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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 {\it J. L. Friedmann}, {\it K. Schleich} and {\it D. M. Witt} [Phys. Rev. Lett. 71, 1486-1489 (1993)] without the assumption of asymptotic flatness. He assumes that $(M, g)$ is a spacetime with timelike boundary $\partial M=T$, where $T$ is diffeomorphic to $\bbfR \times S^2$, and that for each $t$, $\Sigma_t:= \{t \}\times S^2$ 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 $J^+ (T)\cap J^- (T)$ 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 $\{T_\alpha \}_\alpha$ and $(M, g)$ is globally hyperbolic, then $J^+ (T_a) \cap J^- (T_b)= \emptyset$, provided $a\ne b$. A possible interpretation of this theorem is that in globally hyperbolic spacetimes there are no wormholes connecting different asymptotic regions.

53Z05Applications of differential geometry to physics
83C75Space-time singularities, cosmic censorship, etc.
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