13.2: Lorentz Transformation Matrix and Metric Tensor

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:

$$\overline{\boldsymbol{x}}^{\prime}=L \overline{\boldsymbol{x}}\label{13.2.1}$$

Here $L$ is a $4 \times 4$ matrix:

$$L=\left( \begin{array}{cccc}{\gamma(u)} & {-\gamma(u) \frac{u}{c}} & {0} & {0} \\ {-\gamma(u) \frac{u}{c}} & {\gamma(u)} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {1}\end{array}\right)\label{13.2.2}$$

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 $\boldsymbol{v}_{i}$ for the $i$th component of vector $v$, and $A_{i j}$ for the $i$th row, $j$th column of matrix $A$), and those of Minkowski-space vectors and matrices with Greek letters - so we write $x_{\mu}$ for the $\mu$th component of the four-vector $\overline{\boldsymbol{x}}$, where $\mu$ can be 0, 1, 2, or 3.

We can also write Equation $\ref{13.1}$ in index form: