+ \frac{\dd^{2} g_{\mu\tau}}{\dd x_{\rho}\, \dd x_{\sigma}} - \frac{\dd^{2} g_{\mu\sigma}}{\dd x_{\rho}\, \dd x_{\tau}} - \frac{\dd^{2} g_{\rho\tau}}{\dd x_{\mu}\, \dd x_{\sigma}}\right). \Tag{(52.4)} \] The rest of~$B_{\mu\tau\sigma\rho}$ is omitted because it consists of products of two vanishing factors ($3$-index symbols), so that after differentiation by~$\dd x_{\nu}$ one vanishing factor always remains. By the double interchange $\sigma$ for~$\tau$, $\rho$~for $\nu$, two terms in~\Eq{(52.4)} cancel out, leaving \[ \frac{1}{\sqrt{-g}}\, \frac{\dd}{\dd x_{\nu}} (G_{\mu}^{\nu} \sqrt{-g}) = \tfrac{1}{2} g^{\nu\tau} g^{\sigma\rho}\, \frac{\dd}{\dd x_{\nu}} \left( \frac{\dd^{2} g_{\rho\sigma}}{\dd x_{\mu}\, \dd x_{\tau}} - \frac{\dd^{2} g_{\rho\tau}}{\dd x_{\mu}\, \dd x_{\sigma}}\right). \Tag{(52.51)} \] Similarly \begin{align*} \tfrac{1}{2} g^{\alpha\beta}\, \frac{\dd G_{\alpha\beta}}{\dd x_{\mu}} &= \tfrac{1}{2} g^{\nu\tau}\, \frac{\dd G_{\nu\tau}}{\dd x_{\mu}} = \tfrac{1}{2} g^{\nu\tau}\, \frac{\dd}{\dd x_{\mu}} (g^{\sigma\rho} B_{\nu\tau\sigma\rho}) \\ &= \tfrac{1}{4} g^{\nu\tau} g^{\sigma\rho}\, \frac{\dd}{\dd x_{\mu}} \left(\frac{\dd^{2} g_{\rho\sigma}}{\dd x_{\nu}\, \dd x_{\tau}} + \frac{\dd^{2} g_{\nu\tau}}{\dd x_{\rho}\, \dd x_{\sigma}} - \frac{\dd^{2} g_{\nu\sigma}}{\dd x_{\rho}\, \dd x_{\tau}} - \frac{\dd^{2} g_{\rho\tau}}{\dd x_{\nu}\, \dd x_{\sigma}}\right) \\ &= \tfrac{1}{2} g^{\nu\tau} g^{\sigma\rho}\, \frac{\dd}{\dd x_{\mu}} \left( \frac{\dd^{2} g_{\rho\sigma}}{\dd x_{\nu}\, \dd x_{\tau}} - \frac{\dd^{2} g_{\rho\tau}}{\dd x_{\nu}\, \dd x_{\sigma}}\right), \Tag{(52.52)} \end{align*} since the double interchange $\sigma$ for~$\tau$, $\rho$~for $\nu$, causes two terms to become equal to the other two. Comparing \Eq{(52.51)} and \Eq{(52.52)} we see that the required result is established for coordinates chosen so as to have the property~\Eq{(52.3)} at the point considered; and since it is a tensor equation it must hold true for all systems of coordinates. \Section{53.}{The material energy-tensor} \index{Energy-tensor of matter}% \index{T@$T_{\mu\nu}$ (energy-tensor)}% Let $\rho_{0}$~be the proper-density of matter, and let $dx_{\mu}/ds$ refer to the motion of the matter; we write, as in~\Eq{(46.8)}, \[ T^{\mu\nu} = \rho_{0}\, \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}. \Tag{(53.1)} \] Then $T^{\mu\nu}$ (with the associated mixed and covariant tensors) is called the \emph{energy-tensor} of the matter. \PageSep{117} For matter moving with any velocity relative to Galilean coordinates, the coordinate-density~$\rho$ is given by \[ \rho = \rho_{0} \left(\frac{dt}{ds}\right)^{2}, \Tag{(53.2)} \] for, as explained in~\Eq{(14.2)}, the FitzGerald factor $\beta = dt/ds$ appears twice, once for the increase of mass with velocity and once for the contraction of volume.