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Consider the line element of the sphere of radius a: ds2 = a2(dθ2 + sin2 θ dØ2 ). The only non-vanishing Christoffel symbols are rθ ØØ = -sinθcosθ, rØ θØ = rØ Øθ = 1/tanθ a) Write down the metric and the inverse metric, and use the definition rpµv = 1/2gpσ(δµgvσ + δvgµσ - δσgµv) = rpvµ to reproduce the results written above for rθ ØØ and rØ θØ. b) Write down the two components of the geodesic equation. c) The geodesics of the sphere are great circles. Thinking of θ = 0 as the North pole and θ = π as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator

Consider the line element of the sphere of radius a: ds2 = a2(dθ2 + sin2 θ dØ2 ). The only non-vanishing Christoffel symbols are rθ ØØ = -sinθcosθ, rØ θØ = rØ Øθ = 1/tanθ a) Write down the metric and the inverse metric, and use the definition rpµv = 1/2gpσ(δµgvσ + δvgµσ - δσgµv) = rpvµ to reproduce the results written above for rθ ØØ and rØ θØ. b) Write down the two components of the geodesic equation. c) The geodesics of the sphere are great circles. Thinking of θ = 0 as the North pole and θ = π as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator

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Consider the line element of the sphere of radius a:
ds2 = a2(dθ2 + sin2 θ dØ2 ).
The only non-vanishing Christoffel symbols are

rθ ØØ = -sinθcosθ, rØ θØ = rØ Øθ = 1/tanθ

a) Write down the metric and the inverse metric, and use the definition
rpµv = 1/2gpσ(δµgvσ + δvgµσ - δσgµv) = rpvµ
to reproduce the results written above for rθ ØØ and rØ θØ.
b) Write down the two components of the geodesic equation.
c) The geodesics of the sphere are great circles. Thinking of θ = 0 as the North pole and θ = π as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and
also the solution corresponding to the equator.
 

Consider the line element of the sphere of radius a: ds²a² (do²+ sin² 0 do ²). The only non-vanishing Christoffel symbols are го = = sin 0 cos 0, ФФ ГР = rø ГФ00 = ГФ = 2.900 a) Write down the metric and the inverse metric, and use the definition 1 to reproduce the results written above for rº (@ugvo + avguo dogur) = rº vp - ΦΘ and r op = 00* 1 tan 0 b) 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 T as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator.
Transcribed Image Text:Consider the line element of the sphere of radius a: ds²a² (do²+ sin² 0 do ²). The only non-vanishing Christoffel symbols are го = = sin 0 cos 0, ФФ ГР = rø ГФ00 = ГФ = 2.900 a) Write down the metric and the inverse metric, and use the definition 1 to reproduce the results written above for rº (@ugvo + avguo dogur) = rº vp - ΦΘ and r op = 00* 1 tan 0 b) 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 T as the South pole, find a set a solutions to the geodesic equation corresponding to meridians, and also the solution corresponding to the equator.
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