Again \begin{align*} \delta (\kappa_{\alpha} \kappa^{\alpha} \sqrt{-g}) &= \kappa_{\alpha} \kappa_{\beta}\, \delta(g^{\alpha\beta} \sqrt{-g}) + g^{\alpha\beta} \sqrt{-g} (\kappa_{\alpha}\, \delta\kappa_{\beta} + \kappa_{\beta}\, \delta\kappa_{\alpha}) \\ &= \kappa_{\alpha} \kappa_{\beta} \sqrt{-g} (\delta g^{\alpha\beta} + \tfrac{1}{2} g^{\alpha\beta} g^{\mu\nu}\, \delta g_{\mu\nu}) + 2g^{\alpha\beta} \sqrt{-g}\, \kappa_{\beta}\, \delta\kappa_{\alpha}) \\ &= \kappa_{\alpha} \kappa_{\beta} \sqrt{-g} (-g^{\mu\alpha} g^{\nu\beta} + \tfrac{1}{2} g^{\alpha\beta} g^{\mu\nu})\, \delta g_{\mu\nu} + 2\kappa^{\alpha} \sqrt{-g}\, \delta\kappa_{\alpha} \\ &= \sqrt{-g} (-\kappa^{\mu} \kappa^{\nu} + \tfrac{1}{2} g^{\mu\nu} \kappa_{\alpha} \kappa^{\alpha})\, \delta g_{\mu\nu} + 2\kappa^{\alpha} \sqrt{-g}\, \delta\kappa_{\alpha}. \end{align*} Hence \begin{align*} \frac{\Ham}{\Ham g_{\mu\nu}} (\kappa_{\alpha} \kappa^{\alpha}) &= (-\kappa^{\mu} \kappa^{\nu} + \tfrac{1}{2} g^{\mu\nu} \kappa_{\alpha} \kappa^{\alpha}), \Tag{(90.41)} \\ \frac{\Ham}{\Ham \kappa_{\alpha}} (\kappa_{\alpha} \kappa^{\alpha}) &= 2\kappa^{\alpha}. \Tag{(90.42)} \end{align*} Hamiltonian derivatives of the other terms in~\Eq{(90.3)} have already been found in \Eq{(60.43)}, \Eq{(79.31)} and~\Eq{(79.32)}. Collecting these results we have \begin{align*} \frac{1}{8\lambda}\, \frac{\Ham A}{\Ham g_{\mu\nu}} &= -(G^{\mu\nu} - \tfrac{1}{2} g^{\mu\nu} G) - 6(\kappa^{\mu} \kappa^{\nu} - \tfrac{1}{2} g^{\mu\nu} \kappa_{\alpha} \kappa^{\alpha}) - \lambda g^{\mu\nu} - 2\beta E^{\mu\nu} \\ &= 8\pi T^{\mu\nu} - 2\beta E^{\mu\nu} - 6(\kappa^{\mu} \kappa^{\nu} - \tfrac{1}{2} g^{\mu\nu} \kappa_{\alpha} \kappa^{\alpha}) \Tag{(90.51)} \end{align*} by~\Eq{(54.71)}; and \[ \frac{1}{8\lambda}\, \frac{\Ham A}{\Ham \kappa_{\mu}} = 12\kappa^{\mu} + 4\beta J^{\mu}. \Tag{(90.52)} \] \PageSep{211} If the hypothesis~\Eq{(90.2)} is correct, these must vanish. The vanishing of~\Eq{(90.51)} shows that the whole energy-tensor consists of the electromagnetic energy-tensor together with another term, which must presumably be identified with the material energy-tensor attributable to the binding forces of the electrons\footnotemark.\footnotetext {I doubt if this is the right interpretation. See the end of \SecRef{100}.} The constant~$2\beta/8\pi$ correlates the natural gravitational and electromagnetic units. The material energy-tensor, being the difference between the whole tensor and the electromagnetic part, is accordingly \[ M^{\mu\nu} = \frac{3}{4\pi} (\kappa^{\mu} \kappa^{\nu} - \tfrac{1}{2} g^{\mu\nu} \kappa_{\alpha} \kappa^{\alpha}). \Tag{(90.61)} \] Hence, multiplying by~$g_{\mu\nu}$, \[ \rho_{0} = M = -\frac{3}{4\pi}\, \kappa_{\alpha} \kappa^{\alpha}. \Tag{(90.62)} \]