The inverse mean curvature flow in ARW spaces--transition from big crunch to big bang

Abstract

We consider spacetimes $N$ satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of an inverse mean curvature flow. Moreover, we define a new spacetime $\hat N$ by switching the light cone and using reflection to define a new time function, such that the two spacetimes $N$ and $\hat N$ can be pasted together to yield a smooth manifold having a metric singularity, which, when viewed from the region $N$ is a big crunch, and when viewed from $\hat N$ is a big bang. The inverse mean curvature flows in $N$ \resp $\hat N$ correspond to each other via reflection. Furthermore, the properly rescaled flow in $N$ has a natural smooth extension of class $C^3$ across the singularity into $\hat N$. With respect to this natural, globally defined diffeomorphism we speak of a transition from big crunch to big bang.