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Unformatted text preview: Homework 4 Ted Jacobson and William D. Linch III March 28, 2005 1.a The null generators are radial and are specified by r ( Î» ) and t ( Î» ), with fixed angles. As they form a null geodesic congruence, they extremize the functional I = 1 2 R ( Ë™ t 2 Ë™ r 2 ). (The angular part is trivially stationary since it is quadratic in the derivatives Ë™ Î¸ = Ë™ Ï† = 0.) The EulerLagrange equations for r following from this functional are 0 = Â¨ r . Hence, r = aÎ» + b is an affine parameter for the null generators. 1.b The area of the spherical surfaces dr = 0 of radius r are Area = 4 Ï€r 2 . Then, on the past light cone 1 Î¸ = d Area dr Area r = r = 2 Â· 4 Ï€r 4 Ï€r 2 = 2 r . (1) 1.c According to the focusing theorem, Î¸ â†’ âˆž at or before an affine parameter time Î» = 2  Î¸  1 . Solving (1) gives r = 2 Î¸ = 2  Î¸  1 . On the cone 0 = t 2 r 2 this corresponds to a time Î» = t = r in agreement with the theorem. 2.a By the spherical symmetry of the metric, initially radial geodesics remain radial. Fur thermore, the square of the 4velocity is conserved along a geodesic. Therefore, if the geodesic is radial and null to begin with, it remains radial and null. This is a rigorous argument but perhaps it is helpful to do it explicitly (if not, skip to 2.b ): Geodesics extremize the functional I = 1 2 R f Ë™ t 2 g Ë™ r 2 r 2 Ë™ Î© 2 dÎ» , id est they satisfy the equations d dÎ» ( f Ë™ t ) = 0 f Ë™ t 2 g Ë™ r 2 + 2 d dÎ» ( g Ë™ r ) 2 r Ë™ Î© 2 = 0 1 The sign comes from the fact that...
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 Fall '09
 Work, General Relativity, dv dv dv, Ted Jacobson, radial null rays, aï¬ƒne parameter

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