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The definition of a trapped surface in Sean Carroll's "Spacetime and Geometry" is as follows.

"A compact spacelike, two dimensional submanifold with the property that outgoing future directed light rays converge in both directions everywhere on the submanifold."

The light rays emanating from two spheres inside an event horizons would evolve to smalled values of radius for both outgoing and ingoing rays. I don't understand the intuition behind this explanation. If something is moving outwards how do they converge in the future and merge into singularity.

Qmechanic
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Khushal
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1 Answers1

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If we consider a Schwarzschild black hole, a light cone inside the event horizon tilts in such a way that either an outgoing or an ingoing ray move towards the singularity. Reason is that the $g_{tt}$ and $g_{rr}$ components of the metric tensor change sign. Technically the $t$ direction and the $r$ direction exchange each other, i.e. the $t$ direction from time-like (outside the horizon) becomes space-like (inside the horizon) and the $r$ direction from space-like becomes time-like. So, inside the horizon the light cone gets rotated by $\pi /2$ relative to outside. A material body, being constrained within the light cone, share the same destiny.

Michele Grosso
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