A student is unconvinced that fails to transform as a tensor

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

The Geometry of Gravity, Ch The covariant derivative, parallel transport and geodesics

Problem

A student is unconvinced that Γλμν\Gamma^\lambda{}_{\mu\nu} fails to transform as a tensor, and argues: "The Cartesian components of the flat plane's connection are all zero, Γabc=0\Gamma^a{}_{bc}=0. If Γ\Gamma transformed as an ordinary tensor, its components in polar coordinates would be Γ′λ′μ′ν′=∂x′λ′∂xρ∂xσ∂x′μ′∂xτ∂x′ν′ Γρστ,\Gamma'^{\lambda'}{}_{\mu'\nu'}=\frac{\partial x'^{\lambda'}}{\partial x^\rho}\frac{\partial x^\sigma}{\partial x'^{\mu'}}\frac{\partial x^\tau}{\partial x'^{\nu'}}\,\Gamma^\rho{}_{\sigma\tau}, and since every Γρστ=0\Gamma^\rho{}_{\sigma\tau}=0 on the right, every Γ′λ′μ′ν′\Gamma'^{\lambda'}{}_{\mu'\nu'} must vanish in polar coordinates too. But the chapter states Γrϕϕ=−r\Gamma^r{}_{\phi\phi}=-r, which is not zero. One of these must be wrong." ((a)) Identify exactly which step in this reasoning is the error. ((b)) Using x=rcos⁡ϕx=r\cos\phi, y=rsin⁡ϕy=r\sin\phi, write down the correct, complete transformation law for Γ\Gamma from Cartesian to polar coordinates, and use it, starting from the vanishing Cartesian symbols alone, to derive Γ′rϕϕ\Gamma'^r{}_{\phi\phi} and Γ′ϕrϕ\Gamma'^\phi{}_{r\phi} as functions of rr. ((c)) Evaluate both symbols at r=3.0 mr=3.0\ \mathrm{m}.

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