A navigator argues: "Two ports lie on the parallel; since that circle looks perfectly…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch The metric, the connection and geodesics

Problem

A navigator argues: "Two ports lie on the 45∘N45^\circ\mathrm{N} parallel; since that circle looks perfectly round and symmetric on the globe, sailing along it must be the shortest, straight-line route." Take the sphere with colatitude θ\theta (so θ=90∘−latitude\theta = 90^\circ - \text{latitude}) and the standard Christoffel symbol Γθϕϕ=−sin⁡θcos⁡θ\Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta. ((a)) Evaluate Γθϕϕ\Gamma^\theta{}_{\phi\phi} at θ=45∘\theta = 45^\circ (the parallel in question) and at θ=90∘\theta = 90^\circ (the equator). ((b)) Using the geodesic equation for a curve of constant colatitude, decide whether the 45∘45^\circ parallel is a geodesic, and contrast with the equator.

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