these two approaches to $5D$ relativity are mathematically equivalent. However, they are different as regards physical interpretation and motivation [@physical; @motivation]. In brane theory there is a singular hypersurface that defines spacetime, and the properties of matter in that hypersurface are, in general, [*not identical*]{} to the ones of induced matter calculated in STM from the effective EMT defined by (\[4D Einstein with T and K\]).

In the cosmological realm, nearly all models assume spatial homogeneity and isotropy, which means that the line element in $5D$ is taken to be an extended version of the conventional Friedmann-Roberson-Walker (FRW) metric in $4D$, namely $$\label{Usual cosmological metric in 5D}
dS^2 = n^2(t, \psi)dt^2 - a^2(t, \psi)\gamma_{i j} dx^{i}dx^{j} + \epsilon \Phi^2(t, \psi) d\psi^2,\;\;\;i, j = 1, 2, 3,$$ where $\gamma_{i j}$ is a maximally symmetric $3$-dimensional metric, with curvature index $k = 0, \pm 1$. In these coordinates the full integration of the vacuum Einstein field equations in $5D$ requires the specification of [*two*]{} additional assumptions. One of them is usually an assumption of geometric nature, e.g., that $\Phi = 1$, or $n = 1$. The second one is usually an equation of state for the matter quantities in $4D$ [@Binetruy], [@JPdeL-isotropicCosm].

