a) Write down the metric and the inverse metric, and use the definitionrpµv = 1/2gpσ(δµgvσ + δvgµσ - δσgµv) = rpvµto reproduce the results written above for rθ ØØ and rØ θØ. Consider the line element of the sphere of radius a:ds²a² (do²+ sin² 0 do ²).The only non-vanishing Christoffel symbols areго = -sin cos 0,ФФГР =røГФ00 = ГФ=2.900a) Write down the metric and the inverse metric, and use the definition1to reproduce the results written above for rº(8μgvo + avguo doguv) = rºvp-ΦΘand rop=00*1tan 0b) Write down the two components of the geodesic equation.=c) The geodesics of the sphere are great circles. Thinking of 0 = 0 as the North pole and Tas the South pole, find a set a solutions to the geodesic equation corresponding to meridians, andalso the solution corresponding to the equator.

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Consider the line element of the sphere of radius a: The only non-vanishing Christoffel symbols are го op ds² = a² (d0² + sin² 0 do ²). == sin cos 0, Γ ГР = 00 to reproduce the results written above for ro ФФ = Γ a) Write down the metric and the inverse metric, and use the definition 1 2 go (@μgvo + avgμo - Jogμv) = 1₂ vp and re ΦΘ – 06. 1 tan 0

Consider the line element of the sphere of radius a: The only non-vanishing Christoffel symbols are го op ds² = a² (d0² + sin² 0 do ²). == sin cos 0, Γ ГР = 00 to reproduce the results written above for ro ФФ = Γ a) Write down the metric and the inverse metric, and use the definition 1 2 go (@μgvo + avgμo - Jogμv) = 1₂ vp and re ΦΘ – 06. 1 tan 0