to be compactified. These are membrane theory [@algebraically] and space-time-matter (or induced matter) theory [@general]. They lead to a great variety of models both in the cosmological context and in the description of local self-gravitating objects (see, e.g., [@review], [@available]). Most of these models have been obtained in coordinates where the metric in $5D$ can be written as[^2] $$\label{General line element in 5D without off-diagonal terms}
dS^2 = g_{\mu\nu}(x^{\rho}, \; \psi)dx^{\mu}dx^{\nu} + \epsilon \Phi^2(x^{\rho}, \; \psi)d\psi^2,$$ in such a way that our $4D$ spacetime can be recovered by going onto a hypersurface $\Sigma_{\psi}: \psi =  \psi_{0} = $ constant, which is orthogonal to the $5D$ unit vector $$\label{unit vector n}
{\hat{n}}^{A} = \frac{\delta^{A}_{4}}{\sqrt{\epsilon g_{44}}}, \;\;\;n_{A}n^{A} = \epsilon,$$ along the extra dimension, and $g_{\mu\nu}$ can be interpreted as the metric of the spacetime.

In this framework, the effective equations for gravity in $4D$ are obtained from dimensional reduction of the Einstein field equations in $5D$. The reduction is based on Campbell’s theorem [@Campbell], [@Seahra] and consists in isolating the $4D$ part of the relevant $5D$ geometric quantities and use them to construct the $4D$ Einstein tensor ${^{(4)}G}_{\alpha \beta}$. The crucial result is that, even in the case where the energy-momentum tensor 