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# hw4sol - Physics 570 Homework No 4 1 Solutions due...

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Physics 570 Homework No. 4 Solutions: due Wednesday, 17 February, 2010 1. Recall the gravitational field of a static, spherically-symmetric mass by Φ, exterior to that mass, that was discussed in the previous problem set: g = ds 2 = 1 H 2 dr 2 + r 2 ( 2 + sin 2 θ dϕ 2 ) - H 2 dt 2 , H ≡ 1 + 2Φ , Φ ≡ - M r . During that problem you a) evaluated the metric-compatible connection forms known as Christ¨offel symbols and wrote down the geodesic equations for a timelike path, and b) created an orthonor- mal set of basis 1-forms. I would now like you to determine the connection 1-forms Γ ˆ α ˆ β associated with this orthonormal basis, and again write down the geodesic equations, but of course this time for the 4 components of the 4-velocity vector relative to the orthonormal basis, i.e., determine the 4 equations d u ˆ α + u β Γ ˆ α ˆ β ( e u ) = 0 that determine geodesic, timelike worldlines in these coordinates, where τ is the parameter along these curves with tangent vector e u . [5 pts] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We first re-iterate the desired orthonormal frame for this metric: ω ˆ r = 1 H dr , ω ˆ θ = r, dθ , ω ˆ ϕ = r sin θ dϕ , ω ˆ t = H dt , and then proceed to determine the affine connection 1-forms, using the “guess” method: 0 = d ω ˆ r = ω ˆ θ Γ ˆ r ˆ θ + ω ˆ ϕ Γ ˆ r ˆ ϕ + ω ˆ t Γ ˆ r ˆ t = Γ ˆ θ ˆ r ω ˆ θ , Γ ˆ ϕ ˆ r ω ˆ ϕ , Γ ˆ r ˆ t ω ˆ t , and then the next one: H r ω ˆ r ω ˆ θ = dr = d ω ˆ θ = ω ˆ r Γ ˆ θ ˆ r + ω ˆ ϕ Γ ˆ θ ˆ ϕ + ω ˆ t Γ ˆ θ ˆ t = Γ ˆ θ ˆ r = H r ω ˆ θ , Γ ˆ θ ˆ ϕ ω ˆ ϕ , Γ ˆ t ˆ θ ω ˆ t .

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Continuing we find H r ω ˆ r ω ˆ ϕ + cot θ r ω ˆ θ ω ˆ ϕ = sin θ dr + r cos θ dθ = d ω ˆ ϕ = ω ˆ r Γ ˆ ϕ ˆ r + ω ˆ θ
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hw4sol - Physics 570 Homework No 4 1 Solutions due...

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