(EMT) in $5D$ is zero, to an observer confined to making physical measurements in our ordinary spacetime, and not aware of the extra dimension, the spacetime is not empty but contains (effective) matter whose EMT, ${^{(4)}T}_{\alpha\beta}$, is determined by the Einstein equations in $4D$, namely $$\begin{aligned}
\label{4D Einstein with T and K}
{^{(4)}G}_{\alpha\beta} = 8 \pi \;{^{(4)}T}_{\alpha\beta} = 
- \epsilon\left(K_{\alpha\lambda}K^{\lambda}_{\beta} - K_{\lambda}^{\lambda}K_{\alpha\beta}\right) + \frac{\epsilon}{2} g_{\alpha\beta}\left(K_{\lambda\rho}K^{\lambda\rho} - (K^{\lambda}_{\lambda})^2 \right) - \epsilon E_{\alpha\beta}, \end{aligned}$$ where $K_{\mu\nu}$ is the extrinsic curvature $$\label{extrinsic curvature}
K_{\alpha\beta} = \frac{1}{2}{\cal{L}}_{\hat{n}}g_{\alpha\beta} = \frac{1}{2\Phi}\frac{\partial{g_{\alpha\beta}}}{\partial \psi};$$ $E_{\mu\nu}$ is the projection of the bulk Weyl tensor ${^{(5)}C}_{ABCD}$ orthogonal to ${\hat{n}}^A$, i.e., “parallel" to spacetime, viz., $$\label{Weyl Tensor}
E_{\alpha\beta} = {^{(5)}C}_{\alpha A \beta B}{\hat{n}}^A{\hat{n}}^B 
= - \frac{1}{\Phi}\frac{\partial K_{\alpha\beta}}{\partial \psi} + K_{\alpha\rho}K^{\rho}_{\beta} - \epsilon \frac{\Phi_{\alpha;\beta}}{\Phi},$$ and $\Phi_{\alpha} \equiv \partial \Phi/\partial x^{\alpha}$. Before going on, it is worthwhile to emphasize that the above dimensional reduction of the field equations in $5D$ is a standard technique that leads to the same effective matter content in $4D$, i.e., ${^{(4)}T}_{\alpha\beta}$, regardless of whether the line element (\[General line element in 5D without off-diagonal terms\]) is interpreted within the context of brane theory with ${\bf Z}_2$ symmetry [@Shiromizu] or space-time-matter (STM) theory [@Wesson; @and; @JPdeL]. In this sense 