Obstructions for trapped submanifolds

Abstract

We introduce the concept of $k-$future convex spacelike/null hypersurface $\Sigma$ in an $n+1$ dimensional spacetime $M$ and prove that no $k-$dimensional closed trapped submanifold (k-CTM) can be tangent to $\Sigma$ from its future side. As a consequence, k-CTMs cannot be found in open spacetime regions foliated by such hypersurfaces. In gravitational collapse scenarios, specific hypersurfaces of this kind act as past barriers for trapped submanifolds. A number of examples are worked out in detail, two of them showing 3+1 spacetime regions containing trapped loops ($k=1$) but no closed trapped surfaces ($k=2$). The use of trapped loops as an early indicator of black hole formation is briefly discussed.