Answer Happy

(15 pts) Problem 4. It is known that the Riemann curvature tensor satisfies the Bianchi identity (5) Γλ Κίρσμπ + ΓρRίσλμ + Γσ Κλρμν = 0, where V represents the covariant derivative. Please answer the following questions with tensor analysis, and you are free to use all the symmetries of the Riemann curvature tensor. (a) (5 pts) In the local inertial frame, the Riemann curvature tensor can be represented as 1 ² go 9pv Ρόρσμπ = 8²q 8²9pp მეთ მეს მე" მეს 2² gov + Əx²x¹ მიმ please use it to proof eq.(5). (b) (5 pts) By repeatedly contracting the indices pairs (v, o) and then (u, A) of eq.(5), please show that VGpμ = Rpμ 7pR=0. 2 (c) (5 pts) One version of Einstein's field equation is 1 (6) Ruv - Rguv + Aguv=8nGTμv, 2 where A is the cosmological constant, G is the Newtonian constant and the T is the energy-momentum tensor. By first contracting the indices (u, v), show that eq.(6) can be rewritten into Ruv = 8πG| (Trv - 2T9μv) + Agnv, (7) where Tg Tuv. END OF QUESTION