General Relativity: An Introduction for Physicists · Problem 3.9

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · non-zero-torsion-geodesic-coordinates

Problem

In a manifold with non-zero torsion, show that the condition Γabc=0\Gamma^a{}_{bc} = 0 implies ∂agbc=0\partial_a g_{bc} = 0 but not vice versa. Show further that, under a coordinate transformation of the form x′a=xa−xPa+12Γabc(P)(xb−xPb)(xc−xPc),x'^a = x^a - x^a_P + \frac{1}{2}\Gamma^a{}_{bc}(P)(x^b - x^b_P)(x^c - x^c_P), the affine connection at the point PP in the new coordinate system is given by Γ′abc(P)=12T′abc(P)\Gamma'^a{}_{bc}(P) = \frac{1}{2}T'^a{}_{bc}(P) and hence the transformation does not yield a set of geodesic coordinates. Is it still possible to define local Cartesian coordinates in a manifold with non-zero torsion?

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