Cosmic censorship of smooth structures

TL;DR

The paper discusses how the assumption of global hyperbolicity in 4-dimensional spacetimes restricts the possible smooth structures, often leading to a unique smooth structure on such manifolds.

Contribution

It demonstrates that global hyperbolicity imposes a censorship effect, limiting smooth structures on certain 4-manifolds to standard forms.

Findings

Every contractible smooth 4-manifold with a globally hyperbolic Lorentz metric is diffeomorphic to .

A 4-manifold homeomorphic to N , with N a closed oriented 3-manifold, and admitting such a metric is diffeomorphic to N .

Global hyperbolicity constrains smooth structures, leading to uniqueness in many cases.

Abstract

It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard $\R^4$. Similarly, a smooth 4-manifold homeomorphic to the product of a closed oriented 3-manifold $N$ and $\R$ and admitting a globally hyperbolic Lorentz metric is in fact diffeomorphic to $N\times \R$. Thus one may speak of a censorship imposed by the global hyperbolicty assumption on the possible smooth structures on $(3+1)$-dimensional spacetimes.