3. The two-sphere has the metrie ds2 = dr2 + sinado2 a) Find all nonzero Christo

Question

3. The two-sphere has the metrie ds2 = dr2 + sinado2 a) Find all nonzero Christoffel symbols. b) Find all independent components of the Riemann o (Hint: With all indices lowered, the Riemann tensor Riy obeys RRjikkRuj This dramatically reduces the number of components to be checked. To see this, go to a frame where the Christoffels vanish and use problem 2. Since Rijki obeys the requisite symmetry in this frame, it does so in any frame.) c) Find the contractions Rik-gijRijkl and R = gikR.. (These are respectively known as the Ricci tensor and the Ricci scalar).

Explanation / Answer

If the form of your metric such that r and phi are varying, all the components of the Christoffel symbols are vanishing and there is no Riemann tensor associated with the metric. It is just a flat space-time metric.

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