For a curve on a manifold

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

For a curve xμ(s)x^\mu(s) on a manifold, the tangent vector at each point, as defined from the directional-derivative operator on curves, has coordinate components Uμ=dxμ/dsU^\mu = \mathrm{d}x^\mu/\mathrm{d}s. The geodesic equation states that this tangent vector is parallel transported along the curve itself, DUμds≡dUμds+ΓμνλUνUλ=0.\frac{\mathrm{D}U^\mu}{\mathrm{d}s} \equiv \frac{\mathrm{d}U^\mu}{\mathrm{d}s} + \Gamma^\mu{}_{\nu\lambda}U^\nu U^\lambda = 0. On the unit sphere (a=1a=1) with coordinates (θ,ϕ)(\theta,\phi), the nonzero Christoffel symbols are Γθϕϕ=−sin⁡θcos⁡θ\Gamma^\theta{}_{\phi\phi} = -\sin\theta\cos\theta and Γϕθϕ=Γϕϕθ=cot⁡θ\Gamma^\phi{}_{\theta\phi} = \Gamma^\phi{}_{\phi\theta} = \cot\theta. Consider the equatorial curve θ(s)=π/2\theta(s) = \pi/2, ϕ(s)=s\phi(s) = s (an arc-length parametrisation since the sphere has unit radius). ((a)) Write down UθU^\theta and UϕU^\phi for this curve, and evaluate DUθ/ds\mathrm{D}U^\theta/\mathrm{d}s. ((b)) Evaluate DUϕ/ds\mathrm{D}U^\phi/\mathrm{d}s, and state whether the equator is confirmed to be a geodesic by this direct check.

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