General Relativity: An Introduction for Physicists · Problem 3.19

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · geodesics-sphere-lagrangian-method

Problem

By considering the 'Lagrangian'

L = g_{ab}\dot{x}^a \dot{x}^b

, derive the equations for an affinely parameterised geodesic on the surface of a sphere in the coordinates

(\theta, \phi)

. Hence show that, of all the circles of constant latitude on a sphere, only the equator is a geodesic. Use your geodesic equations to pick out the connection coefficients in this coordinate system.

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