A cone of half-angle has metric ds^2=dr^2+r^2 ^2 _ c ,d ^2

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 cone of half-angle βc=30∘\beta_{\rm c}=30^\circ has metric ds2=dr2+r2sin⁡2βc dϕ2,r≥0, ϕ∈[0,2π).ds^2=dr^2+r^2\sin^2\beta_{\rm c}\,d\phi^2, \qquad r\ge0,\ \phi\in[0,2\pi). A colleague argues: "This cannot be brought to the constant form dr2+r2dϕ2dr^2+r^2d\phi^2 of the flat plane, because of the fixed factor sin⁡2βc<1\sin^2\beta_{\rm c}<1 multiplying dϕ2d\phi^2 everywhere; the cone is therefore intrinsically curved, unlike the plane." ((a)) Decide whether this argument is correct by finding coordinates in which the metric is constant. ((b)) If the metric can be made constant, compute the angular deficit, in degrees, of the cone compared with the full plane.

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