+ t N(\psi)\right]^{a^2/(a + b)},$$ where $M$ and $N$ are arbitrary differentiable functions of $\psi$. For any other $c_{2} \neq 0$, we obtain $V$ from (\[obtaining V from F in Synchronous coord.\]) after choosing some smooth function $F(t, \psi)$.

Thus, in principle the function $V$ can be determined if we know $F$. At this point the question arises of whether we can single out the function $F$ from “physical" considerations in $4D$. Further analysis of the field equations shows that if we assume an equation of state for the matter in $4D$, then we obtain an extra differential equation connecting $V$ and $F$, which in addition to (\[obtaining V from F in Gaussian coord.\]) or (\[obtaining V from F in Synchronous coord.\]), allows to express the solution (\[light-like metric in t-psi coordinates\]) in terms of $t$ and $\psi$. This is what is required for the $4 + 1$ dimensional reduction of the $5D$ solutions. The general calculations are straightforward, but the equations are notational cumbersome in both STM and braneworld theory. On the other hand, (\[F in Gaussian coordinates, c2 = 0\]) and (\[F in the synchronous coordinates\]) indicate that a great algebraic simplification is attained if $c_{2} = 0$. 