The Galilean values of~$g^{\mu\nu}$ are $g^{44} = 1$, $g^{11} = g^{22} = g^{33} = -1$, and the other components vanish. Hence \Eq{(30.6)}~can be written \[ g^{\mu\nu}\, \frac{\dd^{2} \phi}{\dd x_{\mu}\, \dd x_{\nu}} = 0. \Tag{(30.65)} \] The potential~$\phi$ being an invariant, its ordinary derivative is a covariant vector $\phi_{\mu} = \dd\phi/\dd x_{\mu}$; and since the coordinates are Galilean we may insert the covariant derivative~$\phi_{\mu\nu}$ instead of~$\dd \phi_{\mu}/\dd x_{\nu}$. Hence the equation becomes \[ g^{\mu\nu} \phi_{\mu\nu} = 0. \Tag{(30.7)} \] Up to this point Galilean coordinates are essential; but now, by examining the covariant dimensions of~\Eq{(30.7)}, we notice that the left-hand side is an invariant, and therefore its value is unchanged by any transformation of coordinates. Hence \Eq{(30.7)}~holds for all coordinate-systems, if it holds for any. Using~\Eq{(29.3)} we can write it more fully \[ g^{\mu\nu} \left(\frac{\dd^{2}\phi}{\dd x_{\mu}\, \dd x_{\nu}} - \{\mu\nu, \alpha\}\, \frac{\dd\phi}{\dd x_{\alpha}}\right) = 0. \Tag{(30.8)} \] This formula may be used for transforming Laplace's equation into curvilinear coordinates, etc. It must be remembered that a transformation of coordinates does not alter the kind of space. Thus if we know by experiment that a potential~$\phi$ is propagated according to the law~\Eq{(30.6)} in Galilean coordinates, it follows rigorously that it is propagated according to the law~\Eq{(30.8)} in any system of coordinates in flat space-time; but it does not follow rigorously that it will be propagated according to~\Eq{(30.8)} when an irreducible gravitational field is present which alters the kind of space-time. It is, however, a plausible suggestion that \Eq{(30.8)}~may be the general law of propagation of~$\phi$ in any kind of space-time; that is the suggestion which the principle of equivalence makes. Like all generalisations which are only tested experimentally in a particular case, it must be received with caution. The operator~$\Wave$ will frequently be referred to. In general coordinates it \index{Contracted derivative (divergence)!second derivative ($\Wave$)}% \index{Operators!W@$\Wave$}% is to be taken as defined by \[ \Wave A_{\mu\nu\cdots} = g^{\alpha\beta} (A_{\mu\nu\cdots})_{\alpha\beta}. \Tag{(30.9)} \] Or we may write it in the form \[ \Wave = \bigl((\cdots)_{\alpha}\bigr)^{\alpha}, \] i.e.\ we perform a covariant and contravariant differentiation and contract them. \PageSep{65} \Summary{Summary of Rules for Covariant Differentiation.}% \index{Covariant derivative of vector!of tensor}% \index{Differentiation!covariant, rules for}%