$$\require{cancel}$$

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: