We know that in a two dimensional Riemannian manifold there always is, at least locally, a coordinate system in which $g_{ij} = \lambda^2 \delta_{ij}$ - such coordinates are called isothermal. There are great answers such as here and here explaining the reason for such name, since any such coordinate system ${\bf x}$ satisfies $\triangle {\bf x} = 0$, being a stationary solution to the heat equation.
However, for a two dimensional Lorentz manifold, there is also, locally, a coordinate system in with $g_{ij} = \lambda^2 \eta_{ij}$, where $\eta_{ij}$ is the Minkowski metric, say, $\eta_{11} = -\eta_{22}=1$ and $\eta_{12}=0$. As far as I have seen, such coordinate systems are also called isothermal. This bothers me a bit, since in this case we must have $$\Box {\bf x} \stackrel{\cdot}{=} \frac{\partial^2 {\bf x}}{\partial u^2}-\frac{\partial^2 {\bf x}}{\partial v^2}=0,$$and so ${\bf x}$ is a solution of the wave equation, not the heat equation. Is there a more appropriate name for such coordinates that I somehow missed? Can someone shed some light on this? Thanks.