A cone has half-angle at its apex

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 cone has half-angle α\alpha at its apex, meaning a point at slant distance ρ\rho from the apex lies a perpendicular distance ρsin⁡α\rho\sin\alpha from the axis. Using the slant distance ρ≥0\rho\ge0 and the azimuthal angle ϕ\phi round the axis as coordinates, the induced metric on the cone's surface is ds2=dρ2+ρ2sin⁡2α dϕ2.\mathrm ds^2=\mathrm d\rho^2+\rho^2\sin^2\alpha\,\mathrm d\phi^2. ((a)) Derive the Christoffel symbols Γρϕϕ\Gamma^\rho{}_{\phi\phi} and Γϕρϕ\Gamma^\phi{}_{\rho\phi}, and write the two geodesic equations. ((b)) Show that L≡ρ2ϕ˙L\equiv\rho^2\dot\phi (dot =d/ds=\mathrm d/\mathrm ds) is constant along a geodesic. ((c)) Unroll the cone onto a flat sector using ψ=ϕsin⁡α\psi=\phi\sin\alpha, so the metric becomes ds2=dρ2+ρ2dψ2\mathrm ds^2=\mathrm d\rho^2+\rho^2\mathrm d\psi^2, the flat plane in polar form. A straight line in that plane at perpendicular distance pp from the origin satisfies ρcos⁡(ψ−ψ0)=p\rho\cos(\psi-\psi_0)=p. Use this to find the shape ρ(ϕ)\rho(\phi) of a geodesic on the cone that passes through its point of closest approach to the apex, ρ=ρ0\rho=\rho_0, at ϕ=0\phi=0, moving there purely in the ϕ\phi-direction. ((d)) Take α=30∘\alpha=30^\circ and ρ0=8.0 cm\rho_0=8.0\,\mathrm{cm}. Find the slant distance ρ\rho on this geodesic at ϕ=40∘\phi=40^\circ.

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