Tensor question about Hypersurfaces

[hamjam9]

I've been wracking my brains trying to answer this but it's just really hard. If some one could help me out I would really appreciate it. Thank you so much in advanced!

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 na 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^b = 0.

Interpret your results in terms of waves and rays.

Where _|a denotes partial derivative with respect to a, and _||a denotes the covariant derivative with respect to a.

 

[shoehorn]

This sounds like a homework problem. As stated in the PF guidelines,

You MUST show that you have attempted to answer your question in order to receive help.

As it stands, your question is a pretty standard and basic one on the geometry of hypersurfaces. Show us what you've done to answer the question so far and we'll help you through the rest of it.