The cosmological problem is considered in a five-dimensional (bulk) manifold with two time coordinates, obeying vacuum Einstein field equations. The evolution formalism is used there, in order to get a simple form of the resulting constraints. In the spatially flat case, this approach allows us to find out the general solution, which happens to consist in a single metric. All the embedded Friedmann-Robertson-Walker (FRW) metrics can be obtained from this “mother” metric (“M-metric”) in the bulk, by projecting onto different four-dimensional hypersurfaces (branes). Having a time plane in the bulk allows us to devise the specific curve which will be kept as the physical time coordinate in the brane. This method is applied for identifying FRW regular solutions, evolving from the infinite past (no big bang), even with an asymptotic initial state with nonzero radius (emergent universes). Explicit counter-examples are provided, showing that not every spatially-flat FRW metric can actually be embedded in a $3+2$ bulk manifold. This implies that the extension of the Campbell theorem to the general relativity case works only in its weaker form in this case, requiring as an extra assumption that the constraint equations hold at least in a single four-dimensional hypersurface.

• Received 27 September 2018

DOI:https://doi.org/10.1103/PhysRevD.99.043530