Surfaces and geodesics in General Relativity

[student85]

Hi all. This is one of the problems that I was asked to do for my General Relativity class. I know this may look a little long, but if anyone can help me out with ANYTHING about this problem, I will greatly appreciate it.

Homework Statement

Consider the family of hypersurfaces where each member is defined by the constancy of the function S(x^c) over that hypersurface and further require that each hypersurface be a null hypersurface in the sense that its normal vector field, n_a = S|_a , be a null vector field.
Let ¡ be a member of the family of curves that pierces each such hypersurface orthogonally, meaning that the tangent vector to ¡, say k^a, is everywhere collinear with the vector n^a at the point of piercing. Show that ¡ is a null geodesic and find the condition on the relation between n^a and k^a that allows the geodesic equation to be written in the simple form k^a||_bk^a = 0.
Interpret your results in terms of waves and rays.

Homework Equations

The geodesic equation: $$\ddot{x}$$^e + $$\Gamma$$^e_mb$$\dot{x}$$^m $$\dot{x}$$^b = 0

The Attempt at a Solution

By reading through the problem it is not very hard to get the hang of what it is saying, and it seems pretty clear that $\Gamma$ must be a null surface. But I don't know where to get started in showing that it is a "null geodesic", and how to derive at the simple geodesic equation they give. I'm just very stuck here. If anyone can give me a little hint I would appreciate it. Thanks in advance.

 

[student85]

No relativists here? :S