In empty space the expression for~$\mf{t}_{\mu}^{\nu}$ can be simplified. Since $\mf{G} = 0$, \Eq{(58.8)}~becomes \[ \mf{L} = \frac{\dd}{\dd x_{\nu}} \left(\mf{g}^{\alpha\beta}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}}\right). \] Hence \begin{align*} 16\pi\mf{t}_{\mu}^{\nu} &= \frac{\dd}{\dd x_{\mu}}\left(\mf{g}^{\alpha\beta}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}}\right) - \mf{g}_{\mu}^{\alpha\beta}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}}\displaybreak[0] \\ &= \mf{g}^{\alpha\beta}\, \frac{\dd}{\dd x_{\mu}}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}}\displaybreak[0] \\ &= \mf{g}^{\alpha\beta}\, \frac{\dd}{\dd x_{\mu}} \left[ -\{\alpha\beta, \nu\} + g_{\alpha}^{\nu} \frac{\dd}{\dd x_{\beta}} \log \sqrt{-g}\right] \Tag{(59.6)} \end{align*} by~\Eq{(58.52)}. \Section{60.}{Action} The invariant integral \[ A = \iiiint \rho_{0} \sqrt{-g}\, d\tau \Tag{(60.11)} \] represents the \emph{action} of the matter in a four-dimensional region. By~\Eq{(49.42)}, \begin{align*} A &= \iiiint \rho_{0}\, dW\, ds \\ &= \iint m\, ds, \Tag{(60.12)} \end{align*} where $m$~is the invariant mass or energy. Thus the action of a particle having energy~$m$ for a proper-time~$ds$ is equal to~$m\, ds$, agreeing with the definition of action in ordinary mechanics as energy multiplied by time. By~\Eq{(54.6)} another form is \[ A = \frac{1}{8\pi} \iiiint G\sqrt{-g}\, d\tau, \Tag{(60.2)} \] so that (ignoring the numerical factor) $G\sqrt{-g}$, or~$\mf{G}$, represents the action-density of the gravitational field. Note that material action and gravitational action are alternative aspects of the same thing; they are not to be added together to give a total action. But in stating that the gravitational action and the material action are necessarily the same thing, we have to bear in mind a very peculiar conception which is almost always associated with the term Action. From its first introduction, action has always been looked upon as something whose sole \Foreign{raison \Typo{d'étre}{d'être}} is to be varied---and, moreover, \emph{varied in such a way as to defy the laws \PageSep{138} of nature!} We have thus to remember that when a writer begins to talk about action, he is probably going to consider impossible conditions of the world. (That does not mean that he is talking nonsense---he brings out the important features of the possible conditions by comparing them with impossible conditions.) Thus we may not always \emph{disregard} the difference between material and gravitational action; it is impossible that there should be any difference, but then we are about to discuss impossibilities.