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student85
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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.
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||bka = 0.
Interpret your results in terms of waves and rays.
The geodesic equation: [tex]\ddot{x}[/tex]e + [tex]\Gamma[/tex]emb[tex]\dot{x}[/tex]m [tex]\dot{x}[/tex]b = 0
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.
Homework Statement
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||bka = 0.
Interpret your results in terms of waves and rays.
Homework Equations
The geodesic equation: [tex]\ddot{x}[/tex]e + [tex]\Gamma[/tex]emb[tex]\dot{x}[/tex]m [tex]\dot{x}[/tex]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 [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.