Problem 3. (25 pts) (a) (5 pts) State all symmetries of the Riemann tensor Ruvpo. Use the result to explain why in two-d

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Problem 3 25 Pts A 5 Pts State All Symmetries Of The Riemann Tensor Ruvpo Use The Result To Explain Why In Two D 1 (173.66 KiB) Viewed 21 times

Problem 3. (25 pts) (a) (5 pts) State all symmetries of the Riemann tensor Ruvpo. Use the result to explain why in two-dimensional space, all the components of the Riemann tensor are either zero or equal to R1212. Please argue that R1212 = R2121. (b) Consider a two-dimensional space with coordinates x₁ = and with line element 0 and x2 = ds² = a²dr + a² sin² x dx², (4) where a is a constant. (i) (5 pts) Please write down the metric tensor guv, and the inverse metric tensor gv. (ii) (10 pts) Calculate all the Christoffel symbols, including the zero and nonzero ones. Prove that the only nonzero terms are l'¹22, and 1²12 T221 Employ these results to show that = R1212 a² sin²x₁. = (c) (5 pts) Compute all the components of the Ricci tensor and the Ricci scalar. Show that the Einstein tensor Guv is zero. μν END OF QUESTION 3