13.2: Lorentz Transformation Matrix and Metric Tensor
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this section, we’ve joined space and time in a single four-vector and defined a new inner product on the space of those four-vectors. In Chapter 11 we defined the Lorentz transformations of the space and time coordinates, which are linear transformations. Linear transformations can, of course, be represented by matrices, and for our four-vectors, we can write down the appropriate Lorentz transformation matrix, rewriting equation (11.12) as a vector equation:
¯x′=L¯x
Here L is a 4×4 matrix:
L=(γ(u)−γ(u)uc00−γ(u)ucγ(u)0000100001)
Likewith the four-vectors, we start labeling the rows and columns of L with index 0. To indicate the difference with matrices in regular space, it is conventional to indicate indices of regular-space vectors and matrices with Roman letters (like vi for the ith component of vector v, and Aij for the ith row, jth column of matrix A), and those of Minkowski-space vectors and matrices with Greek letters - so we write xμ for the μth component of the four-vector ¯x, where μ can be 0, 1, 2, or 3.
We can also write Equation ??? in index form: