A torus of revolution is obtained by rotating a circle of radius (the tube radius)…

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

A torus of revolution is obtained by rotating a circle of radius rr (the tube radius) about an axis in its own plane, the axis lying a distance RR from the tube's centre circle, with R>rR \gt r. Points on the torus are parametrised by the poloidal angle θ\theta (angle round the tube's own circular cross-section) and the toroidal angle ϕ\phi (angle round the axis of revolution), with position vector r(θ,ϕ)=((R+rcos⁡θ)cos⁡ϕ, (R+rcos⁡θ)sin⁡ϕ, rsin⁡θ).\mathbf r(\theta,\phi) = \big((R+r\cos\theta)\cos\phi,\ (R+r\cos\theta)\sin\phi,\ r\sin\theta\big). ((a)) Show that the induced metric is ds2=r2 dθ2+(R+rcos⁡θ)2 dϕ2\mathrm ds^2 = r^2\,\mathrm d\theta^2 + (R+r\cos\theta)^2\,\mathrm d\phi^2. ((b)) Derive the nonzero Christoffel symbols Γθϕϕ\Gamma^\theta{}_{\phi\phi} and Γϕθϕ\Gamma^\phi{}_{\theta\phi} of the Levi-Civita connection of this metric. ((c)) Using the geodesic equation, show that every poloidal circle (constant ϕ\phi, θ\theta varying) is a geodesic, but that among the toroidal circles (constant θ\theta, ϕ\phi varying) only two are geodesics, and state which values of θ\theta these are.

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