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.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

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}||_{b}k^{a}= 0.

Interpret your results in terms of waves and rays.

2. Relevant equations

The geodesic equation: [tex]\ddot{x}[/tex]^{e}+ [tex]\Gamma[/tex]^{e}_{mb}[tex]\dot{x}[/tex]^{m}[tex]\dot{x}[/tex]^{b}= 0

3. 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 [itex]\Gamma[/itex] 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.

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# Homework Help: Surfaces and geodesics in General Relativity

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