Next consider variations~$\delta\kappa_{\mu}$, the $g_{\mu\nu}$~remaining constant. We have \begin{align*} \delta (F^{\mu\nu} F_{\mu\nu} \sqrt{-g}) &= 2F^{\mu\nu} \sqrt{-g} · \delta F_{\mu\nu} \\ &= 2F^{\mu\nu} \sqrt{-g} \left(\frac{\dd(\delta\kappa_{\mu})}{\dd x_{\nu}} - \frac{\dd(\delta\kappa_{\nu})}{\dd x_{\mu}}\right) \\ &= 4F^{\mu\nu} \sqrt{-g} · \frac{\dd(\delta\kappa_{\mu})}{\dd x_{\nu}} \intertext{owing to the antisymmetry of~$F^{\mu\nu}$} &= -4\, \frac{\dd}{\dd x_{\nu}} (F^{\mu\nu} \sqrt{-g})\, \delta\kappa_{\mu} + 4\, \frac{\dd}{\dd x_{\nu}} (F^{\mu\nu} \sqrt{-g} · \delta\kappa_{\mu}). \end{align*} The second term can be omitted since it is a complete differential, and yields a surface-integral over the boundary where the variations have to vanish. Hence \begin{align*} \delta \int F^{\mu\nu} F_{\mu\nu} \sqrt{-g}\, d\tau &= -4\int \frac{\dd}{\dd x_{\nu}} (F^{\mu\nu} \sqrt{-g}) · \delta\kappa_{\mu}\, d\tau \\ &= -4\int J^{\mu}\, \delta\kappa_{\mu} · \sqrt{-g}\, d\tau \end{align*} by~\Eq{(73.75)}. This demonstrates~\Eq{(79.32)}. In a region free from electrons \[ T^{\mu\nu} - E^{\mu\nu} = 0. \] Hence by \Eq{(60.43)} and~\Eq{(79.31)} \[ \frac{\Ham}{\Ham g_{\mu\nu}} (G - 4\pi F^{\mu\nu} F_{\mu\nu}) = 0. \Tag{(79.4)} \] In the mechanical theory, neglecting electromagnetic fields, we found that the action~$G$ was stationary in regions containing no matter. We now see that when electromagnetic fields are included, the quantity which is stationary is $G - 4\pi F^{\mu\nu} F_{\mu\nu}$. Moreover it is stationary for variations~$\delta\kappa_{\mu}$ as well as~$\delta g_{\mu\nu}$, since when there are no electrons present $J^{\mu}$~must be zero. The quantity $G - 4\pi F^{\mu\nu} F_{\mu\nu}$ thus appears to be highly significant from the \PageSep{189} physical point of view, in the discrimination between matter (electrons) and electromagnetic fields. But this significance fails to appear in the analytical expression. Analytically the combination of the two invariants $G$ and $F^{\mu\nu} F_{\mu\nu}$---the one a spur, and the other a square of a length---appears to be quite nonsensical. We can only regard the present form of the expression as a stepping-stone to something simpler. It will appear later that $G - 4\pi F^{\mu\nu} F_{\mu\nu}$ is perhaps not the exact expression for the significant physical quantity; it may be an approximation to a form which is analytically simpler, in which the gravitational and electromagnetic variables appear in a more intelligible combination. Whereas material and gravitational actions are two aspects of the same thing, electromagnetic action stands entirely apart. There is no gravitational action associated with an electromagnetic field, owing to the identity $E = 0$. Thus any material or gravitational action is additional to electromagnetic action---if ``addition'' is appropriate in connection with quantities which are apparently of dissimilar nature.