The alternative formula~\Eq{(51.51)} may be used to calculate~$T_{\mu\nu}^{\nu}$, giving \[ \frac{\dd}{\dd x_{\nu}} \mf{T}_{\mu}^{\nu} = \tfrac{1}{2} \mf{T}^{\alpha\beta}\, \frac{\dd g_{\alpha\beta}}{\dd x_{\mu}}. \Tag{(55.6)} \] Retaining on the right only~$\mf{T}^{44}$, we have by comparison with~\Eq{(55.51)} \[ X,\ Y,\ Z = -\frac{1}{2}\, \frac{\dd g_{44}}{\dd x},\ -\frac{1}{2}\, \frac{\dd g_{44}}{\dd y},\ -\frac{1}{2}\, \frac{\dd g_{44}}{\dd z}. \Tag{(55.7)} \] \PageSep{124} Hence, for a static coordinate-system \begin{align*} X\, dx + Y\, dy + Z\, dz &= -\frac{1}{2}\left(\frac{\dd g_{44}}{\dd x}\, dx + \frac{\dd g_{44}}{\dd y}\, dy + \frac{\dd g_{44}}{\dd z}\, dz\right) \\ &= -\tfrac{1}{2} dg_{44}, \end{align*} so that $X$, $Y$, $Z$ are derivable from a potential \index{Potential!gravitational}% \[ \Omega = -\tfrac{1}{2} g_{44} + \text{const.} \] Choosing the constant so that $g_{44} = 1$ when $\Omega = 0$ \[ g_{44} = 1 - 2\Omega. \Tag{(55.8)} \] Special cases of this result will be found in~\Eq{(15.4)} and~\Eq{(38.8)}, $\Omega$~being the potential of the centrifugal force and of the Newtonian gravitational force respectively. Let us now briefly review the principal steps in our new derivation of the laws of mechanics and gravitation. We concentrate attention on the world-tensor~$T_{\mu}^{\nu}$ \emph{defined} by \[ T_{\mu}^{\nu} = -\frac{1}{8\pi} (G_{\mu}^{\nu} - \tfrac{1}{2} g_{\mu}^{\nu} G). \] The question arises how this tensor would be recognised in nature---what names has the practical observer given to its components? We suppose tentatively that when Galilean or natural coordinates are used $T_{4}^{4}$~is recognised as the amount of mass or energy per unit volume, $T_{1}^{4}$, $T_{2}^{4}$, $T_{3}^{4}$ as the negative momentum per unit volume, and the remaining components contain the stresses according to the detailed specifications in~\Eq{(53.91)}. This can only be tested by examining whether the components of~$T_{\mu}^{\nu}$ do actually obey the laws which mass, momentum and stress are known by observation to obey. For natural coordinates the empirical laws are expressed by $\dd T_{\mu}^{\nu}/\dd x_{\nu} = 0$, which is satisfied because our tensor from its definition has been proved to satisfy $(T_{\mu}^{\nu})_{\nu} = 0$ identically. When the coordinates are not natural, the identity $T_{\mu\nu}^{\nu} = 0$ gives the more general law \[ \frac{\dd}{\dd x_{\nu}} \mf{T}_{\mu}^{\nu} = \frac{1}{2}\, \frac{\dd g_{\alpha\beta}}{\dd x_{\mu}}\, \mf{T}^{\alpha\beta}. \] We attribute an abstract Galilean geometry to these coordinates, and should accordingly identify the components of~$T_{\mu}^{\nu}$ as before, just as though the coordinates were natural; but owing to the resulting confusion of unit mesh with unit natural volume, the tensor-densities $\mf{T}_{1}^{4}$, $\mf{T}_{2}^{4}$, $\mf{T}_{3}^{4}$, $\mf{T}_{4}^{4}$ will now be taken to represent the negative momentum and energy per unit volume.