we construct a class of non-static vacuum solutions on the brane. We also develop [*static*]{} pancake-like distributions where the matter is concentrated in a thin surface (near $z = 0$), similar to those proposed by Zel’dovich for the shape of the first collapsed objects in an expanding anisotropic universe. The solutions discussed here can be applied in a variety of physical situations.'
author:
- |
    J. Ponce de Leon[^1]\
    Laboratory of Theoretical Physics, Department of Physics\
    University of Puerto Rico, P.O. Box 23343, San Juan,\
    PR 00931, USA
title: '$4D$ spacetimes embedded in $5D$ light-like Kasner universes'
---

PACS: 04.50.+h; 04.20.Cv

[*Keywords:*]{} Cosmological models; Kasner universes; 5D Models; Braneworld theory; Induced matter theory; Exact solutions; General Relativity.

Introduction
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In recent years there has been an increased interest in theories that envision our spacetime as embedded in a universe with more than four large dimensions. There are several reasons that justify this interest, among them that extensions of four-dimensional general relativity to five and more dimensions seem to provide the best route to unification of gravity with the interactions of particle physics [@Davidson; @Owen]-[@particle; @physics]. In $5D$ there are two versions of relativity where the extra dimension is not assumed 