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 (8μgvo + 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 = 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 (8μgvo + 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 = 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.
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
Related questions
Question
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 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
(8μgvo + 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 = 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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