Originally published at 狗和留美者不得入内. You can comment here or there.

I can remember the velocity addition formula corresponding to Lorentz transformations along a single dimension, , off the top of my head. It might mistakenly give the impression that the Lorentz group in general is abelian. It is not for four dimension. A Lorentz boost along one direction and one along another direction do not commute. The Lorentz boost matrix is given by

Moreover, tells us that to obtain the inverse, we flip the sign of the entries in the boost matrix such that one index corresponds to time coordinate and the other index corresponds to a space coordinate. This is given formulaically by

wherein is the matrix corresponding to Minkowski metric. In Einstein notation, we denote with and raising the lower index and then lowering the upper index in it gives

Raising an index corresponds to applying Minkowski matrix from the left, in which case for each entry in the result, we iterate along the entries of some column of the Lorentz matrix, which is along the row index. Lowering an index then corresponds to applying Minkowski matrix from the right, which demonstrates the equivalence of and , with . equates towhich if appearing in expression involving Lorentz transformations of tensors of higher rank conveniently cancels out to the identity matrix. Letting , we obtain

where is the matrix corresponding to the transform of the contravariant four-vector for a given Lorentz transformation.