A vector is parallel transported once around the circle of colatitude on a two-sphere…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

A vector is parallel transported once around the circle of colatitude θ0=45∘\theta_0 = 45^\circ on a two-sphere of radius aa, starting and ending at the same point. Using the two coupled transport equations for the components Vθ,VϕV^\theta, V^\phi along this circle, dVθdϕ=sin⁡θ0cos⁡θ0 Vϕ,dVϕdϕ=−cot⁡θ0 Vθ,\frac{dV^\theta}{d\phi} = \sin\theta_0\cos\theta_0\,V^\phi, \qquad \frac{dV^\phi}{d\phi} = -\cot\theta_0\,V^\theta, find the angle through which the transported vector has rotated, relative to the local coordinate basis, after one complete circuit (ϕ\phi increasing by 2π2\pi).

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