describe non-static solutions, and apply it to the braneworld theory under investigation. In Sec. \[static-case\], we return to the issue of static solutions with arbitrary spatial curvature using conveniently chosen variables. Finally, in Sec. \[conclusions\], we present our conclusions.

Basic equations {#basic}
===============

In this paper, we consider the case where the braneworld is the time-like boundary of a five-dimensional purely gravitational Lorentzian space (bulk), which is equivalent to the case of a brane embedded in the bulk with $\mathbb{Z}_2$ symmetry of reflection with respect to the brane. The theory is described by the action [@CHS]: $$\label{act}
S=-M^3\left[\int_{\rm bulk}(\mathcal{R}-2\Lambda_{\rm b})-2\int_{\rm brane}
K\right]-\int_{\rm brane}(m^2R-2\sigma)+\int_{\rm brane} L(h_{ab},\phi)\, .$$ Here, $\mathcal{R}$ is the scalar curvature of the metric $g_{ab}$ in the five-dimensional bulk, and $R$ is the scalar curvature of the induced metric $h_{ab}=g_{ab}-n_{a}n_{b}$ on the brane, where $n^{a}$ is the vector field of the inner unit normal to the brane, and the notation and conventions of [@Wald] are used. The quantity $K=K_{ab}h^{ab}$ is the trace of the symmetric tensor of extrinsic curvature $K_{ab}=h^{c}_{a}\nabla_{c}n_{b}$ on the brane. The symbol $L(h_{ab}, \phi)$ denotes the Lagrangian density of the four-dimensional matter fields $\phi$ whose dynamics is restricted to the brane so that they interact only 