Ordinary flat three-dimensional space is described by spherical coordinates with metric

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

Ordinary flat three-dimensional space is described by spherical coordinates (r,θ,ϕ)(r,\theta,\phi) with metric ds2=dr2+r2dθ2+r2sin⁡2θ dϕ2ds^2 = dr^2 + r^2d\theta^2 + r^2\sin^2\theta\,d\phi^2. Work at the point r=3.0r=3.0 m, θ=45∘\theta=45^\circ. ((a)) The tangent vector V=(1.0 m) ∂r+2∂θ\mathbf V = (1.0\ \mathrm{m})\,\partial_r + 2\partial_\theta acts on f(r,θ)=rcos⁡θf(r,\theta) = r\cos\theta. Evaluate V(f)\mathbf V(f). ((b)) Find the length of the arc traced by keeping rr and θ\theta fixed at these values while ϕ\phi increases from 00 to 90∘90^\circ. ((c)) At this point a covector has component ωϕ=9.0 m2\omega_\phi = 9.0\ \text{m}^2. Use the inverse metric gμνg^{\mu\nu} to find ωϕ\omega^\phi. ((d)) How many independent components does a totally antisymmetric tensor with three lower indices have in this three-dimensional space?

Browse the problem library