hw4sol - Physics 570 Homework No. 4 Solutions: due...

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Unformatted text preview: Physics 570 Homework No. 4 Solutions: due Wednesday, 17 February, 2010 1. Recall the gravitational field of a static, spherically-symmetric mass by , exterior to that mass, that was discussed in the previous problem set: g = ds 2 = 1 H 2 dr 2 + r 2 ( d 2 + sin 2 d 2 )- H 2 dt 2 , H 1 + 2 , - M r . During that problem you a) evaluated the metric-compatible connection forms known as Christoffel 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 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 d u + u ( e u ) = 0 that determine geodesic, timelike worldlines in these coordinates, where is the parameter along these curves with tangent vector e u . [5 pts] ........................................................................................... We first re-iterate the desired orthonormal frame for this metric: r = 1 H dr , = r,d , = r sin d , t = H dt , and then proceed to determine the affine connection 1-forms, using the guess method: 0 = d r = r + r + t r t = r , r , r t t , and then the next one: H r r = dr d = d = r r + + t t = r = H r , , t t . Continuing we find H r r + cot...
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This note was uploaded on 03/08/2010 for the course PHYSICS AN 570 taught by Professor Davids.king during the Spring '10 term at Caltech.

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hw4sol - Physics 570 Homework No. 4 Solutions: due...

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