Transition from big crunch to big bang in brane cosmology

We consider branes $N=I\times \text{so}$, where $\text{so}$ is an $n$\ndash dimensional space form, not necessarily compact, in a Schwarzschild-AdS_{(n+2)} bulk $\mc N$. The branes have a big crunch singularity. If a brane is an ARW space, then, under certain conditions, there exists a smooth natural transition flow through the singularity to a reflected brane $\widehat{N}$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS_{(n+2)} bulk $\widehat{\text{mc}N}$. The joint branes $N\text{uu}\widehat{N}$ can thus be naturally embedded in $R^2\times \text{so}$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\text{uu}\widehat{N}$ forms a $C^{\mathrm{\infty }}$-hypersurface in $R^2\times \text{so}$. This last notion of a smooth transition also applies to branes that are not ARW spaces, allowing a wide range of possible equations of state.