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
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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.
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 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
(8μgvo + avguo doguv) = 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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