where $t$ is assumed to be the timelike coordinate; $\psi$ the “extra" coordinate; $F$ and $V$ are, in principle, arbitrary differentiable functions of their arguments, except from the condition that the Jacobian of the transformation must be nonzero.

With this transformation we obtain

$$\label{dudv}
du dv = \left(\dot{F}\dot{V} dt^2 + F' V' d\psi^2\right) + \left(\dot{F} V' + F' \dot{V}\right)dt d\psi,$$

where dots and primes denote derivatives with respect to $t$ and $\psi$, respectively. We can choose the coordinates $t, \psi$ in such a way that the $5D$ metric be diagonal. This requires $$\label{diagonal 5D metric}
V' = - \frac{F' \dot{V}}{\dot{F}}.$$ As a consequence, the line element (\[light-like 5D metrics\]) becomes $$\label{light-like metric in t-psi coordinates}
dS^2 = \bar{{\cal{A}}}(t, \psi)\dot{F}\dot{V}dt^2 - {\bar{{\cal{B}}}}_{i j}(t, \psi)dx^{i} dx^{j} - \bar{{\cal{A}}}(t, \psi)\frac{F'^2 \dot{V}}{\dot{F}} d\psi^2,$$ where $\bar{{\cal{A}}}(t, \psi) \equiv {\cal{A}}(F, V)$ and ${\bar{{\cal{B}}}}_{i j}(t, \psi) \equiv {{\cal{B}}}_{i j}(F, V)$. A couple of points should be noticed here. Firstly, that the physical requirement $g_{00}  > 0$ demands $\psi$ to be spacelike. Secondly, that the line element (\[light-like metric in t-psi coordinates\]) contains two arbitrary functions, which are not present in the original solution (\[new solution\]). The question is, why? Is this a mathematical, or gauge, artifact?

The 