space-time_interval}\)) invariant. The matrix representation of this transformation tensor is given (in full) as \[\begin{equation} (\Lambda^\mu_\nu) = \begin{pmatrix} \Lambda^0_0 & \Lambda^0_1 & \Lambda^0_2 & \Lambda^0_3 \\ \Lambda^1_0 & \Lambda^1_1 & \Lambda^1_2 & \Lambda^1_3 \\ \Lambda^2_0 & \Lambda^2_1 & \Lambda^2_2 & \Lambda^2_3 \\ \Lambda^3_0 & \Lambda^3_1 & \Lambda^3_2 & \Lambda^3_3 \end{pmatrix} \thinspace , \end{equation}\] and the requirement of an invariatn space-time interval leads to the following constraint on its components: \[\begin{equation} \Lambda^\alpha_\mu % \eta_{\alpha \beta} % \Lambda^\beta_\nu % = \eta_{\mu \nu} % \thinspace . \end{equation}\] The entries of the transposed matrix are given by \[\begin{equation} (\Lambda^\text{T})^\mu_\nu = \Lambda^\nu_\mu % \thinspace . \end{equation}\]

The inverse transformation is written as \[\begin{equation} x^\mu = (\Lambda^{-1})^\mu_\nu x'^\nu % \thinspace , \end{equation}\] and its components can be calculated through \[\begin{equation} (\Lambda^{-1})^\mu_\nu \equiv \Lambda_\nu^\mu = \eta_{\nu \alpha} \eta^{\mu \beta} \Lambda^\alpha_\beta \thinspace . \end{equation}\] For the components of the transformations and their inverses, the equations \[\begin{equation} \Lambda^\mu _\alpha % \Lambda_\nu ^\alpha % = \delta^\mu _\nu \end{equation}\] and \[\begin{equation} \Lambda_\mu ^\alpha % \Lambda^\nu _\alpha % = \delta_\mu ^\nu \end{equation}\] hold (precisely because they are each other’s inverse), whis means that the components of \(\Lambda^{-1}\) are given by \[\begin{equation} (\Lambda_\mu ^\nu) = \begin{pmatrix} \Lambda^0_0 & -\Lambda^1_0 & -\Lambda^2_0 & -\Lambda^3_0 \\ -\Lambda^0_1 & \Lambda^1_1 & \Lambda^2_1 & \Lambda^3_1 \\ -\Lambda^0_2 & \Lambda^1_2 & \Lambda^2_2 & \Lambda^3_2 \\ -\Lambda^0_3 & \Lambda^1_3 & \Lambda^2_3 & \Lambda^3_3 \end{pmatrix} \thinspace . \end{equation}\]

The 4-gradient (cfr. equation \(\eqref{eq:4-gradient}\)) can be used to write the components of the Lorentz transformations (and their inverses) as \[\begin{equation} \Lambda^\mu_\nu = \partial_\nu x^{' \mu} \end{equation}\] and \[\begin{equation} \Lambda_\mu^\nu = \partial'_\mu x^{\nu} % \thinspace . \end{equation}\]

A rank-\(n\) Lorentz tensor is finally defined as an \(n\)-index object whose \(4^n\) components transform contravariantly or covariantly, depending on if the indices are superscripts or subscripts. An example is the following rank-\(2\) tensor: \[\begin{equation} {T'}^\mu _\nu = % \Lambda^\mu _\alpha % \Lambda_\nu ^\beta % T^\alpha _\beta% \thinspace . \end{equation}\] Specifially, the Minkowski metric is a rank-\(2\) tensor which is identical in each intertial system: \[\begin{equation} \eta'_{\mu \nu} = \eta_{\mu \nu} % \thinspace . \end{equation}\]