prob5 - A.V. Manohar Ph225A: General Relativity Problem Set...

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Unformatted text preview: A.V. Manohar Ph225A: General Relativity Problem Set 5 1. Compute the Riemann tensor for the metric ds2 = e2A(r) dr2 + r2 d2 - e2B(r) dt2 using the Cartan method, and then compute the Ricci tensor and Ricci scalar. 2. Prove the geodesic deviation equation (u u y) = -R u u y where u is the tangent vector of a geodesic. Do this by considering a geodesic x (s) and a neighboring geodesic x (s) + y (s), where y is infinitesimal. Assume torsion vanishes. 3. Show that for Riemann normal coordinates at a point P0 (a) x (b) (P0 ) = 0 (c) x =- P0 = e (P0 ) P0 1 R + R 3 P0 If there is a metric, and g(e , e ) = , then one has in addition (d) g (P0 ) = (e) g, (P0 ) = 0 (f) 1 g, (P0 ) = - (R + R )P0 3 (g) g, (P0 ) - g, (P0 ) = R (P0 ) 1 ...
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This note was uploaded on 04/19/2008 for the course PHYS 225 taught by Professor Manohar during the Fall '07 term at UCSD.

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