Answer Happy

Special Relativity and Riemannian Geometry
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FORMULASHEET
QUESTION 4 Consider the surface that can be parameterized as x(u, v) = u cos v usin v 2 (u, v) = u for u, v € (0, 2). a) Find the line element for the surface. (9) b) Let x1 = u and r2 = v. What is the metric tensor and the dual metric tensor? Explain how you determine these. (5) c) The only non-zero Christoffel coefficients for this surface are l'1, 1122, 1²2 and 13. Calculate these. (10) d) What is the value of the component R 212 of the Riemann curvature tensor? Make sure you simplify your answer. (9)

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