On a sphere of radius m

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Relativity & Gravitation · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Manifolds, vectors, tensor fields and the metric

Problem

On a sphere of radius a=2.0a=2.0 m, consider the covector ω=a2sin⁡θcos⁡θ dθ\boldsymbol\omega=a^2\sin\theta\cos\theta\,d\theta and the vector V=∂θ+∂ϕ\mathbf V=\partial_\theta+\partial_\phi, both at θ=45∘\theta=45^\circ. A student argues: "dθd\theta is an infinitesimally small change in θ\theta. Since V\mathbf V moves in both the θ\theta direction and the ϕ\phi direction at once, and on a curved surface moving in ϕ\phi can drag θ\theta along with it by a small amount, evaluating ω(V)\boldsymbol\omega(\mathbf V) needs to know how much that drag is, which needs a rule for comparing nearby tangent spaces, something this chapter has not yet built. So ω(V)\boldsymbol\omega(\mathbf V) cannot be computed exactly from what we have so far." ((a)) Find the specific error in this argument. ((b)) Give the correct value of ω(V)\boldsymbol\omega(\mathbf V) at θ=45∘\theta=45^\circ, showing that no rule for comparing tangent spaces is needed.

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