From five dimensional flat spacetime to our four dimensional braneworld via kaluza-klein

In five dimensional cosmological models, the convention is to include the fifth dimension in a way similar to the other space dimensions. In this work we attempt to introduce the fifth dimension in a way that a time dimension would be introduced. In our metric ansatz we take the scale factor of three dimensional space, the $x, y, z$ coordinates, to depend on both time and internal space. We allow time and internal space, the extra dimension, to share the same metric coefficient that depends on both dimensions. Our internal space governs the evolution of four dimensional spacetime and time governs the evolution of all four spatial dimensions. As such time and internal space play similar roles. From such a metric, we obtain a five dimensional flat, meaning both Ricci flat and conformally flat, spacetime into which all relevant four dimensional cosmologies can be locally embedded. Different cases, such as radiation, matter or dark energy dominated cosmologies, correspond to different choices of the free parameters. Each choice is a different frame. We argue about which frame might be the cosmological frame. The choice that is simplest and most convenient in terms of dimensional analysis, amounts to a linearly expanding universe.