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.

39 pages, a pdf version can also be retrieved from http://www.math.uni-heidelberg.de/studinfo/gerhardt/imcf-arw.pdf and bibtex data from http://www.math.uni-heidelberg.de/studinfo/gerhardt/bibtexcgimcf-arw.html, v2: minor changes in Section 9: assumptions clarified