A a compact, orientable, spacelike surface always has 2 independent forward-in-t

Question

A a compact, orientable, spacelike surface always has 2 independent forward-in-time pointing, lightlike, normal directions. For example, a (spacelike) sphere in Minkowski space has lightlike vectors pointing inward and outward along the radial direction. The inward-pointing lightlike normal vectors converge, while the outward-pointing lightlike normal vectors diverge. It can, however, happen that both inward-pointing and outward-pointing lightlike normal vectors converge. In such a case the surface is called trapped.-------- from Wikipedia: http://en.wikipedia.org/wiki/Apparent_horizon

Now a null vector is parallel to and perpendicular to itself at the same time. So the tangent plane on the concerned point on the spacelike surface should be a null surface.

What is the formal explanation for this?

Explanation / Answer

The surface is codimension two in spacetime, it is two dimensional inside a four dimensional space. This means that the tangent space to the surface does not consist of all vectors perpendicular to the normal to the surface, just as in three dimensions, the one-dimensional vector tangent space is not the collection of vectors perpendicular to some perpendicular line.

The tangent space to the surface does not include any null vectors, it just is what it is, some 2-dimensional space-like vector space sitting inside the manifold tangent space. To give a simple example, if the surface is the x-y plane at t=0, the future pointing perpendicular null directions are along the z axis going to the right and towards the future (1,0,0,1) and to the left and to the future (1,0,0,-1). The collection of all vectors perpendicular to (1,0,0,1) is all vectors of the form (a,b,c,a) (where a,b,c are real numbers), and includes more vectors than the tangent space of (0,b,c,0). The collection of vectors perpendicular to the other null vector, (1,0,0,-1), are (a,b,c,-a). The collection of all vectors perpendicular to both the outgoing light rays is the tangent space of the surface.

The same holds in curved space, since this is the situation locally.

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