Accordingly the difference between the relativity treatment and the classical treatment is as follows. In both theories it is recognised that in certain cases $\mf{T}_{\mu}^{\nu}$~is conserved, but that in the general case this conservation breaks down. The relativity theory treats the general case by discovering a more exact formulation of what happens to~$\mf{T}_{\mu}^{\nu}$ when it is not strictly conserved, viz.\ $\mf{T}_{\mu\nu}^{\nu} = 0$. The classical theory treats it by introducing a supplementary energy, so that conservation is still maintained but for a different quantity, viz.\ $\dd\mf{S}_{\mu}^{\nu}/\dd x_{\nu} = 0$. The relativity treatment adheres to the physical quantity and modifies the law; the classical treatment adheres to the law and modifies the physical quantity. Of course, both methods should be expressible by equivalent formulae; and we have in our previous work spoken of $\mf{T}_{\mu\nu}^{\nu} = 0$ as the law of conservation of energy and momentum, because, although it is not formally a law of conservation, it expresses exactly the phenomena which classical mechanics attributes to conservation. The relativity treatment has enabled us to discover the exact equations, and we may now apply these to obtain the corresponding exact expression for the quantity~$\mf{S}_{\mu}^{\nu}$ introduced in the classical treatment. It is clear that $\mf{t}_{\mu}^{\nu}$ and therefore~$\mf{S}_{\mu}^{\nu}$ cannot be tensor-densities, because $\mf{t}_{\mu}^{\nu}$~vanishes \index{Pseudo-energy-tensor}% when natural coordinates are used at a point, and would therefore always vanish if it were a tensor-density. We call~$\mf{t}_{\mu}^{\nu}$ the pseudo-tensor-density of potential energy. The explicit value of~$\mf{t}_{\mu}^{\nu}$ must be calculated from the condition~\Eq{(59.3)}, or \begin{align*} \frac{\dd\mf{t}_{\mu}^{\nu}}{\dd x_{\nu}} &= -\frac{\dd\mf{T}_{\mu}^{\nu}}{\dd x_{\nu}} \\ &= -\tfrac{1}{2}\mf{T}^{\alpha\beta} \frac{\dd g_{\alpha\beta}}{\dd x_{\mu}} \quad\text{by~\Eq{(55.6)}} \\ &= -\frac{1}{16\pi}\, \frac{\dd}{\dd x_{\nu}} \biggl\{\mf{g}_{\mu}^{\alpha\beta}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}} - g_{\mu}^{\nu} \mf{L}\biggr\}\quad\text{by~\Eq{(58.93)}.} \end{align*} Hence \[ 16\pi \mf{t}_{\mu}^{\nu} = g_{\mu}^{\nu} \mf{L} - \mf{g}_{\mu}^{\alpha\beta}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\nu}^{\alpha\beta}}. \Tag{(59.4)} \] \PageSep{136} This may remind us of the Hamiltonian integral of energy \[ -h = L - \sum q'\, \frac{\dd L}{\dd q'} \] in general dynamics. We can form a pseudo-scalar-density by contraction of~\Eq{(59.4)} \begin{align*} 16\pi\mf{t} &= 4\mf{L} - \mf{g}_{\mu}^{\alpha\beta}\, \frac{\dd\Typo{L}{\mf{L}}}{\dd\mf{g}_{\mu}^{\alpha\beta}} \\ &= 2\mf{L}\quad\text{by~\Eq{(58.72)}.} \end{align*} Thus we obtain the interesting comparison with~\Eq{(54.4)} \[ \left. \begin{aligned} \mf{L} &= 8\pi\mf{t}\Add{,} \\ \mf{G} &= 8\pi \mf{T}. \end{aligned} \right\} \Tag{(59.5)} \]