Someone argues: 'On the two-sphere of radius

PHYSICS · Subjective · 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

Someone argues: 'On the two-sphere of radius aa, the connection component Γϕθϕ=cot⁡θ\Gamma^\phi{}_{\theta\phi} = \cot\theta diverges as θ→0\theta \to 0. Since the Riemann tensor is built from Γ\Gamma and ∂Γ\partial\Gamma, and ∂θ(cot⁡θ)=−csc⁡2θ\partial_\theta(\cot\theta) = -\csc^2\theta also diverges there, the Ricci scalar RR must diverge at the poles. The poles are therefore curvature singularities, and since a genuine singularity cannot be removed by a change of chart, no coordinate change can make the poles regular.' Find the error in this argument, and state the value of RR at the pole of a sphere of radius a=3.00a = 3.00 m.

Browse the problem library