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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 ( ) and t ( ), with fixed angles. As they form a null geodesic congruence, they extremize the functional I = 1 2 R ( t 2- r 2 ). (The angular part is trivially stationary since it is quadratic in the derivatives = = 0.) The Euler-Lagrange equations for r following from this functional are 0 = r . Hence, r = a + 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 r 2 . Then, on the past light cone 1 = d Area dr Area r = r =- 2 4 r 4 r 2 =- 2 r . (1) 1.c According to the focusing theorem, - at or before an affine parameter time = 2 | |- 1 . Solving (1) gives r =- 2 = 2 | |- 1 . On the cone 0 = t 2- r 2 this corresponds to a time = 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 4-velocity 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 t 2- g r 2- r 2 2 d , id est they satisfy the equations d d ( f t ) = 0 f t 2- g r 2 + 2 d d ( g r )- 2 r 2 = 0 1 The sign comes from the fact that...
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