If $M$ is a $n-1$ dimensional Riemannian submanifold in a $1+n$ dimensional space-time manifold $(V,g)$ of pseudo-Riemannian signature $(1,n)$ and $\nabla$ be the Riemann-Christoffel connection on it.

One chooses two future directed null geodesics orthogonal to M say $X^+$ and $X^-$ and defines two rank $2$ tensors $K^+$ and $K^-$ such that their components are,

$K^+ _{ab} = \nabla _a X^+_b$

$K^- _{ab} = \nabla _a X^- _b$

(where $a$ and $b$ run over the indices on $M$)

Let $h_{ab}$ be the components of the induced metric on $M$ from $g$ of $V$. Then one defines the following "null mean curvatures", $\chi ^+$ and $\chi ^-$ along $X^+$ and $X^-$ as,

$\chi ^ + = h^{ab}K^+_{ab}$

$\chi ^ - = h^{ab}K^-_{ab}$

Now apparently $h$ and $g$ can be related as,

$h = g + \frac{1}{2}(X^+ \otimes X^- + X^- \otimes X^+)$

- This is something I am not very clear about. I guess there is some abuse of notation about what is called $h$ since in the two definitions the dimensions of $h$ don't seem to match.

With the above definition one can show that,

$h^{\mu \nu} \nabla _\mu X^+ _\nu = g^{\mu \nu} \nabla _\mu X^+ _\nu$

(where $\mu$ and $\nu$ run over indices of the space-time $V$)

A similar expression holds with $X^-$ and the proof of this crucially needs both the properties that $X$'s are null as well as geodesic. This relates to the notion of ``expansion" of a geodesic congruence and hence shows that null mean curvature is exactly the expansion for the geodesic congruence to which $X$'s would be tangent vectors to.

Now one calls such a $M$ a ``Trapped Surface" if both the null mean curvatures $\chi ^+$ and $\chi ^-$ are negative.

I run into some confusions when I try doing this test on $r=constant$ and $t=constant$ spheres in the Schwarzschild metric,

$ds^2 = -(1-\frac{2M}{r})dt^2 + \frac{dr^2}{(1-\frac{2M}{r})} + r^2(d\theta ^2 + sin^2 \theta d\phi ^2)$

I should get both the null mean curvatures of all spheres inside $r=2M$ to be negative and hence all of them are trapped surfaces. To get this I have to choose the future directed null geodesics orthogonal to the 2-spheres as,

$X^{+/-} = (\frac{+/-1}{(1-\frac{2M}{r})}, -1 , 0, 0)$

- Now I am not very clear as to how to justify that these are both "future directed" inside the event horizon (i.e the surface $r=2M$). (I have some arguments of my own but not very clear) That these are null and orthogonal geodesics to the two spheres is clear.

For the above one gets that $\chi ^{+/-} = -\frac{2}{r}$ and hence justifying that all spheres inside the event horizon are trapped surfaces.

I would also like to know as to what is the coordinate independent definition for a geodesic to be "future directed" and in how general a space-time can a notion of being "future directed" be imposed globally and continuously.

Given a general pseudo-Riemannian metric how does one detect the presence or absence of a trapped surface? How does one find such a surface? (The above definition seems to give a test of being trapped if one is given a surface)