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Unformatted text preview: Physics 523, Introduction to Relativity Homework 4 Due Friday, 28 th October 2011 Hans Bantilan Geodesics on S 2 Consider the 2sphere, with metric expressed in spherical polars x i = ( , ) as g = d 2 + sin 2 d 2 . Given a tangent vector field ( ) of some curve :[ a, b ] S 2 , the geodesic equation = 0 has components given by d 2 d 2 cos sin d d 2 = 0 d 2 d 2 + 2 cot d d d d = 0 . a. We are to verify that curves c ( s ) with = const are geodesics, and that a curve c ( t ) with = const is a geodesic iff. = / 2. Let us parametrize by arc length so that k c ( s ) k = const and k c ( t ) k = const. By inspection, we see that c ( s ) satisfies the geodesic equation when d 2 ds 2 = 0, which holds since we have k c ( s ) k = const d ds = const . Similarly, we see that c ( t ) satisfies the geodesic equation when cos sin d dt 2 = 0 and d 2 dt 2 = 0. The first equality holds 1 iff. = / 2. The second equality holds since we have k c ( t ) k = const d dt = const . b. Consider a geodesic ( ) with = const, and a vector V = T (0) M . Let us parametrize by = . We are to find the parallel transport V ( ) of V along ( ). The parallel transport V ( ) satisfies the equation V ( ) = 0, with initial condition V (0) = . The components of the equation satisfied by V ( ) are given by dV d cos sin V = 0 dV d + cot V...
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