Working with polar coordinates on the ordinary flat plane,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Invariance, Ch Curvature

Problem

Working with polar coordinates (r,ϕ)(r,\phi) on the ordinary flat plane, ds2=dr2+r2 dϕ2\mathrm{d}s^2 = \mathrm{d}r^2 + r^2\,\mathrm{d}\phi^2, someone argues: "The non-zero Christoffel symbols are Γrϕϕ=−r\Gamma^r{}_{\phi\phi} = -r and Γϕrϕ=Γϕϕr=1/r\Gamma^\phi{}_{r\phi} = \Gamma^\phi{}_{\phi r} = 1/r. Since these do not vanish, parallel transport in this coordinate system changes a vector's components as it moves, so the plane described in polar coordinates must be curved, and the curvature must blow up as r→0r \to 0 because Γϕrϕ=1/r\Gamma^\phi{}_{r\phi} = 1/r diverges there." ((a)) Identify the error in this argument. ((b)) Compute the component RϕrϕrR^\phi{}_{r\phi r} of the Riemann tensor for this metric using Rρσμν=∂μΓρνσ−∂νΓρμσ+ΓρμλΓλνσ−ΓρνλΓλμσR^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}, and use the result to settle whether the plane in polar coordinates is curved.

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