1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Suppose we have a spacetime where the only possible direction to move is outwards i.e the two spheres can move only in the direction of increasing radius. Do we call these also trapped surfaces? Example of such a spacetime is Reissner Nordstrom Black hole. There are two types of surfaces one which has decreasing radius and the other kind have increasing radius. This can be realized from the Conformal diagram of this metric. $\endgroup$ – Khushal Apr 15 '18 at 6:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.