Unformatted text preview: 3. Using the deﬁnition of the Riemann curvature in terms of the commutator of covariant derivatives, show that it acts as a tensor in its 3rd argument. More precisely, show that R ( e u, e v )( f e w ) = f R ( e u, e v )( e w ) . 4. Probably for about the last time, let us consider a diﬀerent, 4dimensional space, with coordinates { χ,θ,ϕ,t } and metric, where K is a constant: g ≡ K 2 { cos 2 t [ dχ 2 + sinh 2 χ ( dθ 2 + sin 2 θ dϕ 2 )]dt 2 } . Please choose an appropriate orthonormal basis set, and determine the curvature 2forms for this metric, and separate out the various invariant parts, i.e., the Ricci tensor, the Ricci scalar, and the selfdual and antiselfdual parts of the Weyl tensor. Although the algebra is a bit lengthy, I believe this is a worthwhile calculation, since this is a form of a metric important for cosmological considerations....
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 Spring '10
 DavidS.King
 General Relativity, Ricci, Ricci tensor, appropriate orthonormal basis, Cartan curvature 2forms

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