A small moon is treated as a sphere of radius km with metric in the usual colatitude…

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

A small moon is treated as a sphere of radius a=1500a=1500 km with metric ds2=a2(dθ2+sin⁡2θ dϕ2)ds^2 = a^2(d\theta^2+\sin^2\theta\,d\phi^2) in the usual colatitude and longitude (θ,ϕ)(\theta,\phi). Work at the point θ0=75∘\theta_0=75^\circ. ((a)) The tangent vector V=5∂θ+2∂ϕ\mathbf V = 5\partial_\theta + 2\partial_\phi acts on the coordinate function f=θf=\theta. Evaluate V(f)\mathbf V(f). ((b)) Find the length of the full circle of latitude at θ0\theta_0. ((c)) Find the angle between the tangent vectors U=∂θ+∂ϕ\mathbf U=\partial_\theta+\partial_\phi and W=∂θ−∂ϕ\mathbf W=\partial_\theta-\partial_\phi at θ0\theta_0. ((d)) At θ0\theta_0 a covector has component ωθ=2.0×106 km2\omega_\theta = 2.0\times10^6\ \text{km}^2. Use the inverse metric gμνg^{\mu\nu} to find ωθ\omega^\theta. ((e)) How many independent components does a totally antisymmetric tensor with four lower indices have in four dimensions?

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