On the Earth

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 the Earth, treated as a sphere of radius a=6371a = 6371 km, use the usual colatitude and longitude (θ,ϕ)(\theta,\phi) with metric ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2 = a^2(d\theta^2 + \sin^2\theta\,d\phi^2). Work at the point θ0=60∘\theta_0 = 60^\circ (latitude 30∘30^\circN). ((a)) The tangent vector V=2∂θ\mathbf V = 2\partial_\theta acts on the function f=sin⁡θf = \sin\theta. Evaluate V(f)\mathbf V(f) at θ0\theta_0. ((b)) Find the length of the full circle of latitude at θ0\theta_0. ((c)) Find the angle between the tangent vectors U=∂θ\mathbf U = \partial_\theta and W=2∂θ+∂ϕ\mathbf W = 2\partial_\theta + \partial_\phi at θ0\theta_0. ((d)) At θ0\theta_0 a covector has component ωϕ=1.0×107 km2\omega_\phi = 1.0\times10^7\ \text{km}^2. Use the inverse metric gμνg^{\mu\nu} to find ωϕ\omega^\phi. ((e)) How many independent components does a totally antisymmetric tensor with three lower indices have in four dimensions?

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