Problem 3. (20 pts) (a) (5 pts) State all symmetries of the Riemann tensor Rapo. Use the result to explain why in two-di

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Problem 3 20 Pts A 5 Pts State All Symmetries Of The Riemann Tensor Rapo Use The Result To Explain Why In Two Di 1 (149.3 KiB) Viewed 21 times

Problem 3. (20 pts) (a) (5 pts) State all symmetries of the Riemann tensor Rapo. Use the result to explain why in two-dimensional spacetime, all the components of the Riemann tensor are either zero or equal to ±Ro101. Please argue that R0101 = R1010- (b) Consider a two-dimensional spacetime with coordinates xo = t and x₁ = r and with line element ds² = t−² (−dt² + dr²), where a is a constant. (i) (2 pts) Please write down the metric tensor guv, and the inverse metric tensor gv. (ii) (8 pts) Calculate all the Christoffel symbols, including the zero and nonzero ones. Prove that the only nonzero terms are o rº, and I¹¹01¹10. Employ these results to show that = 11) R0101 1 t4 (c) (5 pts) Compute all the components of the Ricci tensor and the Ricci scalar. Show that the Einstein tensor G is zero. HV END OF QUESTION 3