The Principle of Relativity · Albert Einstein
(B)
(44) $$ B_{\mu\nu} = R_{\mu\nu} + S_{\mu\nu} $$ $$ R_{\mu\nu} = - \frac{\partial}{\partial x_{\alpha}} \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} + \begin{Bmatrix}\mu & & \alpha\ \beta&\end{Bmatrix} \begin{Bmatrix}\nu & & \beta\ \alpha&\end{Bmatrix} $$ $$ S_{\mu\nu} = \frac{\partial \log \sqrt{-g}}{\partial x_{\mu} \partial x_{\nu}} - \begin{Bmatrix}\mu & & \nu\ \alpha&\end{Bmatrix} \frac{\partial \log \sqrt{-g}}{\partial x_{\alpha}} $$ _Remarks upon the choice of co-ordinates._—It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that √(-_g_) = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. It is specially true for the tensor B_{μν}, which plays a fundamental rôle in the theory. By this simplification, S_{μν} vanishes of itself so that tensor B_{μν} reduces to R_{μν}. I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special case.
Text from Project Gutenberg, public domain in the United States.