Roll a flat sheet into a cone

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor

Problem

Roll a flat sheet into a cone. Measured along the surface from the tip, with θ\theta the angle round the axis, the cone has the metric ds2=dr2+α2r2 dθ2,r>0, θ∈[0,2π),ds^2 = dr^2 + \alpha^2r^2\,d\theta^2,\qquad r>0,\ \theta\in[0,2\pi), where α=0.80\alpha = 0.80 is a constant fixed by how much sheet was cut away. ((a)) Find the non-zero Christoffel symbols and show that Rrθrθ=0R^r{}_{\theta r\theta} = 0 at every r>0r>0. What does this say about every curvature invariant of the cone away from its tip? ((b)) Parallel-transport a vector once round the circle r=r0r = r_0, starting from V=∂rV = \partial_r at θ=0\theta = 0. Through what angle, in degrees, has it turned when it comes back? ((c)) Show that your answer to (b) does not depend on r0r_0, and use the small-loop result of this chapter to decide whether the tip r=0r = 0 can be an ordinary point of a smooth surface, even though every invariant is zero right up to it.

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