hw4soln - Homework 4 Ted Jacobson and William D. Linch III...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 4

hw4soln - Homework 4 Ted Jacobson and William D. Linch III...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online