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## 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 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||b}k

^{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.

## Homework Equations

## 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.