= Solution
For each fixed coordinate function $x^\alpha$, the expression $\Box x^\alpha$ is a scalar: $\alpha$ labels four chosen functions and is not a free <tensor> index. The equation is therefore coordinate invariant when those functions are held fixed. However, the demand that the currently chosen coordinates themselves be <harmonic coordinates> is a chart condition, not a four-vector equation invariant under arbitrary changes of coordinate functions.
Indeed, for new functions $\bar x^\alpha=f^\alpha(x)$ the <d'Alembert operator> gives
$$
\Box\bar x^\alpha=f^\alpha{}_{,\beta}\Box x^\beta+g^{\beta\gamma}f^\alpha{}_{,\beta\gamma}.
$$
The second term prevents a vector transformation law. An affine redefinition preserves <harmonic coordinates>; an arbitrary nonlinear one does not. For example, a Cartesian spatial coordinate $x$ in <Minkowski spacetime> has $\Box x=0$, whereas $\Box(x^2)=2$, and $x^2$ is a valid new coordinate on $x>0$. \b[The equations are scalar equations for specified functions, while harmonic gauge is chart dependent.]
Back to article page
