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13.2: Lorentz Transformation Matrix and Metric Tensor

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  • 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: