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A ‘finite infinity’ version of topological censorship. (English) Zbl 0853.53069

The author gives a version of the topological censorship theorem of J. L. Friedmann, K. Schleich and D. M. Witt [Phys. Rev. Lett. 71, 1486-1489 (1993)] without the assumption of asymptotic flatness. He assumes that $\left(M,g\right)$ is a spacetime with timelike boundary $\partial M=T$, where $T$ is diffeomorphic to $ℝ×{S}^{2}$, and that for each $t$, ${{\Sigma }}_{t}:=\left\{t\right\}×{S}^{2}$ is spacelike. He further assumes that for each ${{\Sigma }}_{t}$, the null second fundamental forms corresponding to (any) inward (respectively, outward) pointing null vector field are negative (respectively, positive) definite. Each ${{\Sigma }}_{t}$ is supposed to be acausal in $M$ and the null convergence condition is assumed to hold. Under these conditions, global hyperbolicity of ${J}^{+}\left(T\right)\cap {J}^{-}\left(T\right)$ implies that this set is simply connected.

In the asymptotically flat case, there exist timelike tubes $T$ near infinity which satisfy the assumptions above and the theorem can be applied to the complement of the asymptotic region bounded by $T$.

He also shows that if the boundary of $M$ consists of several timelike tubes ${\left\{{T}_{\alpha }\right\}}_{\alpha }$ and $\left(M,g\right)$ is globally hyperbolic, then ${J}^{+}\left({T}_{a}\right)\cap {J}^{-}\left({T}_{b}\right)=\varnothing$, provided $a\ne b$. A possible interpretation of this theorem is that in globally hyperbolic spacetimes there are no wormholes connecting different asymptotic regions.

##### MSC:
 53Z05 Applications of differential geometry to physics 83C75 Space-time singularities, cosmic censorship, etc.