Geometric obstructions of C0-extendibility of gravitational collapse spacetimes

In practice, hypersurfaces are defined via functions of coordinates. As a result, causal classification of hypersurfaces bounding charts can be frame-dependent. We prove that if there exists any frame such that, at some (limiting) point, the metric is diagonal and a basis vector lies on a null hypersurface (defined by that frame) then the metric is degenerate and [Formula: see text]-inextendible there. We apply this to Schwarzschild spacetimes. We show that volume-forms of all 1-submanifolds vanish in the limit of approaching the event horizon, then prove that any point of a 1-submanifold at which volume-forms would vanish (by continuity) cannot be contained in the 1-submanifold, or the manifold itself. This yields a geometric obstruction to the [Formula: see text]-extendibility of curves and the manifold from the black hole exterior to the event horizon. The Ingoing Eddington–Finkelstein coordinate map is shown to fail to be a diffeomorphism at the event horizon, and thus cannot be used to extend the exterior. We discuss the implications for gravitational collapse spacetimes, and show that the geometry of curves inherited from the manifold precludes worldlines from ‘leaving the manifold’, providing new physical intuition for their incompleteness. Finally, the framework is shown not to further constrain extendibility of FLRW spacetimes.