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Astrophysics ASTR3415 Homework Assignment No.2 (Due 10/24/05, 5pm) 1. In a particular coordinate system, the components of a (2,0) tensor T are symmetric (meaning that T ij T ji for all i, j). Show that the components of T must be symmetric in any coordinate system. [4] 2. An orthogonal metric is one for which g 0 and g 0 unless . As you saw in the lectures, the general expression for the Christoffel symbols in terms of the metric and its partial derivatives is 12 g g , g , g , , where the repeated upper and lower index indicates summation over and commas denote partial differentiation. By eliminating the terms in the above expression which are zero, show that for an orthogonal metric 0 if , , are all different 12 g g, 12 g g , 12 g g , (note: the repeated usage of and in the above expressions does not denote the Einstein summation convention. If you’re unsure about the notation here, come and ask me). [8] 3. The invariant interval, expressed in polar coordinates x r cos , y r sin , for 2-D Euclidean space is given by ds 2 dr 2 r 2 d 2 Use the results of Q.2 to show that rrr 0 , r r and rr 0 rr 0 , 0 and r 1 r [8] 4. Consider the vector field V with Cartesian components V x ,V y x 2 3 y, y 2 3x . a. Using the transformation law for a (1,0) tensor and the results of Q.7 of Homework Assignment 1, show that the components V r ,V of the vector field with respect to the usual polar coordinate basis er , e may be written as V r r 2 cos 3 sin 3 6r cos sin V 3 cos 2 sin 2 r cos sin sin cos [6] b. Using the formula for the covariant derivative V i ; j V i , j ijkV k , verify that the covariant derivative of V has the following components with respect to the Cartesian basis V x ; x 2 x, V x ; y 3, V y ; x 3, V y ; y 2 y [4] c. The covariant derivative of a (1,0) tensor transforms as a (1,1) tensor. Using the transformation law for the components of the covariant derivative V ;j i x i x l k k V ;l x x j show that V ;r 2 cos sin sin cos 3r cos 2 sin 2 [6] d. Now, using the results given in Q.3 and Q.4a, apply the explicit formula for the covariant derivative (as given in Q.4b only this time in a polar coordinate system) and verify that you get the same result for V ;r as you did in Q.4c. (This will also be the case for all the other components of the covariant derivative). [6] 5. In relativity we frequently measure time in units of length – e.g. we replace 1 second by the distance (in metres) travelled by light in this time. This conversion is equivalent to setting c 1. a. Given that in SI units G 6.67 10 11 N m 2 kg 2 and 1 N 1 kg ms 2 , show that G 7.41 10 28 m kg 1 [3] b. Hence show that setting G 1 is equivalent to measuring mass in units of length, such that 1 m 1.34 10 27 kg . [2] c. The Schwarzschild radius of a star of mass M is given by RS or (setting 2GM c2 c 1 and G 1) RS 2M . Using the conversion factor from Q.5b, calculate the Schwarzschild radius of: i. the Earth (mass 6 10 24 kg ); ii. the Sun (mass 2 10 30 kg ); iii. a supermassive black hole (mass = 10 7 solar masses) [3] Total: [50]