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Claus Gerhardt

Publications and source records attributed to Claus Gerhardt.

48 records · Page 3Linked to original sources
arXiv2006

The mass of a Lorentzian manifold

We define a physically reasonable mass for an asymptotically Robertson-Walker (ARW) manifold which is uniquely defined in the case of a normalized representation.

math.DG↗
arXiv2004

Transition from big crunch to big bang in brane cosmology

We consider branes $N=I\times\so$, where $\so$ 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 $\hat N$, which has a big bang singularity and which can be viewed as a brane in a reflected Schwarzschild-AdS_{(n+2)} bulk $\hat{\mc N}$. The joint branes $N\uu \hat N$ can thus be naturally embedded in $R^2\times \so$, hence there exists a second possibility of defining a smooth transition from big crunch to big bang by requiring that $N\uu\hat N$ forms a $C^\infty$-hypersurface in $R^2\times \so$. 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.

gr-qc↗
arXiv2004

The inverse mean curvature flow in Robertson-Walker spaces and its application to cosmology

We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing that in general the transition flow is only of class $C^3$.

gr-qc↗
arXiv2004

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

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.

math.DG↗