hw5 - 3 Using the definition of the Riemann curvature in...

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Physics 570 Homework No. 5 due Wednesday, 24 February, 2010 1. After your experience with the 2-sphere, let’s consider the 3-sphere, as a subset of 4-space, with the following metric, in the coordinates { ψ,θ,ϕ } : g = 2 + sin 2 ψ ( 2 + sin 2 θ dϕ 2 ) . Again choose an appropriate orthonormal basis, work out the connection 1-forms, and determine the Cartan curvature 2-forms. Use those to show that R ˆ ρ ˆ σ ˆ μ ˆ ν Rg ˆ ρ μ g ˆ σ ˆ ν ] , where R is the so-called Ricci scalar, i.e., the complete trace of the curvature tensor: R g ˆ ρ ˆ μ g ˆ σ ˆ ν R ˆ ρ ˆ σ ˆ μ ˆ ν . What is the (numerical) constant of proportionality? I note that such a relationship can be shown to be true for every manifold of maximal symmetry. Also please determine the relationship between the Ricci tensor and the metric. 2. Using the orthonormal basis and connections for the metric involving the (spherically-symmetric) gravitational potential, please determine the various curvature 2-forms, Ω ˆ μ ˆ ν , and from that the components of the Ricci tensor.
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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, 4-dimensional 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 2-forms for this metric, and separate out the various invariant parts, i.e., the Ricci tensor, the Ricci scalar, and the self-dual and anti-self-dual 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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