Consider the family of hypersurfaces where each member is defined by the constancy of the function S(xc) over that hypersurface and further require that each hypersurface be a null hypersurface in the sense that its normal vector field, na = 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 ka 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 na and ka that allows the geodesic equation to be written in the simple form ka||bkb = 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.
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 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.