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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 = ( θ, φ ) as g = dθ 2 + sin 2 θdφ 2 . Given a tangent vector field ˙ γ ( λ ) of some curve γ :[ a, b ] → S 2 , the geodesic equation ∇ ˙ γ ˙ γ = 0 has components given by d 2 θ dλ 2 cos θ sin θ dφ dλ 2 = 0 d 2 φ dλ 2 + 2 cot θ dθ dλ dφ dλ = 0 . a. We are to verify that curves c φ ( s ) with φ = const are geodesics, and that a curve c θ ( t ) with θ = const is a geodesic iff. θ = π/ 2. Let us parametrize by arc length so that k ˙ c φ ( s ) k = const and k ˙ c θ ( t ) k = const. By inspection, we see that c φ ( s ) satisfies the geodesic equation when d 2 θ ds 2 = 0, which holds since we have k ˙ c φ ( s ) k = const ⇔ dθ ds = const . Similarly, we see that c θ ( t ) satisfies the geodesic equation when cos θ sin θ dφ dt 2 = 0 and d 2 φ dt 2 = 0. The first equality holds 1 iff. θ = π/ 2. The second equality holds since we have k ˙ c θ ( t ) k = const ⇔ dφ dt = const . b. Consider a geodesic γ ( λ ) with θ = const, and a vector V = ∂ θ ∈ T γ (0) M . Let us parametrize by λ = φ . We are to find the parallel transport V ( λ ) of V along γ ( λ ). The parallel transport V ( λ ) satisfies the equation ∇ ˙ γ V ( λ ) = 0, with initial condition V (0) = ∂ θ . The components of the equation satisfied by V ( λ ) are given by dV θ dλ cos θ sin θV φ = 0 dV φ dλ + cot θV...
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 '09
 FRANSPRETORIUS
 Work, General Relativity, Riemannian geometry, Geodesic, R∗, dr dt dt

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