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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)

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Maybe I am misunderstanding. However, I believe the expression for the Schwarzschild metric you wrote down in coordinates only holds outside the event horizon (i.e. r>2M). My guess is if you use coordinates that extend past the event horizon (if I recall correctly an example of these are called advanced and retarded coordinates) then things should work out in a straight forward manner. –  Rbega Oct 15 '10 at 22:24

1 Answer 1

Quite a lot of questions you have there. First a bit of nitpick:

  • You didn't give enough information to define the null second fundamental forms. Just knowing a geodesic is not enough. You need a whole family of them. To be precise, you need $X^\pm$ to be null vector fields along $M$ orthogonal to $TM$, and then you extend them by parallel transport along themselves in the null direction.

For a bit more about the geometry, look at Galloway's Beijing Lecture Notes or B. O'Neill's Semi-Riemannian Geometry.

About the relationship between $h$ and $g$: the induced metric on a Riemannian submanifold of a semi-Riemannian manifold can be treated as a projection operator on the total tangent space restricted to over the submanifold. In other words, let $p\in M\subset V$, then because $M$ is Riemannian, you can split $T_pV = T_pM \oplus (T_pM )^\perp$, then $h$ uniquely lifts from $Sym_2(T_pM)$ to $Sym_2(T_pV)$ by extending trivially in the perpendicular directions. This lift coincides with $g + X^-\otimes X^+ + X^+\otimes X^-$ (if you normalize $g(X^-,X^+) = -1$). (Note, this fixing of normalization is important. You didn't specify it in your question.)

Now, you should get out of the habit of working in Schwarzschild coordinates for the trapped region. That coordinate system is only good for thinking about the exterior of the black hole. Inside the trapped region, the surfaces of constant $r$ are in fact space-like... The picture would be much clearer if you think in terms of Kruskal coordinates.

About "future directed":

  • Looks like you are missing a bit of basic causal theory? A space-time is called time-orientable if it admits a continuous, non-vanishing time-like vector field $\tau$. For time-orientable space-times, the notion of future and past are well-defined: a causal vector $v$ is said to be future oriented if $g(v,\tau) < 0$, and past-oriented if $g(v,\tau) > 0$. There exists Lorentzian manifolds which are not time-orientable. See any General Relativity textbook for more details and examples (esp. Hawking-Ellis and O'Neill).

Detection of trapped surface:

  • Outside of spherical symmetry, where you can detect trapped surfaces by using the metric dual of the one form $dr$, where $r$ is the area-radius (if $dr$ is space-like, then the sphere is non-trapped), there's no general mechanism to detect trapped surfaces, other than a priori knowledge that both of its null mean curvature are negative (as in, this is an open problem). If you restrict to a spatial slice, then there are some work in the Riemannian geometry literature concerning this problem by looking at the Jang equation (I think Marcus Khuri worked on it for a bit, don't know if he is still on it). There are also related work on dynamical horizons and marginally outermost trapped surfaces by Ashtekar, Galloway, Lars Andersson, and others. But if you do find a general way to detect presence of trapped surfaces without local computations, do let me know, because you'd be about a third way there toward solving weak cosmic censorship.
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Also, a geometrically better way of thinking about trapping is to ignore this whole geodesic and null vector field business, and think about the mean curvature vector. The second fundamental form of a submanifold $M\subset V$ measures the failure of $TM$ to be parallel under transport by $\nabla$. Thus the proper way to see the second fundamental form, is that it is a symmetric two tensor on $TM$ that takes value in the normal bundle for $M$. Then taken the $g$ (or the $h$, doesn't matter) trace of this tensor you get a section of the normal bundle over $M$. This is the mean curvature vector. –  Willie Wong Oct 16 '10 at 1:14
If the mean curvature vector is space-like, $M$ is not trapped (near that point). If it is time-like, then it is either trapped (future pointing) or anti-trapped (past pointing, like in a white hole). For spherically symmetric space-times, the mean curvature vector of the symmetry spheres are precisely the metric dual of the one form $dr$ that I wrote above, up to a scalar multiple. For more about the second fundamental form as taking values in the normal bundle, see O'Neill's book. –  Willie Wong Oct 16 '10 at 1:19
@Willie Thanks for your replies. I am familiar with the understanding of the null second fundamental form as symmetric rank two tensor on the submanifold taking values in the normal bundle. I skipped writing all that and to keep the question short I directly defined the quantities $K^{+/-}_{ab}$. So you are saying that the tensor whose components are $h_{ab}$ (as defined in the question) and the tensor $h$ (as defined in the question) should be thought of as lifts of each other? –  Anirbit Oct 16 '10 at 7:08
@ Willie But then what does this lift tell us? What good it is when it does not lift to the original metric on the full space-time? Is there a way of explicitly writing down the embedding of $M$ in $V$ along which if $g$ is pulled back it precisely gives the metric ${h_{ab}}$ ? I am familiar with the notion of time-orientability as you defined it. SO I guess you are suggesting that I look in the Kruskal coordinates and identify these symmetry two spheres in those coordinates and also the future directed null geodesics in those coordinates and repeat the calculation? –  Anirbit Oct 16 '10 at 7:12
@Willie Thanks for what you called the "nitpicking". It was important. –  Anirbit Oct 16 '10 at 7:13

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