Special Relativity And Riemannian Geometry 1 (39.71 KiB) Viewed 36 times
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MAYBE YOU CAN USE SOME OF THESE FORMULAE
Special Relativity And Riemannian Geometry 2 (27.6 KiB) Viewed 36 times
QUESTION 5 a) Consider the contravariant tensor 1 [141-(0) 1/u in a space described by the metric tensor uu bul-() [9] 1 2 with z = u and x2 = v. What is the covariant form of [A"]? (5) b) Verify that if a tensor is symmetric in one frame, it will be symmetric in all coordinate frames. That is show that if it is given that X = X* in frame S, then it will be true that " = X1 in a coordinate frame S. (6) c) The Ricci tensor is defined as a contraction of the Riemann curvature tensor R., - ΣR', k and the Riemann curvature tensor has the following symmetries Rijkl = - Rik? Rijk = - Rijk Rijki = Rilij Use this definition and the symmetry properties of the Riemann curvature tensor to show that the Ricci tensor is symmetric, i.e. Rj = Rji Show all your steps carefully. (6)
e-V λει λεε: c+V Γ' = δε a.ck θρα ΤΕ -V υ 1- υπV/c2 dix d.χ.: Σr, d, dλ jk =0 dx2 J υ (1 - V/c2) U. γ (1 – υπV/c2) (ty - y (12 + y2)3/2 R1212 K ik 9 R', OΓ'. dri ΓΙ Γ'. + ack 1jk ΣΓ"Γ' -ΣΓ) Γ' ] τιk τι TH Τμν – (ρ + p/2) Uυν – pg θυ. zva = dra Gμων, Rμων» " A 29 δυα G ξα"R Ru - ERO ==KT Vau + ΣΓΧουλ dr3 A μ ara Σ δχ A 4. Or3 Σ acta A3
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