Trapped surface

Remarks

The expansion $\theta_\pm$ is the rate of change of the area element of $S$ when $S$ is pushed along $\ell_\pm$. Rescaling $\ell_\pm$ by a positive function rescales $\theta_\pm$ by the same function, so the sign condition does not depend on the choice. If $\theta_+ = 0$ for the outgoing direction, the surface is called marginally outer trapped.

Example: in the Schwarzschild metric of mass $m$, extended across $r = 2m$, take a round sphere of area radius $r < 2m$. In the region $r < 2m$ the function $r$ strictly decreases along every future-directed causal curve. The area of the sphere is $4\pi r^2$, so it decreases in both null directions, and $\theta_+ < 0$ and $\theta_- < 0$.

Non-example: in Minkowski space, the round sphere $t = 0$, $\abs{x} = R$ has $\theta_+ = 2/R > 0$ for the outgoing null normal $\partial_t + \partial_r$. It has $\theta_- = -2/R < 0$ for the ingoing one, so it fails the condition $\theta_+ < 0$.