The fundamental equation treated in the usual theory of electromagnetic waves is \[ \left(\frac{\dd^{2}}{\dd t^{2}} - \frac{\dd^{2}}{\dd x^{2}} - \frac{\dd^{2}}{\dd y^{2}} - \frac{\dd^{2}}{\dd z^{2}}\right) \kappa_{\mu} = 0, \Tag{(74.51)} \] which is the form taken by $\Wave\kappa_{\mu} = 0$ in Galilean coordinates. When the region of space-time is not flat we cannot immediately simplify $\Wave\kappa_{\mu}$ in this way; but we can make a considerable simplification by adopting natural coordinates at the point considered. In that case the $3$-index symbols (but not their derivatives) vanish, and \begin{align*} \Wave\kappa_{\mu} &= g^{\alpha\beta}(\kappa_{\mu})_{\alpha\beta} \\ &= g^{\alpha\beta} \left(\frac{\dd^{2}\kappa_{\mu}}{\dd x_{\alpha}\, \dd x_{\beta}} - \frac{\dd}{\dd x_{\alpha}} \{\mu\beta, \epsilon\} · \kappa_{\epsilon}\right). \end{align*} Hence the law of propagation $\Wave\kappa_{\mu} = 0$ becomes in natural coordinates \[ \left(\frac{\dd^{2}}{\dd t^{2}} - \frac{\dd^{2}}{\dd x^{2}} - \frac{\dd^{2}}{\dd y^{2}} - \frac{\dd^{2}}{\dd z^{2}}\right) \kappa_{\mu} = g^{\alpha\beta}\, \frac{\dd}{\dd x_{\alpha}} \{\mu\beta, \epsilon\} · \kappa_{\epsilon}. \Tag{(74.52)} \] At first sight this does not look very promising for a justification of the principle of equivalence. We cannot make all the derivatives $\dd\{\mu\beta, \epsilon\}/\dd x_{\alpha}$ vanish by any choice of coordinates, since these determine the Riemann-Christoffel tensor. It looks as though the law of propagation in curved space-time involves the Riemann-Christoffel tensor, and consequently differs from the law in flat space-time. But the inner multiplication by~$g^{\alpha\beta}$ saves the situation. It is possible to choose coordinates such that $g^{\alpha\beta}\, \dd\{\mu\beta, \epsilon\}/\dd x_{\alpha}$ vanishes for all the sixteen possible combinations of $\mu$ and~$\epsilon$\footnotemark.\footnotetext {According to~\Eq{(36.55)} it is possible by a transformation to increase $\dd\{\mu\beta, \epsilon\}/\dd x_{\alpha}$ by an arbitrary quantity~$a_{\mu\beta\alpha}^{\epsilon}$, symmetrical in $\mu$,~$\beta$ and~$\alpha$. The sixteen quantities $g^{\alpha\beta} a_{\mu\beta\alpha}^{\epsilon}$ ($\mu, \epsilon = 1, 2, 3, 4$) will not have to fulfil any conditions of symmetry, and may be chosen independently of one another Hence we can make the right-hand side of~\Eq{(74.52)} vanish by an appropriate transformation.} For these coordinates \Eq{(74.52)}~reduces to~\Eq{(74.51)}, and the usual solution for flat space-time will apply at the point considered. \PageSep{178}