answer to this question is that the arbitrary functions in (\[light-like metric in t-psi coordinates\]) are [*not*]{} gauge artifacts. They reflect the physical reality that there are many ways of embedding a $4D$ spacetime in $5D$ while satisfying the field equations. If we choose some particular embedding we obtain a differential constraint connecting $V$ and $F$, which allows us to obtain one of them in terms of the other, e.g., $V$ in terms of $F$. Then, the remaining unknown function, e.g. $F$, can be determined from the physics in $4D$. As an illustration of the former assertion, let us consider two common embeddings that arise from the choice of the coordinate/reference system.

#### Gaussian normal coordinate system:

A popular choice in the literature is to use the five degrees of coordinate freedom to set $g_{4 \mu} = 0$ and $g_{44} = - 1$. This is the so-called ‘Gaussian normal coordinate system’ based on $\Sigma_{\psi}$. Consequently, in such coordinates $\dot{V} = ({\dot{F}}/{\bar{{\cal{A}}} F'^2})$ and (\[diagonal 5D metric\]) becomes $V' = - (1/{\bar{{\cal{A}}} F'})$. Now the condition $(\partial \dot{V}/\partial \psi) = (\partial {V'}/\partial t)$ yields $$\label{obtaining V from F in Gaussian coord.}
a^2\left(c_{1}F + c_{2}V\right) F'' - 2c c_{1} F'^2 = 0.$$ 